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).

We will explain the relation between the so-called big Heegner points introduced by Ben Howard and two different anticyclotomic p-adic L-functions: the first one, as appeared in the work of Perrin-Riou and Nekovar on a p-adic variant of the Gross-Zagier formula for Heegner points and Heegner cycles, and the second one, as considered in the more recent work of Bertolini-Darmon-Prasanna, Brakocevic and Hsieh.

Isoperimetric problems are of great interest both in the Euclidean and in the Gaussian setting. A fundamental tool to understand them is the idea of symmetrizations (in the Euclidean setting, the idea is due to Steiner), as it is known that the perimeter behaves monotonically under symmetrization. Natural questions which arise in this context are then the analysis of equality cases and of rigidity for the symmetrization inequalities, where by rigidity we mean the situation when the equality cases are symmetric sets by themselves. We study a geometric characterization of rigidity, focusing mainly on the Gaussian problem. The condition is formulated in terms of a new measure-theoretic notion of connectedness for Borel sets, inspired by Federer’s definition of indecomposable current.

A level set method on non-graded adaptive Cartesian grids is presented. They use quadtree and octree data structures to represent the grid and a simple algorithm to generate a mesh with the finest resolution at the interface.

An insider is a market agent who has privileged information about the future state of the market or, more specifically, the future state of an underlying stock. In the filtration enlargement framework, this additional information yields a drift in the perceived dynamics of the stock. The insider's trading strategy can therefore be thought of as resulting from an optimization problem where the stock has dynamics that are different than those that are publicly observed. One can ask whether it is possible to recover the insider’s perceived drift knowing only his trading strategy. An approach to solving this problem is presented.

In 1955, Cassels gave an upper bound for the least nontrivial integral point of a quadratic form. Many new proofs and generalizations have been found, departing from his original method. We return to his 1955 paper and interpret his calculations in the language of Macaulay resultants and height functions, deriving a theorem applicable to forms of arbitrary degree. In particular, if n-1 homogeneous polynomials in n variables share only finitely many projective zeros, we give an upper bound for a weighted product of the heights of those zeros in terms of a weighted product of the heights of the polynomials. This is a thesis defense.

Abstract: We will remind ourselves of the Boltzmann equation and will explain what creation/propagation of exponential moments means. Then, we will sketch three methods for proving these results, some of which appeared in the literature very recently. Finally, we will mention a related open problem and discuss possible ways of treating it.

A large class of kinetic and fluid problems in plasma physics can be expressed using a Hamiltonian approach. Such systems consist of an advection equation coupled to field equations. The advection speeds are determined from the Hamiltonian and a Poisson bracket, and, in many instances, the fields are computed using elliptic equations. One feature of such systems are the existence of quadratic invariants, like energy and, in context of incompressible fluid flow, enstrophy. In this talk we present extensions of discontinuous Galerkin algorithms to solve such equations. We show that with a proper choice of basis functions for the advection and elliptic equations the DG schemes can conserve energy exactly. Further, with a choice of central fluxes the enstrophy (the L2 norm of the solution) can also be conserved exactly. Extension to the scheme to handle discontinuous Hamiltonians are developed. Application of these schemes to incompressible flow problems, drift wave turbulence and Vlasov-Poisson equations are presented.

We will start with the basics of a finance model as well as the concept of arbitrage to motivate the FTAP. A proof of the theorem will be provided in a discrete-time case. If time permits, we will discuss the continuous case and generalizations.

This works are related to analytical properties the Boltzmann equation, a probabilistic model for interacting rarefied gases. The interacting operator, usually referred as the collision operator, is a nonlinear integral form. After an extensive work on creation and propagation of polynomial moments for the Boltzmann equation, a similar question for exponential moments was raised in late 90's and early 00's for the case on integrable angular cross section in the collisional operator (Grad's cutoff assumption). Recently, some progress was made in the non-integrable case with a controlled singularity (non-cutoff case). We will discuss some of the recent methods developed in the literature so far for proving creation and propagation of exponential moments for the Boltzmann equation. Finally, we will say few words about a related open problem. This is a candidacy exam.

It is a theorem of Jaco and Oertel from 1984 that the question of whether a given 3-manifold contains a closed incompressible surface is algorithmically decidable. However their algorithm is not suitable for implementation on a real world computer because of iterated-exponential complexity in the input size, both in theory and in practice. In this talk I will discuss recent joint computational work with Stephan Tillmann and Ben Burton that overcomes these problems and determines in practice whether a given 3-manifold contains a closed essential surface. We apply our techniques to determine computationally exactly which of the 2977 prime knots with crossing number at most 12 contain a closed essential surface in their complement, and exactly which of the 10,031 closed hyperbolic manifolds in the Hodgson-Weeks census are Haken.

Renormalization and renormalization group is the most important theoretical advance in the 2nd half of the last centrury in physics. At its heart is a mathematical tool for handling singularities, infinities and complex systems involving multiple scales. Yet at the mathematical level, it still remains to be somewhat of a mysterious, ad hoc and even dubious procedure. We will discuss our effort to develop the mathematical foundation of the exact renormalization group flow. We will start with some simple examples (central limit theorem, extreme statistics, homogenization) and then discuss applications to stochastic PDEs, quantum field theory and turbulent transport.

Poisson vertex algebra (PVA) theories can be used to understand integrable Hamiltonian PDEs. When a PVA has two compatible λ-brackets, Lenard-Magri scheme may applicable to find integrable systems and integrals of motion of them. An important example of such PVAs is a classical affine W-algebra (under some assumptions). Similarly, Hamiltonian ODEs can be studied using Poisson algebra theories. In particular, we are interested in the case when the Poisson algebra is a Zhu algebra of some PVA. In this talk, I will introduce integrable Hamiltonian PDEs (resp. ODEs) using PVA (resp. Poisson algebra) theories and show some examples : affine (resp. finite) classical W-algebras. Also, I will explain Zhu algebras and suggest some interesting problems regarding the relations between Hamiltonian PDEs and ODEs.

Pursuing a work that was done with Pierre Pageault on Lyapunov functions for a dynamical system, we study the problem of existence of closed 1-forms for example for flows whose integral along orbits is non-positive, thus generalizing some work of Schwartzman--compare also with the work of Farber, Kappeler, Latschev, and Zehnder. We will in fact consider not necessarily smooth closed 1-forms, but will also consider "forms" whose local primitives are merely continuous or Lipschitz functions.

Finite groups of Lie type arise as Frobenius fixed points of certain linear algebraic groups, and the first part of the talk will serve as an introduction to such finite groups. The second part of the talk will explore the representation theory of these groups; we will focus on the unipotent characters and their relationship to the Weyl group of the original linear algebraic group. Finally, we give an indication of how the representation theory of finite groups of Lie type may be re-interpreted using topological field theory.

Low regularity global well-posedness is an interesting question in dispersive PDEs. The method of Colliander-Keel-Staffilani-Takaoka-Tao, called the I-method, has been successfully used for this question in various settings. In this talk, we show global well-posedness of a nonlinear Schr\"odinger equation with a potential below the energy norm. For this, we develop harmonic analysis tools to run the I-method in the inhomogeneous setting.

On a symplectic manifold one can define many almost-complex structures J. Given one of these, one can study the moduli space of (simple) J-holomorphic curves from some fixed Riemann surface into M. For generic J's, this space is a manifold. I might also state Gromov's compactness theorem so I get to draw some bubbles. All definitions will be given. The talk is supposed to be accessible to anybody interested in geometry/topology.

It is a well known result that the number of points over a finite field on the Legendre family of elliptic curves can be written in terms of a hypergeometric function modulo p. In this talk, I extend this result, due to Igusa, to a family of monomial deformations of a diagonal hypersurface and to some families of K3 surfaces. I also find explicit relationships between the number of points and generalized hypergeometric functions as well as their finite field analogues. If time permits, I will conclude with an ongoing joint project relating these results to arithmetic mirror symmetry.

Log Calabi-Yau manifolds (the complement Y-D of an effective anticanonical divisor D in a projective manifold Y) have a rich symplectic geometry. They are the subject of a mirror symmetry conjecture: roughly, if Y-D is affine, there is a mirror variety M, that is also affine log Calabi-Yau, and the Floer theory of Y-D is reflected in the algebraic geometry of M. For instance, one expects to find the ring of regular functions on M sitting inside the symplectic cohomology of Y-D. I will describe some results towards this conjecture, in complex dimension two. There is a basis for degree zero symplectic cohomology indexed by integral points in a certain SL(2,Z)-manifold (related to the Gross-Hacking-Keel theta functions). We can also compute the product structure in a particular case.

Single particle reconstruction techniques in Cryo-EM typically assume that all data comes from a single molecular conformation. In practice this is rarely the case. This talk introduces a method for dealing with multiple heterogeneous conformations of a molecule. The method reduces to solving the synchronization problem over the symmetric group. This is a candidacy talk.

The first talk for the student run Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Lee Cohn will be an introduction to characteristic classes, roughly following the second chapter of the book.

The second talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Tom Mainiero will be on Clifford algebras, and the construction and properties of the Dirac operator for a Clifford bundle. It will roughly following the third chapter of the book.