Abstract:

The curve complex (or graph if we are only interested in its 1-skeleton) is a wonderful tool for proving theorems about mapping class groups of surfaces. Some of these theorems are also true, or we at least hope are true, for automorphism groups of free groups. Much recent progress has focused on understanding analogous objects to the curve graph in the free group setting. We will give a survey of some recent results. We talk about one reason for the existence of multiple graphs in this setting: there is more than one way of describing what a ‘simple closed curve in a surface’ should be in a free group.

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Given a conic metric along a simple normal crossing divisor, we consider the conical Kahler-Ricci flow which preserves the conic structure. We use approximation method to construct the solutions on the largest time interval. Then we establish high order estimates for such flow solution. Finally we briefly introduce some convergence results.

Abstract:

On a Kahler manifold there is a clear connection between the  complex geometry and underlying Riemannian geometry, which can be used to characterize the Kahler condition. While such a link is not as clear in the non-Kahler setting, one can seek to understand these  characterizations as specific instances of a more general type.  I will address such questions from the perspective  of relationships between the Chern and Riemannian scalar curvatures. This is joint work with Michael Dabkowski.

Abstract:

The recent proof Demailly's conjecture by Witt Niströrm gives another evidence that pluripotential theory play a key role when working with complex Monge-Ampère equations in order to solve problems in differential and algebraic geometry. In this talk we investigate pluripotential tools: we characterize the
Monge-Ampère energy class E in terms of "envelopes". And in order to do that, we develop the theory of weak geodesic rays in a big cohomogy class. We also give a positive answer to an open problem in pluripotential theory. This is a joint work with Tamas Darvas and Chinh Lu.

Abstract:

Compact 6-dimensional nearly Kähler manifolds are the cross-sections of Riemannian cones with G2 holonomy. In particular they are Einstein manifolds with positive scalar curvature and admit real Killing spinors. Viewing Euclidean 7-space as the cone over the round 6-sphere endows the 6-sphere with a nearly Kähler structure which coincides with the standard G2-invariant almost complex structure induced by octonionic multiplication. A long-standing problem has been the question of existence of complete nearly Kähler 6-manifolds besides the four known homogeneous ones. We resolve this problem by proving the existence of an exotic (inhomogeneous) nearly Kähler structure on the 6-sphere and on the product of two 3-spheres. This is joint work with Mark Haskins, Imperial College London.

 

Abstract:

In the seminar I will present some joint work with T. Rivière where we study the Dirichlet energy of moving frames on 2-dimensional tori immersed in the euclidean $3\leq m$-dimensional space.  This functional, called Frame energy, is naturally linked to the Willmore energy of the immersion and on the conformal structure of the abstract underlying surface. I will discuss a ''Willmore Conjecture'' type lower bounds for such energy,  then I will introduce basic tools for doing the calculus of variations and finally I will give applications to regular homotopy classes of immersed tori in R^3.
 

Abstract:

I will present an algebraic characterization of the conformal classes realizing a compact manifold’s Yamabe invariant (if any such classes exist). This characterization is a nonlinear analogue of an observation of Nadirashvili for metrics realizing the maximal first eigenvalue, and of an observation of Fraser and Schoen for metrics realizing the maximal first Steklov eigenvalue.
 

Abstract:

Donaldson's continuity method of deforming cone angles for conical Kahler-Einstein metrics is an important tool in the solution of the Yau-Tian-Donaldson conjecture. We give an alternative proof to the "openness" part of this method by smooth approximation, and generalize this idea to smoothable Q-Fano variety, which allows us to prove the existence and study the deformation of singular KE metrics.

Abstract:

Stratified spaces are singular metric spaces which have been introduced in topology, and then studied from an analytical point of view. They naturally appear in differential geometry as quotients or limits of smooth manifolds. In this talk I will briefly introduce the geometry of stratified spaces with the aim of illustrating the recent developments in the resolution of the Yamabe problem for this singular setting. In order to do this, I will show how some classical results of Riemannian geometry can be recovered by singular theorems for stratified spaces.

Abstract:

To find solutions for the Cauchy problem in general relativity, one must understand the set of solutions to the Einstein constraint equations. The conformal method has proved to be a very useful tool for parametrizing this set, and it has only recently found application on initial data sets with arbitrary mean curvature. In this talk, I will outline the conformal method and describe my recent work in extending this technique to manifolds with asymptotically conformally cylindrical ends.

Abstract:

For a compact Riemannian manifold of dimension at least three, we know that positive Yamabe invariant implies the existence of a conformal metric with positive scalar curvature. As a higher order analogue, we seek for similar characterizations for the Paneitz operator and Q-curvature in higher dimensions. For a smooth compact Riemannian manifold of dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q-curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator. In addition, we also study the relationship between different conformal invariants associated to the Q-curvature. This is joint work with Matt Gursky and Fengbo Hang.

Abstract:

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we characterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean 4-balls.  This is joint work with Alice Chang.

Abstract:

It is a classical question to examine which smooth manifolds admit Riemannian metrics with positive/non-negative sectional curvature. In this talk, I will discuss some joint work with Manuel Amann on this question in the presence of torus symmetry, specifically in small dimensions. It extends work of Dessai, where a number of topological invariants of 8-manifolds with positive curvature and torus symmetry are calculated, both in general and under additional assumptions (e.g., rationally elliptic or homogeneous).

Abstract:

Gradient Ricci expanding solitons are immortal self-similarities to the Ricci flow. We will focus on the possibility of resolving metric cones over a smooth compact manifold by such expanders. More precisely, we will start by giving examples confirming these expectations then we will concentrate on asymptotic estimates, compactness phenomena, and deformations of conical Ricci expanders.

Abstract:

After a brief review of full justification of model equations, we present an application of model justification to the long-time
well-posedness of the cubic "hyperbolic" NLS equation, in which the Laplacian is replaced by an indefinite second order operator-- and for which the long-time existence remains open for large data due to the absence of a coercive Hamiltonian.

Abstract:

We prove strong gradient decay estimates for solutions to the multi-dimensional Fisher-KPP equation with fractional diffusion. It is known that this equation exhibits exponentially advancing level sets with strong qualitative upper and lower bounds on the solution. However, little has been shown concerning the gradient of the solution. We prove that, under mild conditions on the initial data, the first and second derivatives of the solution obey a comparative exponential decay in time. We then use this estimate to prove a symmetrization result, which shows that the reaction front circularizes in renormalized coordinates.

Abstract:

The solutions to many singular SPDEs are obtained as limits of regularised and then renormalised equations. The renormalisation changes the "original" equation via quantities that are typically infinity, but they do have concrete physical meanings. As an example, I will discuss how the Phi^4_3 equation, interpreted after suitable renormalisations, arise naturally as the universal limit of symmetric phase coexistence models. We will also see how this universality can be lost when symmetry is broken.

Abstract:

Poincar\'e inequalities are among the most studied inequalities in probability and functional analysis. In this talk we will discuss the $L_p$ Poincar\'e inequalities with constants $C\sqrt{p}$. They are weaker than the log-Sobolev inequality, but still imply subgaussian concentration and transportation cost inequalities. We use martingale methods and infinite dimensional Brownian motions to prove these inequalities in various contexts, and demonstrate that certain noncommutative techniques are helpful even in some commutative settings.

Abstract:

The first part of my talk will discuss ways of establishing cubic lifespan bounds for quasi-linear dispersive equations via the \emph{modified energy method}. This robust method is an upgrade of the normal form energy method introduced by Shatah in 1983 for the Klein-Gordon equation. For simplicity we will first implement it for the Burger's-Hilbert equation (which is not dispersive), but where it nevertheless works.  The second part of my talk will discuss global existence of solutions provided that the initial data is small and spatially localized via the \emph{testing with wave packets method}. We will show how it works  for one simple example: for the one dimensional cubic NLS.

Abstract:

 The log-Sobolev inequality (LSI) is a very useful tool for analyzinghigh-dimensional situations. For example, the LSI can be used for deriving hydrodynamic limits, for estimating the error in stochastic homogenization, for deducing upper bounds on the mixing times of Markov chains, and even in the proof of the Poincaré conjecture by Perelman. For most applications, it is crucial that the constant in the LSI is uniform in the size of the underlying system. In this talk, we
discuss when to expect a uniform LSI  in the setting of unbounded spin systems. We will also explain a connection to the KLS conjecture.

Abstract:

The wave maps equation is an important example of a geometric wave equation, and generalizes the concept of a free wave to manifold-valued maps. There has been significant progress on understanding the dynamics of wave maps from the Minkowski space to both positively and negatively curved targets, especially in the energy critical setting where the spatial domain dimension is two. In this talk I will discuss some of the new behaviors which result from changing the domain geometry. In particular, I will give a survey of recent works with Andrew Lawrie and Sung-Jin Oh in the case where the domain is the hyperbolic space.

Abstract:

We review the subcritical scattering theory for intercritical NLS.  Specifically, we consider defocusing power-type nonlinear Schr\"odinger equations with a nonlinearity that is mass-supercritical and energy-subcritical and give a proof of scattering for solutions with data in H^1.  The talk will be primarily expository and is meant to serve as an introduction to Morawetz estimates. 

Abstract:

It is well known from the results of A.Zvonkin, A.Veretennikov, N.Krylov, A.Davie, F.Flandoli, J.Mattingly and other probabilists that ordinary differential equations (ODEs) regularize in the presence of noise. Even if an ODE is "very bad" and has no solutions (or has many solutions), then the addition of a random noise leads (almost surely) to a "nice" ODE with a unique solution.  We investigate the same phenomenon for a heat equation with a drift. We prove that for almost all trajectories of random white noise the perturbed heat equation has a unique solution (even if the original heat equation with a drift had many or no solutions).(Joint work with Leonid Mytnik.)

Abstract:

A fundamental problem in PDE is to determine the coefficients of a system from physical measurements. I will introduce the problem of recovering a magnetic Schrodinger operator from the boundary data of solutions and discuss some connections to dispersive PDE. 

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We will provide a direct proof of the observability inequality of backward stochastic heat equations for measurable sets. As an immediate application, the null controllability of the forward heat equations is obtained. Moreover, an interesting relaxed optimal actuator location problem is formulated, and the existence of its solution is proved. Finally, the solution is characterized by a Nash equilibrium of the associated game problem.

Abstract:

We study the stochastic heat equation with multiplicative noises: $\frac {\partial u }{\partial t} =\frac  12 \Delta u  + u \dot{W}$, where $\dot W$ is a mean zero Gaussian noise and $u \dot{W}$ is interpreted  both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutions are obtained under general and different conditions. These formulas are applied to obtain the H\"older continuity of the solutions. They are also applied to obtain the intermittency bounds for the moments of the solutions.

Abstract:

Intermittency usually refers to something occurring irregularly at different scales or tall peaks on small regions, typically as time gets larger.

In the first part of this talk, instead of looking at a large time behavior, we will consider nonlinear noise excitation of a large family of intermittent stochastic partial differential equations (SPDEs). We show that there is a near-dichotomy: “Semi-discrete” equations are nearly always far less excitable than “continuous” equations.

In the second part of this talk, we consider large scale structures of points of tall peaks for various SPDEs such as parabolic Anderson models (which are intermittent) and SPDEs with additive noise (which are not intermittent). We show that parabolic Anderson models are multi-fractal (as expected), but so are SPDEs with additive noise.

This is based on joint works with Davar Khoshnevisan and Yimin Xiao.

Abstract:

Recently there has been much interest in damping phenomena for kinetic equations following the seminal works of Mouhot-Villani on Landau damping and of Bedrossian-Masmoudi on inviscid damping around Couette flow.
In this talk I present some of the main results of my PhD thesis on linear inviscid damping for the 2D Euler equations around general monotone shear  flows in the framework of Sobolev regularity.
Here I consider both the settings of an infinite periodic channel and a finite periodic channel with impermeable walls.
The latter setting is shown to not only be technically more challenging, but to exhibit qualitatively different behavior due to boundary effects.

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We will discuss the classical small divisor problem, and connections to questions concerning Sobolev stability of solutions to the nonlinear Schrodinger equation on the d-dimensional torus. In particular, we will examine results concerning the arbitrarily long-time orbital stability of plane wave solutions under generic perturbations.

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We will study the diophantine properties of random finitely generated subgroups of Lie groups, focusing mainly on the case of nilpotent Lie groups.
 

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We will overview both examples of flat surfaces with amazing and unexpected geometry and dynamics, and limitations to the number and behavior of such surfaces.

Abstract:

Bounded cohomology is a variation of usual cohomology that has many interesting geometric applications: for example it allows to define interesting invariants on some character varieties, and select maximal representations. However it is in general very difficult to compute and still poorly understood.

I will discuss in which sense mapping class groups have enough hyperbolic features to ensure that their third bounded cohomology is infinite dimensional. This is joint work with Roberto Frigerio and Alessandro Sisto.

Abstract:

Given a subset A of a group G, we want to compare the size the product set AAA of elements that can be written as products of three elements of A with the size of A. I will discuss this problem when G is a simple Lie group and the sets A and AAA are measured by their Hausdorff dimension.

Abstract:

The Nahm Transform is a nonlinear analog of the Fourier Transform, used to construct maps between moduli spaces of solutions of dimensional reductions of Yang-Mills equations. I will discuss several examples and open problems related  to this construction, focusing on the moduli space of solutions to Hitchin equations over a Riemann surface. 

Abstract:

In 1940 Siegel proved that for any indefinite integral quadratic form the integral points of its orthogonal group is finitely generated. The aim of this talk is to present an effective upper bound on the norm of a finite generating set. This is a joint work with Professor Gregory A. Margulis, and the proof uses our recent work on the equivalence of integral quadratic forms.

Abstract:

Given a geodesic lamination with finitely many leaves in a closed surface of genus at least 2, I'll construct a very explicit parametrization of the Hitchin component of this surface. In essence, this parametrization is an extension of Thurston's shearing coordinates for the Teichmueller space of a closed surface, combined with Fock-Goncharov's coordinates for the moduli space of positive framed local systems of a punctured surface. This is joint work with Francis Bonahon.

Abstract:

The stable ratio set of a nonsingular action is a notion introduced by Bowen and Nevo to prove pointwise ergodic theorems for measure preserving actions of certain nonamenable groups.

I  prove that stable ratio set of the action of a discrete subgroup of isometries of a CAT(-1) space with finite Bowen-Margulis measure on its boundary with the Patterson-Sullivan measure has numbers other than 0, 1 and \infinity, extending techniques of Bowen from the setting of hyperbolic groups.

I also prove the same result for the action of the mapping class group on the sphere of projective measured foliations with the Thurston measure, using some "statistical hyperbolicity" properties for the (non hyperbolic) Teichmueller metric on Teichmueller space.

Abstract:

In this talk, I will talk about a technical part in the joint paper with Brian Collier about studying asymptotic behavior of Hitchin representations in terms of Higgs bundles. This technical part estimates the asymptotic solution of the coupled Hitchin equations. I will also discuss how the analysis is related to geometry of frames.

Abstract:

(Joint with Caglar Uyanik.) Following a strategy developed by Athreya and Cheung, we compute the gap distribution of the slopes of saddle connections on the octagon by translating the problem to a question about return times of the horocycle flow to an appropriate Poincaré Section. This same strategy was used by Athreya, Chaika, and Lelièvre to compute the gap distribution on the Golden L. The octagon is the first example of this type of computation where the Veech group has two cusps.

Abstract:

In 1970, E.M. Andreev gave a full description of 3-dimensional compact hyperbolic polyhedra with dihedral angles submultiples of pi. We call them hyperbolic Coxeter polyhedra. More precisely, given a combinatorial polyhedron C with assigned dihedral angles, Andreev’s theorem provides necessary and sufficient conditions for the existence of a hyperbolic Coxeter polyhedron realizing C. Since hyperbolic geometry arises naturally as sub-geometry of real projective geometry, we can ask an analogous question for compact real projective Coxeter polyhedra. In this talk, I’ll give a partial answer to this question. This is a joint work with Suhyoung Choi.

Abstract:

We consider the orbits {pu(n^{1+γ})|n ∈ N} in Γ\PSL(2,R), where Γ is a non-uniform lattice in PSL(2,R) and u(t) is the standard unipotent group in PSL(2,R). Under a Diophantine condition on the intial point p, we show that {pu(n^{1+γ})|n ∈ N} is equidistributed in Γ\PSL(2,R) for small γ>0, which generalizes a result of Venkatesh.

Abstract:

A complex projective structure is a geometric structure on a surface modeled on the Riemann sphere. Then a complex projective structure has a holonomy representation from the fundamental group of the surface into PSL(2, C), which is not necessarily discrete.

We discuss about ``neck-pinching'’ type degeneration of complex projective structures when their holonomy representations converge in the character variety.

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In his seminal paper from 1967, Furstenberg proved that for every irrational x, the set 2^{n}3^{m}x is dense modulo 1.
I will show a couple of generalizations of this result, which imply density of much sparser sequences, using earlier works of D. Meiri and M. Boshernitzan, following the proof of Bourgain-Lindenstrauss-Michel-Venkatesh.

Abstract:

The semi-direct product of a finite-rank free group with the integers can often be expressed as such a product in infinitely many ways. This talk will explore this phenomenon and work towards 1) describing the structure of the family of such splittings of a given group, and 2) looking for for relationships between the splittings themselves. Along the way, we'll study dynamical systems ranging from graph maps and cross sections of semi-flows to newly introduced polynomial invariants tying these all together. Time permitting, I'll discuss geometric properties of more general free group extensions. This represents work with Ilya Kapovich and Christopher Leininger, and separately with Samuel Taylor.

Abstract:

Let $F=\{g_t\}$ be a one-parametr diagonal subgroup of $SL_n(\mathbb R)$.
We assume  $F$ has no nonzero invariant vectors in $\mathbb R^n$.
Let $x\in X, \varphi\in C_c(X)$ and $\mu$ be the probability Haar measure
on $X$. For certain proper subgroup $U$ of the unstable horospherical  subgroup
of $g_1$ we show that for almost every $u\in U$
\[
\frac{1}{T}\int_0^T\varphi({g_tux})dt \to \int_X\varphi d\mu.
\]
If $\varphi$ is moreover smooth, we can get an  error rate of the convergence.
The error rate is ineffective due to the use of Borel-Cantelli lemma.

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Abstract:

An almost-Fuchsian manifold is a quasi-Fuchsian manifold which contains an incompressible minimal surface with principal curvatures less than one everywhere.  The topological entropy of the geodesic flow on the minimal surface defines a function on the space of almost-Fuchsian manifolds.  We will explain how this function can be used to give a metric on Fuchsian space.  Furthermore, we will also discuss the relationship between this entropy and the Hausdorff dimension of limit sets of quasi-Fuchsian groups.

Abstract:

The quantum ergodicity theorem says that on a compact Riemannian manifold with ergodic geodesic flow, for any orthonormal basis of eigenfunctions of the Laplacian in L2, the modulus squared of these eigenfunctions converge weakly as probability measures to the uniform measure, in the limit of large eigenvalues and up to a subsequence of density 0.
On the sphere the geodesic flow is not ergodic and it is possible to find subsequences of eigenfunctions with positive density that do not satisfy the conclusion of the theorem. However, it holds almost surely for random eigenbasis.
We will present a quantum ergodicity theorem on the sphere for joint eigenfunctions of the Laplacian and an averaging operator over a finite set of rotations. The proof is based on a new argument for quantum ergodicity on regular graphs.
 

Abstract:

The Kac polynomial counts the number of representations of a quiver over a finite field which are indecomposable over the algebraic closure of this field. Recently Hausel, Letellier, and Rodriguez-Villegas proved the Kac conjecture which states that the coefficients of the Kac polynomial are nonnegative. I will talk about this and related results.

Abstract:

A representation of a reductive p-adic group has its character as a distribution on the group. Its asymptotic behavior near the identity is given by a finite-term local character expansion of Harish-Chandra. In this talk, we state a result giving a few terms in the local character expansions for certain supercuspidal representations of a ramified unitary group. The numbers are related to the number of rational points on certain covers of hyperelliptic curves. We'll then talk about how endoscopy transfer for these characters is related to geometric identities regarding H^1 of these curves. A side goal will be to demonstrate possible similarity between such phenomenon and the work of Bhargava-Gross on arithmetic invariant theory of $SO_{2n+1}$ on $\text{Sym}^2$.

Abstract:

I will present some results about the geometry of the stack of G-bundles on an elliptic curve and how one can use this to construct simple summands of spherical Eisenstein sheaves. If time permits I will discuss a conjectural description of all the simple summands of spherical Eisenstein sheaves.

Abstract:

The eigencurve is a rigid analytic curve over Q_p parametrizing all finite slope overconvergent modular eigencurve. It is a conjecture of Coleman-Mazur that the eigencurve has "no holes". In other words, the eigencurve is proper over the weight space. We prove that the conjecture is true.

Abstract:

In this talk, we will explain the link between the Vinberg's semigroup and the Hitchin group like fibration, that was introduced by Frenkel and Ngô for SL_{2}. This fibration appears as a nice object to get orbital integrals for the spherical Hecke algebra and a good understanding of the orbital side of the trace formula.

Abstract:

A theorem of Coleman asserts that if f is an overconvergent U_p-eigenform of weight k>1 such that the valuation of its U_p-eigenvalue is <k-1, then f is classical modular form. In this talk I will discuss variations of Coleman's proof of this theorem, with an eye towards ideas that generalize to higher-dimensional Shimura varieties. Part of this is joint work with Vincent Pilloni.

Abstract:

We discuss a level raising result mod p=2 for weight 2 modular forms, where some extra phenomena happens compared to the p odd case. We then apply this to study 2-Selmer groups of modular forms in level raising families.

This is joint work with Chao Li.

Abstract:

We're going to try and describe the ideas that go into Beilinson and Drinfeld's main construction from their book "Quantization of Hitchin's integrable system and Hecke eigensheaves."

Abstract:

I will discuss a class of integrable connections associated to root systems and describe their monodromy in terms of quantum groups. These connections come in three forms, rational form, trigonometric form, and the elliptic form, which lead to representations of braid groups, affine braid groups, and elliptic braid group respectively.

For the rational connection, I will discuss in detail two concrete incarnations: the (Coxeter) Knizhnik-Zamolodchikov connection and the Casimir connection.

The first takes values in the Weyl group W. Its monodromy gives rise to an isomorphism between the Hecke algebra (with generic parameters) of W and the group algebra C[W] of the Weyl group. The second is associated to the semisimple Lie algebra g, and takes values in the universal enveloping algebra of g. Its monodromy is described by the quantum Weyl group operators of the quantum group. The trigonometric and the elliptic analog will also be discussed.

The elliptic part is joint work with Valerio Toledano Laredo.

Abstract:

Kisin showed that the generic fibers of potentially semi-stable (framed) deformation rings of p-adic Galois representations valued in GL_n are generically smooth, and he computed their dimensions.  I will explain how to extend these results to Galois representations valued in an arbitrary connected reductive group G.  If time permits, I will give an example showing that the corresponding schemes can have singular components.  The key tool is the geometry of the nilpotent cone.

Abstract:

I will present a sketch of a topological proof of the Kashiwara-Vergne problem in Lie theory. This is a special case of a general method which provides several interesting examples of close relationships between quantum topology and algebra, in particular equations in graded spaces.

Abstract:

A theorem of Weyl asserts that the number of eigenvalues of the Laplacian less than X on a compact Riemann surface of dimension d is asymptotic to a constant multiple of X^{d/2}. A similar statement is true for the number of cuspidal automorphic representations with bounded infinitesimal character of G(\R)/K for G a split adjoint semisimple group and K a maximal compact subgroup of G(\R) (Selberg, Miller, Müller, Lindenstrauss-Venkatesh). Instead of just counting automorphic forms, it is also of interest to weight this counting by traces of Hecke operators. An asymptotic for this problem together with a bound on the error term has applications in the theory of families of L-functions. I want to explain the automorphic Weyl law, and some recent results for the problem involving Hecke operators in the case of GL(n).

Abstract:

I will explain how the Springer correspondence (and some generalizations) arise naturally when considering adjoint functors of parabolic induction and restriction for sheaves on a reductive group.

Abstract:

Shimura varieties attached to unitary groups of signature (n-r,r) have integral models described by moduli spaces of certain principally polarized abelian schemes. We will consider the case r=0, and show that the corresponding moduli stack can be described as a categorical tensor product of the stack of CM elliptic curves with a category of rank-n positive-definite hermitian modules.

Abstract:

For elliptic curves E/Q whose L-function L=L(E,s) vanishes to order one at s=1, the rank of E(Q) is also known to be one. This is the first prediction of the Birch and Swinnerton-Dyer conjecture, and the main ingredient of the proof is the formula of Gross and Zagier relating the heights of modularly-constructed points on E to the central derivative of L. The second prediction of BSD is a formula for the central leading term of L. This is only implied by the Gross-Zagier formula up to a nonzero rational number. One way to go on and study the BSD formula up to p-integrality is provided by a p-adic analogue of the Gross-Zagier formula due to Perrin-Riou and Kobayashi. I will explain this circle of ideas as well as its generalization to totally real fields. Time permitting, I will also discuss the representation-theoretic context.

 

The talk is meant to be accessible to a broad audience.

Abstract:

We will explain how to obtain effective Mordell-Lang and Manin-Mumford using jet space techniques in the characteristic zero function field setting.

Abstract:

Homotopy theory can be thought of as the study of geometric objects and continuous deformations between them, and then iterating the idea as the deformations themselves form geometric objects. One result of this iteration is that it replaces morphism sets with topological spaces, thus remembering a lot more information. There are many examples to show that the approach of replacing sets with spaces in a meaningful way can lead to remarkable developments. In this talk, I will explain some of my recent work in the case of implementing homotopy theory in arithmetic in a way which produces new results and relationships between some classical notions of duality in both fields.

Abstract:

This talk--aimed for a general audience of neither topologists nor model theorists--will discuss applications of cobordisms to Kontsevich's mirror symmetry conjecture. We'll begin by stating a rough version of the
conjecture, which builds a bridge between symplectic geometry on one hand, and on the other hand, algebraic geometry over the complex numbers. We then discuss how the theory of cobordisms, which studies when two manifolds can be the boundary of another manifold, sheds light on how to generalize the mirror symmetry conjecture, while giving us information about objects in symplectic geometry. (For example, two Lagrangians related by a compact cobordism are equivalent in the Fukaya category.)

Abstract:

We will explore the relationships between Galois theory, groups acting on spaces, and motivic homotopy theory. Ultimately, for R a real closed field, we will discover that that there is a full and faithful embedding of the stable Gal(R[i]/R)-equivariant homotopy category into the stable motivic homotopy category over R.

Abstract:

Homotopy theorists try to gain geometric information and insight through the use of algebraic invariants. Specifically, these invariants are useful in determining whether or not two spaces can be equivalent. We will begin with an example to demonstrate the usefulness of cohomology and some of the extra structure it possesses, such as cup products and power operations. This extra structure provides a very strong invariant of the space. As these invariants are representable functors, this extra structure is coming from the representing object. Indeed, cohomology theories possess products and power operations when they are represented by objects called commutative ring spectra. We then shift focus to studying commutative ring spectra on their own and try to detect what maps of commutative ring spectra might look like.

Abstract:

In this talk, we will be working in with the theory of differentially closed fields of characteristic zero; essentially this theory says that every differential equation which might have a solution in some field extension already has a solution in the differentially closed field. After introducing this theory in a bit of detail, we will sketch a proof of the strong minimality of the differential equation satisfied by the classical $j$-function starting from Pila's modular Ax-Lindemann-Weierstrass theorem. This resolves an open question about the existence of a geometrically trivial strongly minimal set which is not $\aleph _0$-categorical. If time allows, we will discuss some finiteness applications for intersections of certain sets in modular curves. This is joint work with Tom Scanlon.

Abstract:

One of the great insights of model theory is the observation that very mundane-looking "combinatorial configurations" carry a huge amount of geometric information about a structure. In this talk, I will explain what we mean by "combinatorial configuration," and then I will sketch out how configurations can be "smoothed out" to yield Ramsey classes, which can themselves be analyzed using model-theoretic tools. I will also discuss the kinds of model-theoretic dividing lines that can be defined just through the interaction of structures with Ramsey classes.

Abstract:

The study of category theory was started by Eilenberg and MacLane, in their effort to codify the axioms for homology. Category theory provides a language to express the different structures that we see in topology, and in most of mathematics. Categories also play another role in algebraic topology. Via the classifying space construction, topologists use categories to build spaces whose geometry encodes the algebraic structure of the category. This construction is a fruitful way of producing important examples of spaces used in algebraic topology. In this talk we will describe how this process works, starting from classic examples and ending with some recent work.

Abstract:

I will describe a topologists' perspective on the history of the study of an object that Mumford called "the tautological ring" and its generalizations.

The tautological ring was originally defined as a subring of the cohomology of the moduli space of Riemann surfaces, but can also be studied as a ring of characteristic classes of topological bundles. This point of view led to a proof of Mumford's conjecture, stating that the tautological ring coincides with the entire cohomology of the moduli space in a "stable range", as well as to some generalizations of this result. If time permits, I will explain what we know about the tautological ring outside the stable range.

Abstract:

In algebraic topology, one key way of understanding group actions on spaces is by considering families of fixed points under subgroups.  In this talk, we will discuss this basic structure and its fundamental role in understanding equivariant algebraic topology.  I will then describe some recent joint work with A. Osorno that builds on fixed point information to create equivariant cohomology theories.

Abstract:

I will present a combinatorial property---finite VC-dimension---which appeared independently in various parts of mathematics.

In model theory it is called "NIP" and is used notably in the study of ordered and valued fields. In probability theory, it is related to "learnable classes". In combinatorics, classes of finite VC-dimension behave a lot like families of convex subsets of euclidean space. I will also talk about Banach spaces and topological dynamics.

The talk will be accessible to postdocs of both programs.

Abstract:

Speaker: Romeo Awi

Abstract:

A new method, currently under development, is brought forward to establish an upper bound on the growth of any finitely generated group, using a variant of monoid categories and analagous CW-complexes.

Abstract: I'll describe the implementation of Hiary's O(t1/3) algorithm and the computations that we have been running using it. Some highlights include the 10^32nd zero (and a few hundred of its neighbors, all of which lie on the critical line), values of S(T) which are larger than 3, and values of zeta larger than 14000.

Abstract: In this talk, we will consider a free boundary problem with a
very general free boundary condition and analyze the non-degeneracy of the
largest subsolution near the free boundary.

Abstract: Consider the problem of imaging a reflector (target) from recordings of the echoes resulting from probing the medium with waves emanating from an array of transducers (the array response matrix). We present an algorithm that selectively illuminates the edges or the interior of an extended target by choosing particular subspaces of the array response matrix. For a homogeneous background medium, we characterize these subspaces in terms of the singular functions of a space and wave number restricting operator, which are also called generalized prolate spheroidal wave functions. We discuss results indicating what can be expected from using this algorithm when the medium fluctuates around a constant background medium and the fluctuations can be modeled as a random field.

Abstract: Connections are exposed between integrable equations for spectral gap probabilities of unitary invariant ensembles of random matrices (UE) derived by different --- Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) --- methods. Simple universal relations are obtained between these probabilities and their ratios on one side, and variables of the approach using resolvent kernels of Fredholm operators on the other side. A unified description of UE is developed in terms of universal, i.e. independent of the specific probability measure, PDEs for gap probabilities, using the correspondence of TW and ASvM variables. These considerations are based on the three-term recurrence for orthogonal polynomials (OP) and one-dimensional Toda lattice (or Toda-AKNS) integrable hierarchy whose flows are the continuous transformations between different OP bases. Similar connections exist for coupled UE. The gap probabilities for one-matrix Gaussian UE (GUE) or joint gap probabilities for coupled GUE satisfy various PDEs whose number grows with the number of spectral endpoints. With the above connections serving as a guide, minimal complete sets of independent lowest order PDEs for the GUE and for the largest eigenvalues of two-matrix coupled GUE are found.

Abstract: We show that on a smooth compact Riemann surface with boundary (M0, g) the Dirichletto- Neumann map of the Schrödinger operator â g + V determines uniquely the potential V . This seemingly analytical problem turns out to have connections with ideas in symplectic geometry and differential topology. We will discuss how these geometrical features arise and the techniques we use to treat them. This is joint work with Colin Guillarmou of CNRS Nice. The speaker is partially supported by NSF Grant No. DMS-0807502 during this work.

Abstract: The problem of Electrical Impedance Tomography (EIT) with partial boundary measurements is to determine the electric conductivity inside a body from the simultaneous measurements of direct currents and voltages on a subset of its boundary. Even in the case of full boundary measurements the non-linear inverse problem is known to be exponentially ill-conditioned. Thus, any numerical method of solving the EIT problem must employ some form of regularization. We propose to regularize the problem by using sparse representations of the unknown conductivity on adaptive finite volume grids known as the optimal grids. Then the discretized partial data EIT problem can be reduced to solving the discrete inverse problems for resistor networks. Two distinct approaches implementing this strategy are presented. The first approach uses the results for the EIT problem with full boundary measurements, which rely on the use of resistor networks with circular graph topology. The optimal grids for such networks are essentially one dimensional objects, which can be computed explicitly. We solve the partial data problem by reducing it to the full data case using the theory of extremal quasiconformal (Teichmuller) mappings. The second approach is based on resistor networks with the pyramidal graph topology. Such network topology is better suited for the partial data problem, since it allows for explicit treatment of the inaccessible part of the boundary. We present a method of computing the optimal grids for the networks with general topology (including pyramidal), which is based on the sensitivity analysis of both the continuum and the discrete EIT problems. We present extensive numerical results for the two approaches. We demonstrate both the optimal grids and the reconstructions of smooth and discontinuous conductivities in a variety of domains. The numerical results show two main advantages of our approaches compared to the traditional optimization-based methods. First, the inversion based on resistor networks is orders of magnitude faster than any iterative algorithm. Second, our approaches are able to correctly reconstruct the conductivities of very high contrast, which usually present a challenge to the iterative or linearization-based inversion methods.