Seminars & Events for 2008-2009

February 26, 2009
4:30pm - 6:30pm
Bounding sup-norms of cusp forms

Given an $L^2$-normalized cusp form $f$ on a modular curve $X_0(N)$, what can be said about pointwise bounds for $f$? For Hecke eigenforms, we will prove the first non-trivial bound in terms of the level $N$ as well as hybrid bounds in terms of the level and the Laplacian eigenvalue. Similar techniques work for functions on other spaces, e.g. quotients of quaternion algebras.

Speaker: V. Blomer, University of Toronto
Location:
IAS Room S-101
February 27, 2009
3:00pm - 5:00pm
Ricci flow on ALE spaces
Speaker: Xianzhe Dai, University of California, Santa Barbara
Location:
Fine Hall 314
March 2, 2009
2:00pm - 4:00pm
Trouble with a chain of stochastic oscillators

I will discuss some recent (but modest) results showing the existence and slow mixing of a stationary chain of Hamiltonian oscillators subject to a heat bath. Such systems are used as simple models of heat conduction or energy transfer. Though the unlimite goal might be seen to under stand the "fourier" like law in this setting, I will be less ambitious.

Speaker: Jonathan Mattingly, Duke University
Location:
Fine Hall 214
March 3, 2009
2:00pm - 4:00pm
What makes the ergodic theory of Markov chains in infinite dimensions different (and difficult)?

I will discuss how Markov chains in infinite dimensions generically have typically have properties which make their ergodic theory difficult. Such properties are very pathological in finite dimensions, but in some sense generic in infinite dimensions. I will draw examples from stochastically forced PDEs and stochastic delay equations.

Speaker: Jonathan Mattingly, Duke University
Location:
Fine Hall 401
March 3, 2009
4:30pm - 6:30pm
Algebraic surfaces and hyperbolic geometry

The intersection form on the group of line bundles on a complex algebraic surface always has signature $(1,n)$ for some $n$. So the automorphism group of an algebraic surface always acts on hyperbolic $n$-space.

Speaker: Burt Totaro, Cambridge University
Location:
Fine Hall 322
March 4, 2009
4:30pm - 6:30pm
Internal aggregation Models: From Diaconis-Fulton addition to a free boundary problem

Start with $n$ particles at each of $k$ points in the $d$-dimensional lattice, and let each particle perform simple random walk until it reaches an unoccupied site. The law of the resulting random set of occupied sites does not depend on the order in which the walks are performed, as shown by Diaconis and Fulton.

Speaker: Yuval Peres, University of California - Berkeley
Location:
Fine Hall 314
March 5, 2009
12:30pm - 2:30pm
Lie Groups: Decomposition and Exponentiation

A manifold with a smooth group structure is called a Lie group. Most of the information about Lie groups is captured by the tangent space at the identity and its Poisson bracket. The map relating these two structures is the exponential map (which in the compact case is the same as the geodesic exponential map).

Speaker: Kevin Wilson, Princeton University
Location:
Fine Hall 314
March 5, 2009
2:00pm - 4:00pm
Hénon Renormalization

The geometry of strongly dissipative infinite renormalizable Hénon maps of period doubling type is surprisingly different from its one-dimensional counterpart. There are universal geometrical properties. However, the Cantor attractor is not geometrically rigid. Typically, it doesn't have bounded geometry. The average Jacobian is a topological invariant of the global attractor.

Speaker: Marco Martens, SUNY at Stony Brook
Location:
Fine Hall 401
March 5, 2009
2:30pm - 4:30pm
New bounds on the size of Kakeya sets in finite fields

A Kakeya set is a set in $(F_q)^n$ (the $n$ dimensional vector space over a field of $q$ elements) which contains a line in every direction. In this talk I will present a recent result which gives a lower bound of $(q/2)^n$ on the size of such sets. This bound is tight to within a multiplicative factor of two from the known upper bounds.

Speaker: Zeev Dvir, IAS
Location:
Fine Hall 224
March 5, 2009
4:30pm - 6:30pm
Subgroup classification in $Out(F_n)$

We prove that for every subgroup $G$ of $Out(F_n)$, one of two alternatives holds: either there is a finite index subgroup $H<G$ and a nontrivial proper free factor $A$ of $F_n$ such that each element of $H$ fixes the conjugacy class of $A$; or there is an element $g\in G$ such that no nontrivial power of $g$ fixes the conjugacy class of any nontrivial proper free factor of $F_n$.

Speaker: Lee Mosher, Rutgers University, Newark
Location:
Fine Hall 314
March 5, 2009
4:30pm - 6:30pm
A "Relative" Langlands Program and Periods of Automorhic Forms

Motivated by the relative trace formula of Jacquet and experience on period integrals of automorphic forms, we take the first steps towards formulating a "relative" Langlands program, i.e. a set of conjectures on H-distinguished representations of a reductive group G (both locally and globally), where H is a spherical subgroup of G. We prove several results in this direction.

Speaker: Yiannis Sakellaridis, University of Toronto
Location:
Fine Hall 214
March 6, 2009
1:00pm - 3:00pm
A generalization of compact operators and its application to the existence of local minima without convexity

We will introduce a certain property for a continuous (non-linear) operator that allows for the existence of local minima for functionals when the derivative complies with such a condition, without the need to check either weak lower semicontinuity or convexity. It turns out that this property is a generalization of the standard compactness for a continuous, non-linear operator.

Speaker: Pablo Pedregal, Universidad de Castilla-La Mancha
Location:
Fine Hall 224
March 6, 2009
3:00pm - 5:00pm
Greatest lower bounds on the Ricci curvature of Fano manifolds

On Fano manifolds we study the supremum of the possible t such that there exists a metric in the first Chern class with Ricci curvature bounded below by t. For the projective plane blown up in one point we show that this supremum is 6/7.

Speaker: Gabor Szekelyhidi, Columbia University
Location:
Fine Hall 314
March 9, 2009
4:00pm - 6:00pm
Compressive Optical Imaging

Recent work in the emerging field of compressed sensing indicates that, when feasible, judicious selection of the type of image transformation induced by imaging systems may dramatically improve our ability to perform reconstruction, even when the number of measurements is small relative to the size and resolution of the final image.

Speaker: Rebecca Willett, Duke University
Location:
Fine Hall 214
March 10, 2009
4:30pm - 6:30pm
Compactified Jacobians and Abel maps for singular curves

We will discuss the problem of extending the construction of the classical Abel maps for smooth curves to the case of singular curves. The construction of degree-1 Abel maps will be shown, together with an approach for constructing higher degree Abel maps.

Speaker: Eduardo Esteves, Instituto Nacional de Matemática Pura e Aplicada, Brasil
Location:
Fine Hall 322
March 11, 2009
4:30pm - 6:30pm
Quantum Unique Ergodicity and Number Theory

A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds.

Speaker: K. Soundararajan, Stanford University
Location:
Fine Hall 314
March 12, 2009
12:30pm - 2:30pm
Morse theory

Morse theory gives the cell structure of a manifold in terms of the critical points of a 'random' real-valued function on this manifold. Besides being clever and pleasing to the eye, it has given us Bott periodicity, counts of geodesics, periodic orbits of dynamical systems, Heegard Floer homology, the foundations of Mirror Symmetry and many many more riches.

Speaker: Michael McBreen, Princeton University
Location:
Fine Hall 314
March 12, 2009
2:00pm - 4:00pm
Random walks with memory and statistical mechanics

This talk will review some results and conjectures about history dependent random walks. For example, edge reinforced random walk (ERRW) is a random walk which prefers to visit edges it has visited in the past. Diaconis showed that ERRW can be expressed as a random walk in a random environment. This environment is highly correlated and is described in terms of statistical mechanics.

Speaker: Thomas Spencer, IAS
Location:
Fine Hall 401
March 12, 2009
2:30pm - 4:30pm
Inverse Littlewood-Offord theory, Smooth Analysis and the Circular Law

A corner stone of the theory of random matrices is Wigner's semi-circle law, obtained in the 1950s, which asserts that (after a proper normalization) the limiting distribution of the spectra of a random hermitian matrix with iid (upper diagonal) entries follows the semi-circle law.

Speaker: Van Vu, Rutgers University
Location:
Fine Hall 224
March 12, 2009
4:30pm - 6:30pm
Congruence subgroup problem for mapping class groups

I will discuss the congruence subgroup problem for mapping class groups, a problem that generalizes the classical one for arithmetic groups. I will discuss an unpublished proof by W. Thurston for an affirmative answer to this problem for genus zero mapping class groups. Time permitting, I will discuss the current state of this problem.

Speaker: Ben McReynolds, University of Chicago
Location:
Fine Hall 314

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