Mathematics Department Colloquium


(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.
(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 quasihomogeneous 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[x_{1}, …, x_{n}]/∂ W. The later is the subject of singularity theory or LandauGinzburg 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 LandauGinzburg side of W. The effort to connect two subject leads to LandauGinzburg/CalabiYau 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 3manifolds is by surgery on a knot in the 3sphere; that is, we remove a neighborhood of a knot, and reglue it in a different way. What 3manifolds 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.
(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 L^{Xb}_{ 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 RaySinger 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 mideighties 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 lowdimensional 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 KnizhnikZamolodchikov equations of Conformal Field Theory. More recently, I proved that quantum groups also describe the monodromy of the socalled 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 KacMoody algebra g.
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