We introduce a new method to study the large time behavior for general classes of Hamilton-Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.

Understanding nonequilibrium phenomena occurring in aerothermodynamic flows is of fundamental importance for many applications. Typical examples, taken from the aerospace domain, are the design of heat shields for space vehicles and interpretation of experiments carried out in high enthalpy wind tunnels. Progresses made in the development physical models and computational methods for aerospace applications could also be used in other areas (such as plasma enhanced combustion and electric propulsion) where the occurrence of nonequilibrium must be taken into account as well. Di fferent regimes can be distinguished in the frame of nonequilibrium flows. When the local Knudsen number does not exceed values of the order 0:01, a hydrodynamic (or continuum) description of the flow can be used. For higher values of the local Knudsen number, rarefied gas eff ects start to occur. The former can be correctly described only by adopting a kinetic description of the flow. The seminar will be divided into two parts. Thefirst will focus on the development of macroscopic physical models for molecular dissociation and internal energy excitation within the context of a continuum flow description. The second will be devoted to the development of computational methods for the solution of kinetic equations for rarefied flows.

Cobordisms (a favorite topic of topologists) can tell us something about counting holomorphic disks (a rich topic for math-physicists and symplectic geometers). For example, cobordisms between Lagrangian submanifolds can detect equivalences in the Fukaya category. In this talk I'd like to discuss a project with David Nadler in which we construct a category, Lag(M), whose objects are Lagrangians in M and whose morphisms are cobordisms between them. This category has many of the algebraic properties of Fukaya(M) -- it has a zero object, it has kernels and cokernels, and it also has a "shift" functor just like the category of chain complexes, or of spectra. We in fact conjecture that our category of cobordisms is equivalent to Fukaya(M) in some cases. If time allows I'd like to discuss a non-trivial functor from Lag(M) to Fukaya(M) which may realize this equivalence. Note the special time.

This will be a gentle introduction to Mathematical Optimal Control Theory, the science of controlled dynamics. We will be motivated by examples and aim to explain some important mathematical questions.

My plan is to show you a classification theorem for a family of calibrated complex structures on the space of Hermitian matrices (in ) with a fixed spectrum of distinct eigenvalues. The real part of the usual Hermitian product of is an inner product, its imaginary part is a skew-symmetric bilinear form, and you can get it by doing the computation Re. The space of Hermitian matrices (in ) with a fixed spectrum of distinct eigenvalues comes equipped with a standard inner product for each tangent plane, we say that a complex structure is calibrated with respect to it if the situation described in the first sentence holds for each tangent plane. In this talk hopefully I will show you to construct some complex structures calibrated with the natural inner product of the previously described space, and sketch the proof of a classification theorem for a family of them that is based on the Euler angles used to describe spinning tops.

My plan is to show you a classification theorem for a family of calibrated complex structures on the space of Hermitian matrices (in ) with a fixed spectrum of distinct eigenvalues. The real part of the usual Hermitian product of is an inner product, its imaginary part is a skew-symmetric bilinear form, and you can get it by doing the computation Re. The space of Hermitian matrices (in ) with a fixed spectrum of distinct eigenvalues comes equipped with a standard inner product for each tangent plane, we say that a complex structure is calibrated with respect to it if the situation described in the first sentence holds for each tangent plane. In this talk hopefully I will show you to construct some complex structures calibrated with the natural inner product of the previously described space, and sketch the proof of a classification theorem for a family of them that is based on the Euler angles used to describe spinning tops.

The appearance of the first works by L. Caffarelli and L. Silvestre on nonlinear problems involving the fractional Laplacian on R^n opened the door to the study of these kind of operators in widely different contexts. Certainly "fractional power of an operator" is a well-known object from the Functional Analysis and Operator Theory point of view. Nevertheless, properties needed in PDE theory, like Schauder estimates or Harnack inequalities, are not covered by the general theory. In this talk we will show how the operator semigroup language, together with some ideas from Harmonic Analysis, can be helpful to understand fractional operators. In particular, it permits us to find pointwise formulas, to solve the Caffarelli--Silvestre extension problem, to derive Holder and Schauder estimates and to prove interior and boundary Harnack inequalities. These ideas will be fixed in a concrete setting: the fractional Laplacian on the torus. In this case the starting points are the Fourier series and the classical result that says that the conjugate function is bounded on H\"older--Zygmund spaces.

Let M be a surface bundle over a circle with pseudo-Anosov monodromy. The stable and unstable foliations associated with the pseudo-Anosov mapping class induce a singular Sol structure on M. Under some additional assumptions on the mapping, we construct (nearly collapsed) singular hyperbolic structures on M that approximate the Sol structure using data from the pseudo-Anosov element.

Geometric methods have revolutionized the field of image processing and image analysis. I will review some of these classical methods including image snakes, total variation minimization, image segmentation methods based on curve minimization, diffuse interface methods, and state of the art fast computational algorithms that include ideas from compressive sensing. Recently some of these concepts have shown promise for problems in high dimensional data analysis and machine learning on graphs. I will give an overview of some of the classical results from imaging and ongoing work in high dimensional and network data.

Geometric methods have revolutionized the field of image processing and image analysis. I will review some of these classical methods including image snakes, total variation minimization, image segmentation methods based on curve minimization, diffuse interface methods, and state of the art fast computational algorithms that include ideas from compressive sensing. Recently some of these concepts have shown promise for problems in high dimensional data analysis and machine learning on graphs. I will give an overview of some of the classical results from imaging and ongoing work in high dimensional and network data.

The Haar measure is a magic thing that powers more theorems in more ways than many appreciate. This talk will focus on a survey of applications of the existence and uniqueness clauses for the Haar measure. We'll prove some of the easier theorems and try and focus on how how the Haar magic makes the wheel turn, as well as ways to exploit it in theorems that (perhaps) you want to prove. Time permitting we'll look at how this relates to some fascinating problems in modern number theory.

A new formalism for the transport properties of partially ionized plasmas is investigated from a computational point of view. Well-posedness of the transport properties is established, provided that some conditions on the kinetic data are met. The mathematical structure of the transport matrices is readily used to build transport algorithms that rely either on a direct linear solver or on convergent iterative Krylov projection methods, such as the conjugate gradient. Air and carbon dioxide mixtures in local thermodynamic equilibrium at atmospheric pressure serve as benchmark to assess the physical model and numerical methods. Superiority of the conjugate gradient method with respect to the direct solver and approximate mixture rules found in the literature is demonstrated in terms of accuracy and computational cost.

We will continue our efforts to understand arXiv:1212.5605.

The intensity of radiation is inversely proportional to the square of the distance from the source. In particular, at large distances from the source, the radiation intensity is distributed over larger surfaces and therefore the intensity per unit area decreases as the distance from the surface to the source increases. We introduce a model to construct reflectors that take into account this law and prove existence of solutions. This is joint work with Ahmad Sabra.

The main goal of this talk is to introduce the audience to a handful of knot invariants that have useful applications to the study of knot concordance. First, we'll introduce some classical invariants coming from the Seifert form of a knot. Next, we'll discuss the construction of the branched double cover of the 3-sphere along a knot (or the 4-ball along a surface) and the relationship between these manifolds and the classical knot invariants. Finally, we will see how modern invariants, namely the correction terms coming from the Heegaard Floer homology of the branched double cover a knot, give us even more information than the classical invariants.

In 1984 the speaker constructed a continuum of groups of intermediate growth, answering a question of Milnor. This continuum can be viewed as a Cantor subset X of the space of marked groups with a natural map T:X----X which preserves many group properties. In fact, the dynamical system (X,T) is topologically conjugate to the one sided-shift. We will show that for any reasonable T-invariant measure \mu on X, \mu-a.e. group G has an upper bound on its growth by a function of the form e^&ob;n^&ob;\alpha&cb;&cb; with \alpha 1. However there is a co-meagre subset Y \subset X of groups with a very different type of behavior, oscillating growth which will be defined during the talk.

The study of harmonic analysis on a reductive Lie group was pioneered by Harish-Chandra. He gave a description of the character of the regular representation as an integral over the space of unitary representations of (the Plancherel formula). Central to his work was a procedure for inducing representations from smaller reductive groups. In this talk, I will give an overview of the analogue of some of these ideas in the context of -modules on (complex reductive). I will explain how certain -modules can be understood in terms of the torus and the Weyl group (a generalization of Springer Theory), and describe some work in progress on applying this to understand the convolution of -modules (in particular, identifying idempotent character sheaves). If time permits, I hope to discuss how this relates to joint work with Ben-Zvi and Nadler on the character theory TFT and the cohomology of character varieties. I will not assume familiarity with -modules or reductive groups!

SYZ mirror symmetry is a geometric approach to understanding mirror symmetry, which essentially is a relationship between the symplectic structure of one manifold to the complex structure its "mirror." It purports that the two manifolds should admit dual special Lagrangian torus fibrations over some base manifold. I'll explain this relationship using the example of the complement of an anti-canonical divisor on a surface. In particular, I'll explain how Gross, Hacking, and Keel, motivated by ideas of Auroux, give an explicit construction of a mirror family with a canonical basis of global sections.

Over fields of positive characteristic, elliptic curves come in two flavors, ordinary and supersingular. Thus, the goal of this talk will be to develop a basic understanding of the meaning of these terms, in part via some explicit examples and computations (over finite fields). We will first remind the audience of some general properties of elliptic curves over an arbitrary field .

In classical mechanics the states of a physical system can modeled in R^{6n}, here n is the number of particles and for each particle we need three dimensions for position and other three for velocity. Newton's formulation of mechanics describes the evolution of such systems by curves adapted to vector fields tangent to R^{6n}. However, in practice many systems can be described in smooth subsets R^{6n} by using the Lagrangian formulation of mechanics, for example a pendulum can be described in a sphere of convenient radius and its evolution corresponds to extremal curves in the sphere minimizing a certain functional. In this talk I will sketch the proof of a theorem related to the convergence of solutions to the Newton's equation in R^{6n} to solutions of the Lagrange equations in smooth subsets when a certain parameter (for example pendulum's string tension) tends to infinity. Finally, If I have time I will show some applications of this result to the movement of tops.

I will start by reviewing the existing results in the study of forward performance processes, which serve as the optimality criteria for investment problems with multiple time horizons. I, then, show that the existing definition of a forward performance lacks an important component. Having added the missing part of the definition, I develop an axiomatic justification for this theory. In this modified setting, under the additional Markovian assumption on the market, the optimality criterion, as well as the solution to the associated optimization problem, is characterized by the Hamilton-Jacobi-Bellman equation on a semi-finite time interval. The main difficulty in analyzing the latter equation stems from the fact that it ”has time running in a wrong direction”: we need to solve this (backward) parabolic PDE forward in time. Using the existing results on the form of the minimal elements of a Martin boundary of a space-time diffusion, I provide an explicit characterization of all positive solutions to some classes of these (ill-posed) equations. In addition, I show how these results extend the classical Widder’s theorem on the positive solutions of a time-reversed heat equation.

I will define and state some basic theorems regarding NNA algebras. Then I will introduce a family of NNA division algebras and explain some results about its classification using a very basic idea regarding Lie Algebras.

This is joint work with M. Bestvina and K. Fujiwara. Using a construction of Brooks we show that a free group has non-trivial second bounded cohomology with coefficients in a uniformly convex Banach space. The method extends to groups that act on a quasi-tree with a free subgroup that acts with weak proper discontinuity. Using a construction from earlier work with Bestvina and Fujiwara a large class of groups have such actions including hyperbolic groups, CAT(0) groups with rank one elements, mapping class groups and Out(F_n). If time permits we'll also discuss how are methods give new results on the stable commutator length in mapping class group.

This talk is about the pivotal role played by topology and geometry in the proofs of arithmetic results about curves in positive characteristic. I will describe some background about the boundary of the moduli space of curves and about the action of fundamental groups on torsion points of abelian varieties. These were key concepts used in results such as Deligne and Mumford’s proof of the irreducibility of the moduli space of curves and Harbater and Raynaud’s proof of Abhyankar’s Conjecture about Galois covers of curves. I will finish by talking about how these concepts were used in my result with Achter about the monodromy of the p-rank strata of the moduli space of curves. The talk will include a lot of background and motivating examples.

The distinction between ordinary and supersingular elliptic curves can be generalized to curves of higher genus. If X is a curve defined over a finite field of characteristic p, its p-rank is the length of the slope 0 segment of the Newton polygon of its L-function. The p-rank measures the number of p-torsion points on the Jacobian of the curve. Using geometric arguments, Faber and Van der Geer proved there exists a smooth curve in characteristic p having genus g and p-rank f (for all primes p, all g 2 and all f between 0 and g). The analogous result is true for hyperelliptic curves by work of myself, Glass, and Zhu. After providing some background, I will discuss results about p-ranks of curves which are an odd cyclic cover of the projective line, starting with Bouw's non-trivial upper bound for the p-rank and ending with a new existence result by Ozman, myself and Weir for trielliptic covers. The proof uses the method of degeneration to the boundary of a Hurwitz space.

In this talk I will describe the geometry of Frobenius manifolds and give some examples (quantum cohomology and Landau-Ginzburg models). Frobenius manifolds are manifolds with a structure of Frobenius algebra on each tangent space. Therefore, they can be viewed as families of 2d TFTs. I will explain the notion of a cohomological field theory (CohFT), and show how to get a Frobenius manifold out of it. Moreover, in nice cases one can reconstruct the whole CohFT from the Frobenius manifold. For example, this allows one to compute Gromov-Witten invariants for all genera from quantum cohomology.

We present new regularity estimates which are propagated by transport equations with rough coefficients. Those estimates provide compactness on the density, which is a key ingredient to obtain existence of solutions for models from fluid mechanics (of the compressible Navier-Stokes type) to chemotaxis. The corresponding spaces are defined at a logarithmic scale in terms of number of derivatives and can thus be seen as intermediary between the usual L^p and Sobolev spaces.

I will define arithmetic manifolds and orbifolds and give a couple of examples. However, to get to the definition, I will first discuss commensurability, invariant trace field, and quaternion algebras over number fields.

This talk will discuss to what extent residually finite groups are determined by their finite quotients. In addition we discuss what properties of residually finite groups are "seen" by the set of finite quotients. All of this can be recast in terms of the profinite completions of the groups, and the methods we use require the technology of profinite groups.

Birch and Swinnerton-Dyer formulated their famous conjecture after looking at numbers on a computer. We will talk about why those numbers led them to make their conjecture, and the role that computation still plays in the theory of elliptic curves.

In this talk I will talk about the level-set based boundary integral method applied on exterior Helmholtz problems. Derivations and regularizations issues will be addressed, and future work including nonsmooth boundary problems and hyper-singular integrals will be discussed.

I will give an informal survey of the mathematical aspects of the surface quasigeostrophic equation. I'll cover a bit of the derivation of the model, explain why it has received so much attention in recent years, and discuss what is known and what is open.

We will begin by defining hypergraphs and introducting Ramsey numbers, and an application with hypergraphs. If time permits we will also talk about how they can be used to organize the number of clubs in towns with peculiar traditions.

The Riemann Problem (RP) is the building block in the theory of hyperbolic systems of conservation laws. The most important such system is that of gas dynamics, which is called Euler equations. Although the RP has been fully understood for a while in one space dimension (the analysis began with Riemann himself), the multi-D analysis is still in infancy. Only some special cases have been solved in reasonable regimes. I shall present results when the gas obeys a Chaplygin equation of state (also called von Karman), where the acoustic fields are linearly degenerate. This yields an interesting elliptic boundary-value problem in the subsonic domain, which displays a Keldysh type degeneracy at the boundary. This boundary-value problem has geometrical interpretations in minimal surface theory.

There are several simplicial complexes that serve as free group analogues of Harvey's curve complex. We will describe some of these complexes and will survey some results known about them.

I will describe work of M. Kontsevich and collaborators presented at the recent Conference on Homological Mirror Symmetry in Miami: (1) Generalized Bogomolov-Tian-Todorov theorems, and (2) a new definition of weak Calabi-Yau algebras.

Given a polynomial equation with rational coefficients, one might ask whether the existence of local solutions for each prime guarantees the existence of a global solution. For quadratic forms, the Hasse Minkowski theorem tells us that this is always the case, and in general we say that the Hasse principle holds for a given variety if such a local global principle is true. Unfortunately, even for cubic forms this principle can fail, and thus mathematicians became interested in classifying obstructions to the Hasse principle. The aim of this talk will be to describe a particular obstruction known as the Brauer-Manin obstruction, and in fact we are interested in the obstruction to integral points. After providing the necessary background, I will explain the obstruction and then provide a concrete example. This is a practice candidacy talk.

I will review the work I did during my graduate studies on the subject of topological glasses, starting with the 2-dimensional model, then moving to the generalization to 3 dimensions. In both cases, the phase space is the set of triangulations of the sphere ( or ), the dynamics are given by the Pachner moves that conserve the number of nodes (T1 in 2d, 2-3 and 3-2 in 3d) and the energy is local and a function of the neighbourhood of a node. I will point out some interesting properties of the phase space and I will show that these models exhibit glassy behaviour.

Luttinger surgery is a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold. The surgery was introduced by Karl Murad Luttinger in 1995, who used it to study Lagrangian tori in R^4. Luttinger's surgery has been very effective tool recently for constructing exoticsmooth structures on 4-manifolds. In this talk, using Luttinger surgery, I will present a new constructions of Lefschetz fibration over 2-sphere whose total space has arbitrary finitely presented group G as the fundamental group. I will also construct infinitely many exotic Stein fillings with the fundamental group G. This is a joint work with B. Ozbagci.

We relate the Cech cohomology of a tiling space to properties of individual tilings. We then show that all topological conjugacies of cut-and-project tilings are equivalent to linear transformations.

I will discuss properties of functions from a number field K to itself obtained by evaluating a polynomial. In particular, I will show that for any f(x) in Q[x], the induced function on Q is at most 6-to-1 outside a finite set. I will also describe the pairs of polynomials in Q[x] whose images have infinite intersection. The proofs use tools from several areas of math, including an intricate application of the classification of finite simple groups.

We investigate prior distributions that are designed to incorporate information about the strength of a regression relationship. The most commonly-used prior distributions for regression models typically assume that coefficients are a priori independent or induce dependence via the empirical design matrix. While these standard priors (and recently-refined versions of them) may exhibit desirable behavior with respect to targeted inferential goals, we should not expect them to distribute probability throughout the entire parameter space in a way that is consistent with all of our prior beliefs. Examination reveals that when we focus on the strength of the regression relationship, standard priors place nearly all of their mass in regions of the parameter space that are not only inconsistent with reasonable prior belief but are nearly certain to clash so greatly with the likelihood that we might question the validity of particular inferences.

We will study two phase problems posed over a two dimensional cone generated by a smooth curve on the unit sphere. We show that when the free boundary avoids the vertex of the cone. When we provide examples of minimizers such that the vertex belongs to the free boundary. This is a joint work with Mark Allen. Reference: http://arxiv.org/abs/1301.6047

I will discuss what a spatial (embedded) graph is, why we should care, and some useful techniques for distinguishing between spatial graphs. In particular, I will define what it means for a graph to be interestingly nonplanar (i.e. a generalized ravel) and provide some results on when one can modify a tangle via vertex insertion and closure to produce a ravel.

A complete affine manifold is the quotient of a vector space (R^n) by a group of affine transformations acting properly discontinuously. In the cocompact case, the group must be virtually solvable (known up to n=6, conjectured for all n). However, in the non-cocompact case, more interesting groups (e.g. free groups) may act properly. I will discuss work in progress towards classifying the topology of affine three-manifolds. The remaining interesting case is that of flat Lorentzian manifolds, in which the linear part of the group action lies in O(2,1). Joint work with Francois Gueritaud and Fanny Kassel.

I will deliver on my promise from last week and explain a definition of weak Calabi-Yau algebras due to Kontsevich and Vlassopoulos, and I will explain how it applies to Fukaya categories.

Algebraic K-theory is an invariant of rings and their generalizations with deep connections to diverse fields of math. In this introductory talk, we will focus on the lower K-theory groups K_0 and K_1 of rings with number-theoretic significance, and locate information about them in more familiar invariants such as zeta functions. Time permitting, we will survey more advanced points of contact between K-theory and arithmetic algebraic geometry.

We consider the entropy of large simple graphs, as a function of fixed edge and triangle densities. We examine the behavior of this function in several regimes and prove that the entropy is not differentiable along a certain curve.

I will describe some work in progress which connects the Grassmannian to "spectral networks," which are certain networks of curves drawn on Riemann surfaces (in this case, the Riemann surface will be the plane). These networks arise naturally in four-dimensional quantum field theories with N=2 supersymmetry.

We give an update to some aspects of theory for solutions to nonlinear elliptic, possibly degenerate, equations involving p-Laplacean type operators. In particular, we focus on equations with assigned datum, which can be a measure in full generality. The main goal of the talk is in describing that pointwise gradient bounds via Riesz potentials - like those available for the linear Poisson equation - actually hold for p-Laplace type equations as well.

In this sequence of talks we outline the proof of the Cyclic Surgery Theorem of Culler, Gordon, Luecke, and Shalen. This talk reviews the backgrounded needed for the statement of the theorem and discusses the broad outline of the proof. We will introduce the SL_2 character variety and explain its role in finding essential embedded surfaces in 3-manifolds. This talk will be accessible to all.

In early 90's Duke, Rudnick and Sarnak make a crucial observation which related certain lattice point counting problems to equidistribution of translates of a closely related periodic orbit; they then used spectral theory to solve the later problem. Soon after, it was noted by Eskin and McMullen that some ideas developed in Margulis' PhD thesis can be used to give a substantially simpler proof of the equdistribution result and hence the counting problem. These lines of ideas have developed much further since; we will address this approach and some recent developments.

Given a polynomial equation with rational coefficients, one might ask whether the existence of local solutions for each prime guarantees the existence of a global solution. For quadratic forms, the Hasse Minkowski theorem tells us that this is always the case, and in general we say that the Hasse principle holds for a given variety if such a local global principle is true. Unfortunately, even for cubic forms this principle can fail, and thus mathematicians became interested in classifying obstructions to the Hasse principle. The aim of this talk will be to describe a particular obstruction known as the Brauer-Manin obstruction, and specifically we will focus on the obstruction for integral points. After providing the necessary background, I will explain the obstruction and then provide a concrete example. This is a candidacy talk.

I will discuss a gluing procedure designed to obtain critical metrics of quadratic Riemannian functionals on connected sums of certain Einstein four-manifolds. Start with two Einstein four-manifolds of positive scalar curvature which are "rigid". Using the Green's function for the conformal Laplacian, convert one of these into an asymptotically flat (AF) scalar-flat metric. A "naive" approximate critical metric is obtained by identifying the boundary of a large ball in the AF metric with the boundary of a small ball in the other compact Einstein metric, using cutoff functions to glue together the AF metric with a suitably scaled compact metric in order to obtain a smooth metric on the connected sum. It turns out that this naive approximate metric is too rough, and must be refined in order to compute the leading term of the Kuranishi map. The main application is an existence result using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on and the product metric on . Using these factors in various gluing configurations, a zero of the Kuranishi map is then found for a specific quadratic Riemannian functional on certain connected sums. The exact functional depends on the geometry of the factors, and also on the mass of the AF metric. Using certain quotients of as one of the gluing factors, several non-simply-connected examples are also obtained. This is joint work with Matt Gursky.

A solution is presented to the problem of provable, fast, reliable, capacity-achieving communications for the additive Gaussian noise channel. This is an important real communications problem for which the fundamental limits of reliability were initially established in the capacity theorem of Shannon in 1948. Not till the 1990s were practical high-rate schemes developed and these form an essential part of the cell phone revolution. However, the demonstration of their reliability is only empirical. The codes we develop here are sparse superposition codes based on a high-dimensional regression model, and they have fast encoders and decoders based on iterative regression fits. In this presentation we describe the framework permitting theoretical demonstration of the desired properties. The error probability is shown to scale favorably with the size of the code, namely, the error probability is exponentially small for every fixed communication rate below capacity. Remaining challenges are discussed in understanding how the complexity scales when the rate approaches capacity. This work is joint with Antony Joseph and Sanghee Cho.

The problem of high-dimensional function estimation is discussed including the need for joint consideration of issues of approximation, estimation and computation, and the role of information theory in understanding the relationships. Flexible accurate function modeling arises by allowance of arbitrary order interactions among explanatory variables or by allowance of general ridge basis expansions. However, the number of candidate basis functions becomes exponentially large in the number of variables, more than can be feasibly computed by standard greedy basis search algorithms. We discuss a class of stochastic search strategies we call adaptive annealling and the prospects for computationally feasible and accurate search with these strategies for certain smoothly parameterized classes of basis functions. This work is joint with Xi Luo and Sabyasachi Chatterjee.

The first part will be about a geometric description of Mirror pairs conjectured by Strominger, Yau and Zaslow. The second part will be about a paralleled story in algebraic geometry called Gross-Siebert program. In the end, I hope I will explain why we care about Gross-Siebert program.

What is so called Nonlinear potential theory about? It essentially aims at replicating, in the setting of nonlinear, possibly degenerate equations, the results which are valid for the linear ones. Although the formulations of the outcomes are in a sense quite classical (continuity of solutions, their regularity and so forth) methods can be quite different. The foundations obviously lie in the classical De Giorgi-Nash-Moser theory. In this talk I will present some recent results in this respect, with special emphasis on potential estimates, both in the elliptic and parabolic case. In particular it will be finally shown that, at least to the C^1 level, the linear and nonlinear theories can be unified by mean of classical linear elliptic and caloric Riesz potentials, while extension to fully nonlinear equations will be also presented.

We analyze a price impact model where a large investor wants to trade an illiquid asset with a market maker who quotes prices for this security. In our model, the market maker's preferences are modeled through an exponential utility function and the price impact of the trading strategy of the large investor is derived endogenously through an equilibrium mechanism. We establish a relationship between the equilibrium mechanism and a two-dimensional BSDE with quadratic growth. This allows us to show that an equilibrium exists under certain conditions on the final payoff of the traded asset, the risk aversion coefficient of the market maker and the trading strategy of the large investor. The relationship between the equilibrium mechanism and the two dimensional quadratic BSDE also allows us to study stability and asymptotic behavior with respect to the parameters of the model. This is a joint project with Dmitry Kramkov.

What rational numbers are the area of a right triangle with rational number sides? This problem has been open for more than a thousand years! I'll talk about how this problem is related to elliptic curves and the Birch and Swinnerton-Dyer conjecture.

In this talk, we investigate connected sum at infinity, an operation introduced by Gompf on non-compact manifolds. In defining connected sum at infinity, rays are chosen in each manifold involved, just as a choice of disc is made in each manifold involved in a connected sum. In connected sum, the choice of disc does not affect the resulting manifold, but in connected sum infinity, a different choice of rays could change the resulting manifold. Such a dependence has been demonstrated for 3-manifolds. We present an infinite family of smooth 4-manifolds for which the choice of ray affects the connected sum at infinity operation, and then generalize this family to produce higher dimensional examples. These examples answer affirmatively a conjecture of Siebenmann.

Given a finite connected graph G, place a pile at each vertex. Two piles are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of piles. In a pair of piles, one pile gets the ball with probability proportional to its current number of balls raised by some fixed power p0. We characterize the limiting behavior of the proportion of balls on the piles when G is regular and p is at most 1. For example, when G is non-bipartite the proportion converges to the uniform measure almost surely. A novel part of the proof consists of analyzing the equilibrium set of a non uniformly hyperbolic dynamical system. Joint work with Itai Benjamini and Jun Chen.

We will watch the video of a recent lecture at MIT. Abstract: This will be a talk aimed at a general mathematical audience, based on the recent work of the speaker with Chen and Sun. A long-standing conjecture, going back to Yau, relates the existence of a Kahler-Einstein metric on a Fano manifold to an algebro-geometric condition of "stability". Our work gives a proof of this conjecture. In the talk, we will review briefly the definitions and earlier work of Tian and others on the problem. We will discuss the difficulties in actually verifying that a manifold is stable. We will outline the strategy of proof, which involves deforming through metrics with cone singularities. If time allows, we will give a little more detail about some aspects of the proof.

Integer lattices of small determinant are fun to work with because there are a lot of calculations you can do to classify them. If you like it when the same characters pop up over and over again in all your stories, then maybe you enjoy that sort of calculation. We'll talk about some ways to enumerate lattices with certain specified invariants and properties. We'll see where that takes us, and maybe talk about what needs to change when the base field is a quadratic extension of the rationals.

A result of Peter May states that a connected topological space is an nth loop space if it has certain algebraic actions. I will describe these actions and relate them to other objects in topology.

We prove the existence of an exponent p2 with the property that the approximate gradient of any local minimizer of the 2-dimensional Mumford-Shah energy belongs to . In turn, this result implies that the set of singular points in the jump set has Hausdorff dimension strictly less than 1.

In this talk, we will conclude our sketch of the proof of the cyclic surgery theorem. This will involve primarily an analysis of the intersections of surfaces which lie within the three manifold in question by constructing so-called "graphs of intersection", a technique made popular in part by this proof. As in the case of the first talk, this will be accessible to all.

Let and be real numbers. We prove estimates for partial sums that are sensitive to the distance from the integer multiples and to the nearest integer. We give both upper and lower bounds for such sums. It turns out that the lower bounds are of the correct order of magnitude whenever the pair , is a counter example to a notorious conjecture of Littlewood in Diophantine approximation. We also consider more general sums involving products of many linear forms in many variables. This is joint work with Thai Hoang Le.

Finite type knot invariants are those invariants vanishing on the nth piece of some natural filtration on the space of knots. This notion was introduced by Vassiliev and it turns out that most of known numerical invariants are of finite type. Kontsevich proved the existence of a "universal" invariant, taking its values in some combinatorial space, of which every finite type invariant is a specialization. This result involves some complicated integrals, but can be made combinatorial using the theory of Drinfeld associators. We will review this construction and explain why the naive generalization of this theory for knot in thickened surfaces fails. We will suggest a general way of overcoming this obstruction, and prove an analog of Kontsevich theorem in this framework for the case M=C^*, i.e. for knots in a solid torus. Time permitting, we will give an explicit construction of specializations of our invariant using quantum groups.

In the first part of this talk we will discuss approaches for modeling the natural history of disease progression using longitudinal data. In particular, we present models for grade progression in active surveillance studies. In the second part of this talk, we investigate the survival benefit implied by prognostic models where the predictor(s) of disease-specific survival are age and/or biomarker level at disease detection. We show that the benefit depends on the rate of biomarker change, the lead time, and the biomarker level at the original date of diagnosis as well as on the parameters of the prognostic model.

With their powers combined, G, M, and N once again save us from the mysterious powers of extended supersymmetry. We will draw the appropriate squiggly arrows to describe the machinery of spectral networks (using a pinch of inspiration from physics), their connection to wall crossing phenomena, and the nonabelianization map; then, perhaps, we will live to see another day.

Abstract: I'll go over the basic definitions of point process, focusing on intuition and drawing parallels to stochastic processes. The Poison Point Process - the most ubiquitous class of Point Processes - will also be presented.

In classic photoacoustic imaging, acoustic pressure boundary data is used to reconstruct an object's optical properties on the interior. The process assumes the acoustic sound speed is known in advance, i.e. reconstructed in a different, separate experiment. We'll discuss a new approach which allows both the sound speed and the optical parameters to be recovered simultaneously in a single experiment, its advantages and disadvantages, and possible extensions of the approach.

We will start by introducing the basic language of first order logic. We will talk about Godel's completeness theorem and some of its implications. Time permitting, we will discuss how these arguments can be used in studying algebraically closed fields (and more). No prior knowledge of the subject will be necessary.

In this talk, I will study a Kirchhoff-type integral, which can be seen as a linear operator mapping angle–azimuth-dependent reflection coefficients along a reflector into reflection data for the acoustic wave equation. I will show that a minor adaptation of a construction of angle–azimuth-dependent images as proposed by Sava and Fomel leads to a left inverse of this operator, which maps primary reflection data to angle–azimuth-dependent reflection coefficients. The new construction naturally leads to a reformulation of the Kirchhoff operator, acting on space-shift-extended images, which can be implemented completely in terms of the fundamental solutions of the wave equation. I will study the composition of this new wave-equation-based Kirchhoff operator with an operator forming space-shift-extended images from data. I will show that these operators are partial inverses of each other, with their compositions being pseudo-differential operators that reconstruct suitably microlocalized versions of primary reflection data and extended images focused at space-shift zero.

The Alexander trick shows that any Dehn twist along the meridian of a torus extends to a homeomorphism of the solid torus. In fact, as a consequence of Johannson's finiteness theorem, this property characterizes the solid torus among orientable, irreducible 3-manifolds with torus boundary. We'll show that this characterization no longer holds at the level of Heegaard Floer homology. This observation is about bordered Heegaard Floer homology, and in particular, appeals to bimodules in bordered theory associated with cable spaces.

The two-player iterated Prisoner's Dilemma game has been used for 30 years to model evolution, especially the emergence of cooperation. But it still holds surprises, even in the simplest case where the game can be modeled as two simple functions on an eight-dimensional cube. Freeman Dyson and I have shown that the game contains extortionate strategies, where a player who knows the secret can extract an "unfair" advantage over a player who doesn't, especially any simple evolutionary player. When both players know the secret, the game is an ultimatum game, where each player sets the other player's score, and each can enforce cooperation. As is tradition in this seminar, we will eat lunch (bring one!) at 12:30 and begin the talk at 12:45

In this talk, I will give an introduction to hypergeometric functions and explain why we should love them. I will define both the classical and finite field versions and some of their more important properties. There are many neat results related to hypergeometric functions, and I will mainly talk about a famous theorem by Beukers and my own thesis work in trying to extend a result by Igusa. If time permits, I will wax-poetic about other cool results and applications.

In this talk we explore the idea of a motive, a universal cohomological object, and how the quest to make this idea rigorous leads to a large-scale importation of the techniques of abstract homotopy theory into geometry and algebra. Our primary emphasis will be on intuition and (non-)examples.

I will give an overview about recent results on the microscopic Bardeen-Cooper-Schriefer theory of superconductivity and its connection to the macroscopic Ginzburg-Landau model.

I will describe the simplest model of a magnet called the XXX spin chain. As Jacques explained last week, it arises in 1-loop computations in planar N=4 super Yang-Mills. I will produce a commuting family of Hamiltonians using a relation with a certain Hopf algebra called the Yangian.

The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schroedinger operator in terms of an norm of the potential. This is dual to a bound on the -norms of a system of orthonormal functions. Here we extend these to analogous inequalities for perturbations of the Fermi sea of non-interacting particles, i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials. (This is joint work with R. Frank, M. Lewin and E. Lieb.)

We study the bordered Floer homology of the 3-manifold with boundary obtained by cutting S^3 along a Seifert surface Sigma for a knot K. In particular, we show that this bordered invariant determines both the Seifert form and Alexander module. This is joint work in progress with Sam Lewallen, Tye Lidman and Liam Watson.

There is a very close relation between the Laplace spectrum and the lengths of closed geodesics on negatively curved manifolds. For hyperbolic surfaces the Laplace spectrum and the length spectrum determine each other. In higher dimensions the connection is more mysterious. In this talk I will describe new results showing that the Laplace spectrum of a compact hyperbolic manifold is determined by its length spectrum, and discuss how much of the length spectrum is actually needed in order to recover the Laplace spectrum.

Let be independent one-dimensional Brownian motions parametrized by the whole real line such that for all . We consider the nth iterated Brownian motion . Such a process first appeared in a paper by Funaki as a probabilistic con- struction of the solution of higher order diffusion equations. The process received some attention by the probability community (Burdzy, Khoshnevisan, Shi, Eisenbaum, Bertoin) in relation to the pathwise properties of this non-semimartingale process. In this talk, we will review some of the fascinating properties of such as Burdzy’s result that, given a path of , one can recover the paths up to a sign. We will then show that has a limit, in a certain very weak sense, as tends to infinity, which is exchangeable and, therefore, by the de Finetti-Hewitt-Savage theorem, a conditionally independent collection of random variables. We identify the object, , we condition on as being the limit of the random occupation measures, on [0, 1], of . It turns out that has almost surely finite support and continuous (random) density. The limiting marginal distributions have some rather curious properties, but only the 1-dimensional ones are explicitly known (from which we can obtain a new characterization of the exponential distribution). We will conclude by some open problems and by providing a glimpse towards the relation of this problem to the work of Beghin and Orsingher on fractional diffusion equations. This talk is based on joint work with Nicolas Curien, Universit\'e Paris VI.

The p-adic local Langlands correspondence is well understood for GL_2(Q_p), but appears much more complicated when considering GL_n(F), where either n2 or F is a finite extension of Q_p. I will discuss joint work with Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas and Sug Woo Shin, in which we approach the correspondence using global methods. We study the relationship between Taylor-Wiles-Kisin patching of completed cohomology and p-adic local Langlands, and give an application to the Breuil-Schneider conjecture.

We investigate the low energy excitation spectrum of a trapped Bose gas. For weak and long range repulsive interactions, we prove that the Bogoliubov approximation is asymptotically correct. The latter predicts that the spectrum is made up of elementary excitations, with linear dispersion relation for small momentum.

I'll introduce S^1-equivariant versions of cohomology and (topological) K-theory with examples and describe how these come up in differential cohomology. I will assume familiarity with singular and de Rham cohomology and not much else.

We give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.

In the past few years, a new method, subspace-based optimization method (SOM), has been proposed to solve inverse problems. It could be applied to both acoustic and electromagnetic inverse scattering problems, such as through wall imaging, reconstruction of the perfectly electric conductors (PEC), or the scatterer's permittivity pro file, etc. The SOM utilizes the spectral information of the operator which maps the induced current to the electric fields on the boundary, which is carried out by means of singular value decomposition. The method shows several strengths, such as robustness against noise, requiring less incidence and high resolution compared to some other methods. Its enhanced version, twofold subspace-based optimization method (TSOM), has been proposed to solve inverse problems in these areas. Compared to the SOM, the TSOM also utilizes the spectral information of the operator which maps the induced current to the electric fields inside the domain of interest besides the boundary measurements. As a result, the TSOM has better stability and better robustness against noise than the SOM.

We will mainly focus on Niven's famous calculus-based half-page proof of the irrationality of . Using the same idea, we can also prove that and are transcendental. If time permits, we will discuss other results and open problems in transcendence theory.

Consider the evolution of N interacting particles in an energy landscape defined by a potential energy V, with additional motion driven by random fluctuations (Brownian motion). This is called Brownian dynamics. (It can be considered a simplification of Langevin dynamics, in an "overdamped" limit.) In applications V typically has many local minima, and it is of interest how the system moves between minima; unfortunately simulating this directly is often computationally prohibitive, especially at low temperature. I discuss an algorithm, called temperature accelerated dynamics (introduced by Art Voter and Mads Sorensen), for speeding up such simulations. Using the notion of quasi-stationary distributions I show how the algorithm can be mathematically analyzed. This is joint work with Tony Lelievre.

In this colloquim, and in follow up talks this week in GADGET and Geometry, I will describe my conjecture, joint with Hacking and Gross, that affine Calabi-Yau manifolds have "theta-functions", a canonical vector space basis of the algebra of regular functions. Informally: if you live on a CY, then there are canonical intrinsic quantities in your world that determine your position (of potential interest e.g. because according to string theory, WE live on a CY.) I will explain our proof, joint with Kontsevich, of much of our conjecture in the case of cluster varieties (where it generalizes the dual basis conjecture of Fock and Goncharov). We get amongst other things a unified construction of (1) A canonical basis for every irreducible representation of SL_n . (2) A canonical basis of functions on the SL_2 character variety of a punctured Riemann surface (3) the classical theory of theta functions for Elliptic curves; all by a conceptually simple construction that has nothing a priori to do with Riemann surfaces or representation theory. I'll aim the colloquim at a very broad audience -- if you know what is meant by the zeros or poles of a meromorphic function, you know enough to follow the essentials.

During this and this week's Geometry seminar I'll explain my conjecture, joint with Gross and Hacking, which gives a synthetic construction of the mirror to a CY manifold with maximal boundary, as a vector space with basis a natural Mori theoretic generalisation of Thurston's space of integer laminations, and structure constants for the multiplication rule positive integers given by counts of tropical discs. Then I'll explain our proof, joint with Kontsevich, of the conjecture for cluster varieties -- where it generalizes a near optimal form of the (corrected) Fock-Goncharov dual basis conjecture. The construction also proves the Fomin-Zelevinski Laurant phenomenon conjecture. One application of our results is a canonical basis for each irreducible representation of SL_n, which we expect is Lusztig's semi-canonical basis, but now obtained by a construction completely different from Lusztig's, one that involves no representation theory at all.

We prove that if f and g are two polynomials in two variables over an algebraically closed field, then under mild conditions their zero loci intersect in at most finitely many points. Time permitting, we will discuss some implications of this fact.

We will continue Pavel's discussion from last time, fitting the XXX spin chain into a larger set of models with a Yangian-module structure. Integrability will follow by introducing a commutative subalgebra of the Yangian and outsourcing the Hamiltonian counting problem to a local elementary school.

Numerical simulation of PDE can play an essential role in informing the analysis of the underlying equations. For example, the output of such simulations can suggest the existence of singularities and guide us towards rigorous results. Simulation has been particularly helpful in the study of nonlinear dispersive wave equations, notably nonlinear Schrödinger equations. In these two talks I will present numerical results on several such equations, including scattering of the energy supercritical nonlinear Schrödinger equation and finite time singularities in a generalized nonlinear Schrödinger equation with derivative. The first talk will present the results and their implications, while the second talk will give an overview of the numerical methods used in these studies.

Consider the moduli space of hyperbolic surfaces of genus g. A well-known compact region of moduli space is the collection X_g of thick surfaces, or surfaces with a lower bound on the injectivity radius. In this talk, I will discuss two notions, called the width and the height of X_g, which naturally measure the maximal combinatorial and geometric difference among points in X_g. I will show that the growth rate of the width and height of X_g have the same asymptotics. This is part of an ongoing project to investigate the shape of X_g.

The combined work of Thurston and Fried shows that the monodromies associated to the fibrations of a hyperbolic 3-manifold are intimately related in important topological, geometric, a dynamical ways. After reviewing this picture, we will describe recent results showing that similar relationships hold for the monodromies associated to splittings of a free-by-cyclic group. In particular, when the free-by-cyclic group is the mapping-torus group of a fully irreducible hyperbolic automorphism of , we will see that the monodromies of all nearby splittings of G as a (f.g. free)-by-(infinite cyclic) group are also fully irreducible and hyperbolic, and that they have stretch factors close to that of . The talk is based on a joint paper with Spencer Dowdall and Chris Leininger.

We will explain how our recent joint work with Greither in equivariant Iwasawa theory leads to a natural construction of explicit l-adic models of Tate sequences for Galois extensions of arbitrary global fields. The applications of this construction include: (new and rather explicit) proofs of the ETNC for Artin motives, under strong hypotheses for number fields and unconditionally for function fields; the proof of a generalization of a conjecture of Darmon on special values of global L-functions, whose particular case pertaining to order of vanishing one L-functions was recently proved by Mazur and Rubin via the theory of (rank one) Kolyvagin systems. The new class of Iwasawa modules constructed and studied in our recent work with Greither plays a crucial role in all these considerations.

Third lecture in the series, presumably to my Ph D students and the few people that owe me money

Let M be a hyperbolic manifold, then two natural questions are how many hyperbolic structures does M admit and what data do we need to determine such a structure. In dimension 2, Teichmuller theory tells us that the lengths of finitely many curves suffices to determine the structure. In dimension 3 the answer is more complicated and involves asymptotic data coming from the ends of M. We will explain how hyperbolic manifolds give rise to ending data and outline the proof of the ending lamination conjecture, which asserts that this data uniquely determines the hyperbolic structure on M.

Numerical simulation of PDE can play an essential role in informing the analysis of the underlying equations. For example, the output of such simulations can suggest the existence of singularities and guide us towards rigorous results. Simulation has been particularly helpful in the study of nonlinear dispersive wave equations, notably nonlinear Schrödinger equations. In these two talks I will present numerical results on several such equations, including scattering of the energy supercritical nonlinear Schrödinger equation and finite time singularities in a generalized nonlinear Schrödinger equation with derivative. The first talk will present the results and their implications, while the second talk will give an overview of the numerical methods used in these studies.

The heterogeneous multiscale method (HMM) is a framework for approximating the effective behavior of multiscale systems that include full microscale models and macroscale models with missing information. In HMM, the unknown macroscale parameters are estimated from local calculations. In this talk I will describe how to reverse this process to determine microstructure from measured macroscopic data. Results from homogenization theory imply that this is in general not possible, but additional assumptions about the properties of the microstructure can guarantee recovery.

First, we'll review results about optimal investment with proportional transaction costs. We'll then describe two newer impulse control models of investment and consumption with fixed transaction costs (i.e. where each trade incurs a fee K 0). Time permitting, we'll discuss prospects and difficulties of obtaining results for these models analogous to those of the proportional case.

The Gross-Pitaevskii hierarchy is an infinite sequence of coupled equations that models a Bose-Einstein condensate. Using the nonlinear Schroedinger equation as motivation, we'll define appropriate Sobolev spaces for solutions of the GP hierarchy and then outline the proof of well-posedness.

A hedge fund manager, who receives performance fees proportional to the fund’s profits, invests optimally for both the fund and his own account, as to maximize the expected power utility of personal wealth. If separate and constant investment opportunities are available for each account, it is optimal for the manager to hold a constant fraction of the fund in risky assets, which corresponds to an effective risk aversion between one and the manager’s own risk aversion. For the personal account, the optimal policy is to accumulate performance fees in safe assets, and invest remaining wealth in a constant portfolio corresponding to the manager’s risk aversion. Under the optimal policy, the manager’s welfare is the maximum between the welfare he would obtain from either keeping fees in safe assets only, or investing his personal wealth alone. This result is robust to correlation between investment opportunities, suggesting that the manager does not tend to hedge exposure to the funds’ performance with personal investments

Deformations of algebraic objects, particularly Lie algebras, arise naturally in the study mathematical physics. Deformation cohomologies are powerful tools that not only allow us to classify certain deformations of an object up to equivalences, but also give explicit descriptions of the possible obstructions to deformation. In this talk I will present some cohomological results that allow us to put any Lie algebra deformation into a normal form; using this I will then give an overview of the proof for uniqueness of Drinfel'd-Jimbo quantum groups. As this is a topic that frequently involves a lot of lengthy calculations, I will be mostly omitting proofs: sl_2 will appear as a consolation prize.

I will discuss ergodic theory over the moduli space of compact Riemann surfaces and its applications to the study of polygonal billiard tables. There is an analogy between this subject and the theory of flows on homogeneous spaces; I will talk about some successes and limitations of this viewpoint. This is joint work with Maryam Mirzakhani and Amir Mohammadi.

In this talk we give a brief history of the conjecture and an overview of the proof. This involves the geometric reformulation of the proof in terms of surfaces in the space of oriented lines – a 4-manifold endowed with a neutral Kaehler metric (of signature (2,2)). By arguments of global analysis, we show how the conjecture follows from the existence of certain holomorphic discs with boundary. The existence of these discs is proven by mean curvature flow with respect to the neutral metric.

We will attempt to motivate and introduce the local Langlands correspondence for GL(2) in families, and discuss how the classical theory of zeta integrals and local constants might work in this setting.

We analyze a spatial birth and death process in the Euclidean space where the birth rate is constant and the death rate of a given point is the shot noise created at his location by the other points of the current configuration. We construct the unique stationary regime of this class of point processes in Euclidean spaces. We establish a hierarchy of balance equations between the stationary moment measures. We prove that the stationary point process exhibits some form of repulsion between its points. We analyze two asymptotic regimes of this class of processes: the fluid regime and the hardcore regime.

We will continue our discussion of integrable systems by studying how these arise in the context of N=2 supersymmetric theories in four dimensions

This is an expository talk aimed at a graduate student audience. I will introduce the nonlinear Schroedinger (NLS) and Korteweg-de Vries equations (KdV), discuss their Hamiltonian structure, and describe the form of special solutions to these equations called solitons. For such unperturbed equations, solitons have straightforward dynamics (straight-line motion). We then pass to a semiclassical rescaling, add a potential to the equations (the perturbation), and consider the effect on the dynamics of the solitons. The perturbed dynamics are interpreted as resulting from symplectic orthogonal projection of the infinite-dimensional perturbed PDE to the finite dimensional submanifold of solitons. The analytical technique for proving the approximation is the introduction of a suitable Lyapunov functional. The results are illustrated with numerical simulations.

Understanding the diffeomorphism types of smooth 4-manifolds is a very difficult problem, and many of the effective tools we have involve complicated analytic machinery and symplectic geometry. The need for these tools and the insufficiency of algebraic topology is part of what makes the world of 4-manifolds so interesting, but also hard to study. In this talk we will look at a cut-and-paste operation called rational blow-down (due to Fintushel and Stern) which can make the algebraic topology of the manifold "smaller" (by removing surfaces representing non-trivial second homology classes) without messing up the analytic/geometric information held by the smooth structure. This construction sparked a deluge of papers giving examples of exotic (homeomorphic but not diffeomorphic) 4-manifolds of small Euler characteristic and simple algebraic topology. The rational blow-down construction can be understood in terms of products and twisted products of surfaces, so you don't need any background on 4-manifolds, just comfort with 2-manifolds and maybe 2-forms.

It turns out that the sum of the Lyapunov exponents of the Teichmueller geodesic flow can be expressed in terms of Siegel-Veech constants. I will spend most of the talk explaining these words, and then a bit on the proof of the formula. It turns out that even though both sides of the formula have origins in dynamics, the proof is a combination of algebraic geometry and analysis (involving in particular the determinant of the Laplace operator). This is joint work with Maxim Kontsevich and Anton Zorich.

These three, seemingly disparate, areas, one in probability, the other in combinatorics and the third in engineering, share a common core. This can be used to learn more about all of them.

Let be a classical modular form on , and assume that its -expansion coefficients are algebraic numbers. If these coefficients are -integral for a prime not dividing then the same holds for the -expansion of at all cusps, and in fact the algebro-geometric interpretation of modular forms identifies with a global section of a suitable line bundle over a specific -integral model of . This is the so-called (algebraic) "-expansion principle" and it has the interesting consequence that from such -integrality of the -expansion at one cusp we can infer -integrality properties when is evaluated at a CM point. The case when divides is more subtle: -integrality at one cusp does &ob;\em not&cb; imply the same at all other cusps. Can we at least (multiplicatively) "bound" in some explicit way (depending on , , and ) the failure of -integrality at the other cusps? (This is useful for applications to evaluations at CM points.) The special case was handled by Deligne and Rapoport using intersection theory. We explain how to generalize their method to any using more recent developments. Surprisingly, the bound can be chosen to depend only on the -part of . We make this explicit when , leading to a conjectural simple bound in general.

Given two Lagrangian spheres in an exact symplectic manifold, we present conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel’s long exact sequence to arbitrary powers of a fixed Dehn twist. Time allowing, we construct families of examples containing such spheres.

We consider the 3D quantum BBGKY hierarchy which corresponds to the -particle Schroedinger equation. We assume the pair interaction is . For interaction parameter , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the limit points satisfy the space-time bound conjectured by Klainerman-Machedon (2008). The needed energy bound was proved to hold for by Elgart-Erdoes-Schlein-Yau (2007). This allows, in the case , for the application of the Klainerman-Machedon uniqueness theorem and hence implies that the limit of BBGKY is uniquely determined as a tensor product of solutions to the Gross-Pitaevskii equation when the -body initial data is factorized. The first result in this direction in 3D was obtained by T. Chen and N. Pavlovic (2011) for and subsequently by X. Chen (2012) for . Overall, this provides an alternative approach to the mean-field program by Erdoes-Schlein-Yau (2008), whose uniqueness proof is based upon Feynman diagram combinatorics.

Abstract: An Insider is an agent in the market who has privileged information about the state of the market (typically the value of stocks at terminal time). We will discuss two ways to model the insider; Kyle's auction model and via initial enlargement of the underlying filtration. As it is essential to the latter, we will also discuss some of Jacod's results on semi-martingales on enlarged filtrations.

I will discuss roughly what we know now and what (in my biased opinion) are some the major open problems. I will also discuss some topics left over from the first two talks.

Maximum likelihood estimation is a fundamental computational task in statistics and involves beautiful geometry. We discuss this task for determinantal varieties (matrices with rank constraints) and show how numerical algebraic geometry can be used to maximize the likelihood function. Our computational results with the software Bertini led to surprising conjectures and duality theorems. This is joint work with Jan Draisma, Jon Hauenstein, and Bernd Sturmfels

[Note the different room on the 7th floor of RLM] The De Giorgi-Nash-Moser Theorem gives regularity for solutions to uniformly elliptic equations in divergence form, and was used to show Hilbert's nineteenth problem. We will discuss the De Giorgi proof, which involves first controlling the norm of a weak solution by its norm, and then using this bound to give regularity.

In this talk we consider a two player, zero sum stochastic differential game based on a formulation given by Fleming and Souganidis. The saddle point property is introduced, and it is proved that the unique uniformly continuous bounded viscosity solution of the upper Isaacs PDE with boundary condition satisfies such a property. Also, it is shown that approximately optimal Markov strategies can be constructed for both players.

Modal logic augments propositional logic with modalities, which qualify how a statement might be true. For example, one might say, "I believe P is true", "P will be true forever", or "it is morally obligatory that P". Since these qualifications have different properties, there are many formalizations of modal logic. This talk will introduce a few and prove soundness and completeness results. The attender should expect to achieve a deeper understanding of a variety of significant topics, including time, provability, and "Chutes and Ladders".

In 1982 Louis Kauffman conjectured that if a knot in S^3 is a slice knot then on any Seifert surface for that knot there exists a (homologically essential) simple closed curve of self-linking zero which is itself a slice knot, or at least has Arf invariant zero. Since that time, considerable evidence has been amassed in support of this conjecture. In particular, many invariants that obstruct a knot from being a slice knot have been explictly expressed in terms of invariants of such curves on the Seifert surface. We give counterexamples to Kauffman’s conjecture, that is, we exhibit (smoothly) slice knots that admit (unique minimal genus) Seifert surfaces on which every homologically essential simple closed curve of self-linking zero has non-zero Arf invariant and non-zero signatures. This is joint work with Christopher Davis.

We introduce models for financial markets and, in their context, the notions of "portfolio rules" and of "arbitrage". The normative assumption of "absence of arbitrage" is central in the modern theories of mathematical economics and finance. We relate it to probabilistic concepts such as "fair game", "martingale", "coherence" in the sense of deFinetti, and "equivalent martingale measure". We also survey recent work in the context of the Stochastic Portfolio Theory pioneered by E.R. Fernholz. This theory provides descriptive conditions under which arbitrage, or "outperformance", opportunities do exist, then constructs simple portfolios that implement them. Finally we explain how, even in the presence of such arbitrage, most of the standard mathematical theory of finance still functions, though in somewhat modified form.

In this talk we introduce the geometry of neutral Kaehler 4-manifolds. These naturally arise on the space of oriented geodesics of 3-manifolds of constant curvature. In the case of Euclidean 3-space, the neutral metric captures the geometry of 2-dimensional submanifolds perfectly and allows for a complete reformulation of the conjecture. Geometric aspects of the proof will be explained, including the notion of the angle matrix of the intersection of two positive definite surfaces, which gives the boundary conditions for mean curvature flow.

We will (quickly) introduce the self-duality equations on a Riemann surface. We will talk about invariant sub-bundles, and introduce the notion of stability. We will then classify the vector bundles that appear in a stable pair. This is intended to be a technical talk and will concentrate on neat ways to prove very technical results. Interested parties might want to glance at the first 10-12 pages of Hitchin's "Self duality equations on a Riemann surface".

Bethe ansatz is a tool to produce eigenvectors of the Bethe subalgebra thus diagonalizing the Hamiltonians. I will describe this procedure for the GL(N) XXX spin chain; time permitting, I will discuss the behavior of solutions in the thermodynamic limit.

We establish quantitative estimates for solutions to the fractional nonlinear diffusion equation, in the whole range of exponents , . The equation is posed in the whole space . We first obtain weighted global integral estimates that allow to establish existence of (very weak) solutions for classes of large data. In the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions, depending only on the norm of the initial data in a certain ball. The estimates take a different form in three exponent ranges: slow diffusion, good range of fast diffusion, and very fast diffusion. Finally, we show existence and uniqueness of initial traces.

We will introduce some of the basic tools for studying smooth 4-manifolds and use them to explore the beautiful knot surgery operation on 4-manifolds, due to Fintushel and Stern. As an application we will produce infinitely many exotic copies of the K3 surface. The talk should be accessible to anyone comfortable with the material in the topology prelims (no Kirby calculus required).

A countable group G is called shift-minimal if all non-trivial measure preserving actions weakly contained in the Bernoulli shift of G are free. I will discuss a number of open questions about the relationship between shift-minimality, cost, the first l^2 Betti-number, and certain properties of the reduced C*-algebra of G. I will also indicate the proof that if a countable group G admits a free measure preserving action of cost strictly greater than 1 then there is a finite normal subgroup N of G such that G/N is shift-minimal.

In this talk, the analytic aspects of the proof of the conjecture will be described. This includes the global argument which relates the classical Riemann-Hilbert problem to umbilic points, as well as mean curvature flow with boundary which establishes the existence of holomorphic discs.

We describe joint work (with David Ben-Zvi and David Nadler) that constructs an equivalence between the derived category of smooth representations of GL_n(Q_p) and a certain category of coherent sheaves on the moduli stack of Langlands parameters for GL_n. The proof of this equivalence is essentially a reinterpretation of K-theoretic results of Kazhdan and Lusztig via derived algebraic geometry. We will also discuss (conjectural) extensions of this work to other quasi-split groups, and to the modular representation theory of GL_n.

Mirror symmetry is a set of conjectures arising from mathematical physics that relate symplectic geometry to algebraic geometry. While initially these conjectures only concerned Calabi-Yau manifolds, over the years it has become clear that mirror symmetry holds in a much wider range of situations. In this talk I will try to give an introduction to mirror symmetry and its successive generalizations, first to Fano varieties then to varieties of general type, from two perspectives: Kontsevich's homological mirror symmetry, and the Strominger-Yau-Zaslow conjecture.

A popular default prior for the general linear mixed model is an improper prior that takes a product form with a flat prior on the regression parameter, and so-called power priors on each of the variance components. I will describe a convergence rate analysis of the Gibbs samplers associated with these Bayesian models. The main result is a simple, easily-checked sufficient condition for geometric ergodicity of the Gibbs Markov chain. [MEETINGS: please contact pmueller@math if you want to meet the speaker in the afternoon]

In this talk, we will present several different types of convergence of stochastic processes, starting from some classical strong convergence theorems to convergence in distribution in C[0,1] and D[0,1], then focusing on Fatou convergence and Meyer-Zheng convergence. I will provide sufficient conditions for each convergence, give some interesting examples, and study when each mode of convergence preserves martingale property, continuity or monotonicity under limiting procedure.

We will look at a bosonic N-particle mean field system with some restrictions on the interaction potential and in the limit of large N. I will show that, assuming that all particles are initially in the same quantum state, the many-body wave function that describes the time evolution of the system factorizes as the product of identical one-particle wave functions. These one-particle wave functions satisfy the Hartree equation with the appropriate initial condition. In other words, in the limit of large N, if your wave function starts out factored it will stay factored.

Cluster algebras were introduced in 2000 by Fomin and Zelevinsky, to quote their abstract, “in an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups”. Our first task will be to make sense of the previous sentence. This should motivate the definition of a cluster algebra. After that, we will look at a few examples of cluster algebras. In the process, we will notice that each “cluster variable” in the examples, while originally expressed as a rational function, can be reduced to a Laurent polynomial with positive integer coefficients. This is related to the “Laurent phenomenon”, which will be primary focus of the remainder of the talk. Time permitting, I'll finish with some brief comments on other contexts in which cluster algebras have turned up recently.

I'll report on work on progress with Thomas Lam on computing the algebraic DeRham cohomology of cluster varieties, with its weight filtration. Full success in this goal would lead to progress in computing extensions of Verma modules and has connections to the work of Hamed et. al. on computing scattering amplitudes. In the first half of the talk, I'll explain the background on deRham cohomology, weight filtrations and cluster algebras. In the second half, I'll explain the partial results we have proved, and the fascinating patterns we have observed and can't prove.

Some geometric objects can be studied `microlocally': instead of just working with their support (the set of points where an object is non-trivial), one can consider their `singular support', which remembers the `direction' of non-trivial behaviour. Examples include the wave front of a distribution, the singular support of a constructible sheaf, and the characteristic variety of a D-module. The goal of this talk is to sketch such `microlocal' theory for coherent sheaves. It turns out that the singular support could be non-trivial only for coherent sheaves on a singular variety. I will define singular support for coherent sheaves on a singular variety that is locally a complete intersection. The singular support measures the `imperfection' of a coherent sheaf: it equals zero if and only if the coherent sheaf has finite Tor dimension (i.e., the sheaf is perfect). In the second half of the talk, I will show how the singular support can be used to enlarge the derived category of a singular variety. In this way, a singular variety carries not one but many derived categories, which differ from each other by a kind of `renormalization'. I also hope to explain the importance of such `renormalized' derived categories for the geometric Langlands program.

I will explain joint work with Emanuele Macri, in which we systematically relate wall-crossing for Bridgeland stability conditions to the birational geometry of moduli spaces of stable sheaves on K3 surfaces. Our results include a description of the nef cone in terms of the lattice of the K3 surface, and a proof of a well-known conjecture on the existence of Lagrangian fibrations. These results are new even in the case of the Hilbert scheme.

Just like real numbers, general p-adic numbers are not finitary objects, and so cannot be exactly specified on a computer. One instead works with them using some scheme of finite approximations. For ring computations (in which no division is involved) one may use fixed-point approximations (i.e., truncation modulo a power of p) but for more general computations it is best to use an analogue of real floating-point arithmetic. When using real or p-adic floating-point arithmetic, one generally sees errors that compound over the course of a computation. But in the p-adic case, certain computations carry enough hidden algebraic structure to cause massive cancellations and drastically reduce the loss of accuracy. One can exhibit many examples of this phenomenon using cluster algebras; as an example, we prove a weak version of a numerical conjecture of David Robbins concerning the numerical stability of the Dodgson condensation algorithm for computing determinants of matrices. Joint work with Joe Buhler (CCR La Jolla).

We consider the ``limiting behavior'' of {\em discriminants}, by which we mean informally the closure of the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the "ring of motives", as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization can be described in terms of the motivic zeta values. The results extend parallel results in both arithmetic and topology. I will also present a conjecture (on ``motivic stabilization of symmetric powers'') suggested by our work. Although it is true in important cases, Daniel Litt has shown that it contradicts other hoped-for statements. This is joint work with Melanie Wood.

We consider the homogeneous Landau equation from plasma physics and show existence and uniqueness of global solutions, assuming only boundedness and spatial decay of the initial distribution. In particular, we can handle discontinuous initial conditions that might be far from equilibrium. This work is joint with Maria Gualdani.

In this talk we describe how the proof of the conjecture can be extended to provide a local index bound for umbilic points on smooth surfaces in Euclidean 3-space. The bound (index less than or equal to 3/2) is weaker than that established by Hans Hamburger for real analytic surfaces in the 1940’s (less than or equal to 1). This leads to the prediction of the existence of smooth, non-real analytic surfaces with “exotic” umbilic points of index 3/2.

Symmetry is pervasive in both nature and human culture. The notion of chirality (or `handedness') is similarly pervasive, but less well understood. In this lecture, I will talk about a number of situations involving discrete objects that have maximum possible symmetry in their class, or maximum possible rotational symmetry while being chiral. Examples include geometric solids, combinatorial graphs (networks), maps on surfaces, dessins d'enfants, abstract polytopes, and even compact Riemann surfaces (from a certain perspective). I will describe some recent discoveries about such objects with maximum symmetry, illustrated by pictures as much as possible.

This is a thesis defense.

I will introduce a formulation of mirror symmetry for abelian varieties proposed by Golyshev, Lunts and Orlov. In the paper, they defined the notion of mirror symmetry for an algebraic pair. Although this should not be regarded as the final version of mirror symmetry for abelian varieties as homological mirror symmetry is not proved, a lot of nice properties suggest that this is at least close to the real mirror symmetry.

In this talk, we study the spread of epidemics (infections) over large networks. In the first part, we study the scenario where the spread is assisted by a small number of external agents: infection sources with bounded spreading power, but whose movement is unrestricted with respect to the underlying network topology. For networks which are ‘spatially constrained’ (e.g., random geometric graphs), we show that the spread of infection can be significantly speeded up even by a few such agents infecting randomly. Further for such networks, we show that simple random, state-oblivious infection-spreading by an external agent is in fact order-wise optimal for dissemination. In the second part, we leverage the network propagation characteristics of epidemics to distinguish between epidemic spreading and "random" failures (e.g., flu vs. allergy, worm attack vs. random computer failures). For instance, a virus that spreads by taking advantage of physical links or user-acquaintance links on a social network can grow explosively if it spreads beyond a critical radius. On the other hand, random infections (that do not take advantage of network structure) have very different propagation characteristics. We address questions like: when can we distinguish between these mechanics of infection? Further, how can this be done efficiently? In this talk, we discuss sufficient conditions and algorithms for different graph topologies, for when it is possible to distinguish between a random model of infection and a spreading epidemic model, with probability of misclassification going to zero.

Continuing our discussion of the Bethe ansatz and its relation to 2d gauge theory, I will discuss the recent paper 1304.0779. This paper explains how to use 3d mirror symmetry to obtain a large number of dualities between integrable systems.

On the first part we will discuss Holder estimates for some non local problems we worked recently with G. Davila. They arise in stochastic optimal control driven by purely jump processes with kernels not necessarily symmetric. On the second part we study two phase problems posed over a two dimensional cone generated by a smooth curve γ on the unit sphere. We show that when length(γ) 2π the free boundary avoids the vertex of the cone. When length(γ) ≥ 2π we provide examples of minimizers such that the vertex belongs to the free boundary.

[Note: non-standard building. CPE is on Dean Keeton and Speedway opposite RLM. Enter on the NW corner of the building.] Differential K-theory is a geometric refinement of topological K-theory for smooth manifolds that roughly pairs topological cocycles and differential forms representing them to capture local geometric as well as global topological information. There is a general construction due to Hopkins-Singer and several geometric models due to Freed-Lott, Bunke-Schick, and Simons-Sullivan. For G a finite or compact Lie group, G-equivariant K-theory is a generalization of topological K-theory for spaces with a G-action. For G a finite group, Ortiz constructs differential G-equivariant K-theory for smooth manifolds with a smooth G-action. For G=T the circle group, we construct differential T-equivariant K-theory. This is a thesis defense.

We say a subfield of has the Northcott property if it has only finitely many points of height , for all , and we say it has the Bogomolov property if there is a positive such that it has no element of positive height . We will discuss some recent results on infinite extensions of that satisfy these properties, as well as some basic questions that remain open.

We propose a novel numerical inversion algorithm for parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The algorithm presented considers a layered medium. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss-Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated computations of the time domain solutions of the forward problem. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments.

Abstract: We will talk about a problem in stochastic impulse control theory that one can understand as an implicit constraint obstacle problem. I will present the basic theory of existence for these problems and discuss the issue of regularity for the solution.

In this presentation, we consider a new mixed finite element method for diffusion equations on general polygonal/polyhedral meshes. Originally the method was invented in 2007.Then it was used in a number of projects supported by ExxonMobil URC. The main idea of the method is based on the approximation of the fluxes by piecewise constant vector functions (PWCF).The normal components of the approximate vector functions are continuous on the interfaces between polyhedral mesh sells. In the interior of mesh cells these vector functions are discontinuous. The error estimates for the special type of meshes are derived. We discuss applications of the PWCF-method in geosciences as well as numerical results.

In this talk, we present some analysis of Ricci flow on noncompact manifolds. Our first result concerns the stability of complex hyperbolic geometry under Ricci flow. Our second result is on the formation of finite-time singularities in Ricci flow on .

We resolve a question proposed in Kardaras and Robertson (2012): how to invest in a robust growth-optimal way in a market where precise knowledge of the covariance structure of the underlying assets is unavailable. Among an appropriate class of admissible covariance structures, we characterize the optimal trading strategy in terms of a generalized version of the principal eigenvalue of a fully nonlinear elliptic operator and its associated eigenfunction, by slightly restricting the collection of non-dominated probability measures.

An insider is a (stock) market agent who has privileged information about the market; one can be interested in precisely quantifying the value of this information. To this end Kyle's auction type model, in both discrete and continuous time, will be presented. An alternative 'small investor' model in the SDE frame work will also be described. Time permitting, the optimal value of insider knowledge will be discussed in both cases.

Beginning with a knot K, one can construct a sequence of links called iterated Bing doubles of K. We will start by constructing minimal genus surfaces in the three-sphere for iterated Bing doubles. We will then broaden our perspective and consider surfaces in the four-ball. In particular, we will discuss the problem of whether an iterated Bing double is slice (that is, whether the components of the link bound disjoint disks in the four-ball).

I'll explain a precise formulation of Kontsevich's homological mirror symmetry conjecture for Calabi-Yau manifolds, which I expect to apply in the context of the Gross-Siebert program. This carries more information than the usual formulations in several ways, but is also based on lower-tech transversality theory. In one dimension, refined HMS has been proved by the speaker with Lekili. Slides are at www.ma.utexas.edu/users/perutz/GADGET1.pdf

I will talk about obtaining the Lind-Lehmer constant for p-groups of the form and for cyclic groups of order less than 892371420. The value for involves determining the smallest non-trivial th power mod and I will mention some upper bounds on this quantity.

In this talk I will present the classification of complex algebraic surfaces, one of the major contributions of the "Italian school of algebraic geometry". As a modern application, I will prove the fact (fundamental to Gross-Hacking-Keel mirror symmetry program) that every log smooth log Calabi-Yau surface can be constructed by blowing-up and blowing-down some toric surface in a combinatorial way--or stated fancily, every smooth affine Calabi-Yau surface appears as a cluster variety.

The group determinant of a finite group G is a homogeneous polynomial in |G| variables with degree |G|. We determine the stabilizer of such polynomial under the action of permuting the variables. There is a connection between the group determinant and Lehmer's problem for finite abelian groups. We prove that the answer to the problem must satisfy a congruence system. In particular, when G is a p-group, we establish upper and lower bounds for the Lehmer's problem.

Graph clustering is the following problem (in words): given a sparse graph, partition the nodes into clusters so that the edge density within a cluster is higher than the density across clusters. It is a widely studied problem, and represents a fundamental step in the pre-processing of large networks for finer-scale analysis. A classical, and natural, statistical setting to formalize this question is the stochastic block model; first a partition is chosen for the nodes, and then a graph is generated at random, with different edge probabilities within and across partitions. The problem now becomes: given just the graph, find the partition. The hardness of the problem is governed by the edge density - both overall, and the differential - and the size of the smallest partition. In this talk we present a new algorithm for this classical problem; and corresponding analytical results which establish that it outperforms the existing literature by polynomial factors. Our algorithm is spectral, but needs an asymmetric low-rank approximation that necessitates the use of convex optimization (as opposed to SVD)

In the first part of the talk I will consider the long time behavior of the solutions of the Schroedinger equation with a weakly random potential, concentrating on the regime when the potential is both spatially and temporally random with slowly decaying correlations. The slow decay of correlations leads to various observables being randomized on different time scales: the phase is randomized on a time scale that is much shorter than the time when the amplitude becomes non-trivial. I will try to argue that this "multitude of time scales" is generic in such media. In the second part of the talk I will describe the time evolution of a particle in a periodic shear flow driven by a fractional (strongly correlated) noise. These results were obtained jointly with G.Bal, T. Komorowski, A. Novikov and S. Olla in various combinations.

Over the past decade, Heegaard Floer homology has proved to be quite useful for studying questions about Dehn surgery and understanding surfaces in three-manifolds. This talk will be motivated by the goal of understanding how the knot Heegaard Floer complex behaves when three-manifolds are cut along surfaces to construct mutant knots. In particular, we will prove that if an -stranded pretzel knot has an essential Conway sphere, then the knot Floer complex contains a summand of rank at least two, and conjecture that this phenomenon occurs more broadly. As a consequence, we can easily classify pretzel knots admitting -space surgeries. We will also construct an infinite family of genus two mutant pairs which share the same total dimension in while being distinguished by as a bigraded or -graded invariant. THIS IS A PRACTICE DEFENSE.

The techniques of compressed sensing allow for the recovery of a high dimensional signal x from linear measurements . While the system may be underdetermined, recovery is still possible given certain assumptions about the sparsity of . In this talk we will consider the problem of one-bit compressed sensing, in which the measurements are quantized to just one bit of information.

Let F(x , y) be an irreducible binary form of degree at least 3 and with integer coefficients. Then by a well-known result of Thue (1909), we know that the equation F(x , y) = m, where m in an integer, has only finitely many solutions in integers x and y. I will talk about the problem of finding an upper bound for the number of integer solutions to such equations.

In 2002, Feldman, Ilmanen, and Knopf constructed the first example of a non-trivial (i.e. non-constant curvature) complete non-compact shrinking soliton, and conjectured that it models a Ricci flow singularity forming on a closed four-manifold. In this thesis, we confirm their conjecture and, as a consequence, show that limits of blow-ups of Ricci flow singularities on closed four-dimensional manifolds do not necessarily have non-negative Ricci curvature. This is a thesis defense.

I will explain Gromov's seminal nonsqueezing theorem from 1985, which constructs the first 2-dimensional invariant of symplectic geometry, the "Gromov Radius". The volume always provides a 2n-dimensional invariant. In 1989 Hofer asked whether there is any symplectic invariant measuring 2k-dimensional information, where 1kn. I will discuss the answer to this question and its relevance in symplectic geometry. This talk will assume no background in symplectic geometry. This is joint work with San Vu Ngoc.

This talk is to design efficient and novel numerical algorithms for a class of deterministic and stochastic dynamical systems with multiple time scales. This is my defense.

Distortion is a property of metrics on Banach spaces. Roughly: one metric is a distortion of another if it is equivalent, and the constant of equivalence can't be improved on any infinite dimensional subspace. Most equivalent metrics encountered in everyday life are not distortions. In fact, it's not easy to see that distortable metrics exist at all. In this talk, we will first see that and are not distortable. Then we will look at Tsirelson's space, which was originally constructed as a space which doesn't contain or . Tsirelson's space was the first space to be shown to be distortable. With luck, we will finish with an inkling of how one might show is distortable, for p not 1 or .

We will discuss the advantages of relating the robust pricing problem to the theory of mass transportation. Mathematically the crucial difference is that transport plans are required to be martingales in our setting. In particular, we will see how the duality theorem from optimal transport can be used to establish new robust super-replication results. This dual viewpoint also provides new insights on classical martingale inequalities. For instance, we establish a (new) sharp version of the classical Doob maximal inequality.

I am taking a cool topics class in optimal transport, so I thought I would share a few things I have learned. We will introduce some basic ideas of optimal transport, state and discuss a major theorem, and use optimal transport to prove the isoperimetic inequality. This talk should be accessible to all Sophex attendees.

In this talk, we present some analysis of Ricci flow on noncompact manifolds. The first result concerns the stability of complex hyperbolic geometry under Ricci flow. The second result is on the formation of finite-time singularities in Ricci flow on . This is a thesis defense.

In this talk I will discuss the k-medoids clustering problem, which seeks to find meaningful groups within data along with a 'best representative' for the group. This is an NP-hard problem and many algorithms have been developed to approximate its solution efficiently. One such algorithm is Affinity Propagation, which is an instance of the max-sum message passing algorithm; I will introduce this family of algorithms, and discuss known theoretical guarantees. In particular, I will present a new observation that relates Affinity Propagation with the linear programming dual of the original problem. Various implications and conjectural implications will follow. This is a candidacy talk.

This talk will be an overview of recent advances in thin position techniques and their applications. Joint work with Scott Taylor.

This is a joint work with Pierre Pageault. For a homeomorphism h of a compact space, a Lyapunov function is a real valued function that is non-increasing along orbits for h. By looking at simple dynamical systems(=homeomorphisms) on the circle, we will see that there are systems which are topologically conjugate and have Lyapunov functions with various regularity. This will lead us to define barriers analogous to the well known Peierls barrier or to the Mañé potential in Lagrangian systems. That will produce by analogy to Mather's theory of Lagrangian Systems an Aubry set which is the generalized recurrence set introduced in the 60's by Joe Auslander (via transfinite induction) and a Mañé set which is essentially Conley's chain recurrent set. No serious knowledge of Dynamical Systems is necessary to follow the lecture.

Over the past decade, Heegaard Floer homology has proved to be quite useful for studying questions about Dehn surgery and understanding surfaces in three-manifolds. This talk will be motivated by the goal of understanding how the knot Floer complex behaves under different types of mutation. We will construct an infinite family of genus two mutant pairs sharing the same total dimension in HFK while being distinguished by HFK as a bigraded or delta-graded invariant. We will also prove that if an n-stranded pretzel knot contains an essential Conway sphere in its complement, then the knot Floer complex contains a summand of rank at least two, and conjecture that this phenomenon occurs more broadly. As a consequence, we can easily classify pretzel knots admitting L-space surgeries. This is a defense.

In part 1 (see www.ma.utexas.edu/users/perutz/GADGET1.pdf), I discussed the Fukaya category of a Calabi-Yau manifold relative to an ample divisor. This week, I'll describe what I think homological mirror symmetry ought to say about such categories near a large complex structure limit, including the proof of a generation theorem which to a large extent reduces the problem to one about the homogeneous coordinate ring. Slides at www.ma.utexas.edu/users/perutz/GADGET2.pdf

Picard-Lefschetz theory is an analogue of Morse theory for complex manifolds. We first give a topological picture of the local model for a quadratic singularity, then describe the monodromy about the critical point. In one case this is the familiar Dehn twist for surfaces. Time permitting, we'll also describe some generalizations in symplectic manifolds.

I will discuss a class of free boundary problems governed by fully nonlinear PDEs as well as the impact they have in the geometric regularity theory for 2nd order elliptic equations.

Convex projective geometry can be viewed as a generalization of hyperbolic geometry. While this geometry retains many nice features from hyperbolic geometry it lacks a notion of Mostow rigidity, which gives rise to an interesting deformation theory. After defining and introducing convex projective geometry we will discuss examples of various rigidity and flexibility phenomena that occur in dimension 3. This is a thesis defense.

This talk is introductory. We explain the role of Carnot groups and their subFinsler geometry in Geometric Group Theory, recalling whole of the terminology. We discuss a link between fine volume estimates, rate of convergence to asymptotic cones, and presence of abnormal geodesics. The results on rate of convergence for nilpotent groups are in collaboration with Breuillard (Paris). The results on abnormal curves in Carnot groups are in collaboration with Leonardi (Modena), Monti (Padova), and Vittone (Padova).