Schedule of talks:

Abstracts

(A. Iosevich): The basic question we ask is, given a sufficiently large subset of a given vector space, where large is determined via cardinality, Lebesgue measure or Hausdorff dimensions, depending on the context, does it contain a congruent copy of a given geometric configuration. In this talk we are going to describe connections between this problem and metric embedding theorems. We will also emphasize the role of curvature in the continuous setting and the analogous arithmetic phenomena in the discrete analogs of the problems. The talk is designed to be accessible to a wide audience.

(K. Soundararajan): In joint work with Robert Lemke Oliver, we discovered a curious phenomenon about consecutive primes: the remainders of consecutive primes modulo q do not like to be the same. I will show striking numerical results exhibiting this phenomenon, and give a conjectural explanation which matches the data very well.

(K. Mischaikow): The mathematical goal of the theory of dynamical systems is to understand the typical dynamics of typical systems and tremendous progress towards achieving this goal has been made over the last half century. However, it is worth remembering that the origins of this goal lie in our inability, in general, to find analytic solutions to nonlinear differential equations. In contrast, outside of mathematics most users of differential equations are interested in specific solutions to specific systems and thus deal with differential equations via numerical simulation.
In this talk I will describe efforts to redo, in a coherent manner, the proofs of basic theorems of ordinary differential equations in such a way that users can efficiently and rigorously determine if the results obtained numerically actually correspond to true solutions. I will also try to point out open problems. The methods are elementary as the necessary prerequisite is the contraction mapping theorem.

(G. Folland): The operations of translation (T_a f(t) = f(t+a)) and modulation (M_b f(t) = e^{2π ibt} f(t)) are basic ingredients of the harmonic analysis of signal processing and related areas. The group of operators on L²(R) generated by a single translation T_a and a single modulation M_b forms a unitary representation of the so-called discrete Heisenberg group, and it is of interest to know how it decomposes into irreducible representations. When ab is rational, finding the answer is a nice exercise in the use of some well-known techniques. But when ab is irrational, this representation provides an accessible and concrete illustration of several "exotic" pathological phenomena in unitary representation theory. We shall review the relevant concepts of representation theory and then discuss both aspects of the problem.

(C. Muscalu): The plan of the talk is to describe a bridge which connects in a natural way two mathematical worlds at the first glance far away from each other: the KdV equation and the absolute Galois group. The pillars of this bridge are given by the analytical objects from the title. The presentation will be "elementary", one does not need to know what is the KdV equation or the absolute Galois group.

(Y. Ruan): Given quasi-homogeneous polynomial W, we can study it in two different areas of mathematics. Namely, we can set W=0 to define a hypersurface X_W of Weight projective space or we can compute its Jacobian ring C[x₁, …, x_n]/∂ W. The later is the subject of singularity theory or Landau-Ginzburg model. An old theorem said that the middle cohomology of X_W can be computed using Jacobian ring. Motivated by physics, we can attach a range of invariants to X_W as well as the Landau-Ginzburg side of W. The effort to connect two subject leads to Landau-Ginzburg/Calabi-Yau correspondence, a famous duality from physics. In the talk, I will survey some of developments about this duality.

(L. DeMarco): In this talk, I will present some connections between recent research in dynamical systems and the classical theory of elliptic curves and rational points. The main goal is to explain the role of dynamical stability and bifurcations in deducing arithmetic finiteness statements. I will focus on three examples: (1) the theorem of Mordell and Weil from the 1920s, presented from a dynamical point of view; (2) a recent result of Masser and Zannier about torsion points on elliptic curves, and (3) features of the Mandelbrot set.

(J. Hom): One way to construct new 3-manifolds is by surgery on a knot in the 3-sphere; that is, we remove a neighborhood of a knot, and reglue it in a different way. What 3-manifolds can be obtained in this manner? We provide obstructions using the Heegaard Floer homology package of Ozsvath and Szabo. This is joint work with Cagri Karakurt and Tye Lidman.

(F. Rodriguez Hertz): In recent years several new advances in the theory of lattice actions have been made. In this talk I will present some of the key ingredients to these advances. I plan to keep the talk at an elementary level so only some basic notions of measure theory and differentiation on manifolds should be needed.

(T. Lam): I will talk about a mirror theorem for flag varieties, which can be formulated as an isomorphism between two D-modules: a quantum connection and an exponential Gauss-Manin connection associated to a Landau-Ginzburg model. In the case of a minuscule flag variety (for example, a Grassmannian), we establish this isomorphism using an instance of the ramified geometric Langlands correspondence, by recognizing the quantum or A-model side as "Galois" and the Landau-Ginzburg or B-model side as "Automorphic". This is joint work with Nicolas Templier.

(J.M. Bismut): In this series of three lectures, I will explain the theory of the hypoelliptic Laplacian. If X is a compact Riemannian manifold, and if X is the total space of its tangent bundle, there is a canonical interpolation between the classical Laplacian of X and the generator of the geodesic flow by a family of hypoelliptic operators LXb| b>0 acting on X. This interpolation extends to all the classical geometric Laplacians. There is a natural dynamical system counterpart, which interpolates between Brownian motion and the geodesic flow.

The hypoelliptic deformation preserves certain spectral invariants, such as the Ray-Singer torsion, the holomorphic torsion and the eta invariants. In the case of locally symmetric spaces, the spectrum of the original Laplacian remains rigidly embedded in the spectrum of the deformation. This property has been used in the context of Selberg's trace formula. Another application of the hypoelliptic Laplacian is in complex Hermitian geometry, where the extra degrees of freedom provided by the hypoelliptic deformation can be used to solve a question which is unsolvable in the elliptic world.

In the first lecture, 'Hypoelliptic Laplacian and analytic torsion', I will give the structure of the hypoelliptic Laplacian, which is essentially a combination of the fibrewise harmonic oscillator and of the geodesic flow. I will also describe its construction in de Rham theory, and explain some properties of the hypoelliptic torsion.

In the second lecture 'Hypoelliptic Laplacian and the trace formula', I will concentrate on the case of symmetric spaces, and on applications to the evaluation of orbital integrals and to Selberg trace formula.

In the third (colloquium) lecture 'Hypoelliptic Laplacian, Brownian motion and the geodesic flow', I will explain the basic elements of the theory and emphasize its connections with dynamical systems.

(V. Toledano Laredo): Quantum groups were introduced in the mid-eighties by Drinfeld and Jimbo as the algebraic backbone of the Quantum Inverse Scattering Method of Statistical Mechanics. They were soon found to have a host of other applications: to low-dimensional topology, representation theory, and algebraic geometry to name a few. This talk will concentrate on one aspect of quantum groups, namely their uncanny ability to describe the monodromy of integrable systems of partial differential equations attached to semisimple Lie algebras. This phenomenon was originally discovered by Drinfeld and Kohno in the early 90s in connection with the Knizhnik-Zamolodchikov equations of Conformal Field Theory. More recently, I proved that quantum groups also describe the monodromy of the so-called Casimir equations of a semisimple Lie algebra g and, in recent joint work with Andrea Appel (USC) that this continues to hold for any symmetrisable Kac-Moody algebra g.

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