In this talk I will discuss smoothing estimates for nonlinear dispersive equations and systems of equations with periodic boundary conditions. The smoothing and decay properties of linear dispersive equations are well known. We examine how these properties carry over into their nonlinear perturbations via the smoothing mechanism of time averaging. We highlight these phenomena by studying almost everywhere convergence of the solution to the initial data, growth of Sobolev norms, dispersive quantization of solutions, the structure of the attracting sets for dissipative dispersive PDE and unconditional uniqueness of the solutions. This is joint work with B. Erdogan.

Let G be a finite group. Artin's theorem says that we can recover the complex representation ring of G from the representations of the cyclic subgroups of G up to torsion, or additive nilpotence. Quillen's F-isomorphism theorem says that we can recover the mod-p cohomology of G from the mod-p cohomology of the elementary abelian p-subgroups of G up to multiplicative nilpotence. This analysis fits into Dress's framework for induction/restriction theory for Mackey functors. These particular Mackey functors appear as the homotopy groups of G-spectra and it is relatively straightforward to lift Dress's framework to G-spectra. Here the central question is: can we recover a G-spectrum from it's underlying H-spectra as H varies over some fixed (small) family of subgroups of G. There are well-known characterizations of when such a result holds, but by asking for a stronger condition we can deduce induction and restriction results for the homotopy groups as well. In particular we can prove generalizations of the theorems of Artin and Quillen. Our theory applies to genuine equivariant complex and real K-theory (extending Artin's theorem and a result of Fausk), and the Borel equivariant cohomology theories associated to mod-p cohomology (extending the result of Quillen), integral cohomology (extending a result of Carlson), complex oriented theories (extending a result of Hopkins-Kuhn-Ravenel), ko, the many variants of topological modular forms, L_n-local spectra, and classical cobordism theories. Along the way, we show that the class of G-spectra such that our strong induction/restriction theorems hold is a triangulated subcategory closed under retracts and (co)tensoring with arbitrary G-spectra. This is joint work with Akhil Mathew and Niko Naumann.

Modern imaging data are often composed of several geometrically distinct constituents. For instance, neurobiological images could consist of a superposition of spines (pointlike objects) and dendrites (curvelike objects) of a neuron. A neurobiologist might then seek to extract both components to analyze their structure separately for the study of Alzheimer specific characteristics. However, this task seems impossible, since there are two unknowns for every datum. Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that high-dimensional signals, which allow a sparse representation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements, by using efficient algorithms. Utilizing the methodology of Compressed Sensing, the geometric separation problem can indeed be solved both numerically and theoretically. For the separation of point- and curvelike objects, we choose a deliberately overcomplete representation system made of wavelets (suited to pointlike structures) and shearlets (suited to curvelike structures). The decomposition principle is to minimize the norm of the representation coefficients. Our theoretical results, which are based on microlocal analysis considerations, show that at all sufficiently fine scales, nearly-perfect separation is indeed achieved. This project was done in collaboration with David Donoho (Stanford University) and Wang-Q Lim (TU Berlin).

This talk will be an introduction to the Coherent-Constructible Correspondence (CCC) of Fang-Liu-Treumann-Zaslow, which says that understanding coherent sheaves on a toric variety is in a sense equivalent to understanding constructible sheaves with specified singular support on a (real) torus. We will take as a guiding example a classical result, Beilinson's theorem on the derived equivalence between projective spaces and certain quivers with relations. The CCC can be thought of as a kind of generalization of this to arbitrary toric varieties, and moreover gives us an appealing way of visualizing Beilinson's theorem. In the worst-case scenario I'll run out of time and the talk will merely be a friendly introduction to the singular support of constructible sheaves. In the best-case scenario I'll have extra time and talk about work in progress with Shende-Treumann-Zaslow relating the CCC to cluster structures on spaces of doubly-periodic monopoles.

We will discuss a variety of exotic Noetherian rings and their constructions.

An exposition of basic results regarding the Jacobian of a complex curve of genus g. This is the second talk in a series building the language necessary to describe the Geometric Langlands Conjecture for GL(1).

Neuronal networks are remarkable in their ability to perform a diversity of coherent tasks such as sensory processing, information transfer and storage. Discovering how information travels through the network over time, and how it is being processed will enable classification of functionality, infer network wiring, and model future dynamics. Typically, such networks are extremely challenging to study using traditional approaches since in on top of their complex structure they exhibit intricate time-dependent dynamics. My talk will focus on our recently developed methods that leverage sampled time-series data from a network. I will describe how to fuse dynamical system theory with data analysis (e.g., phase space analysis, model reduction, optimization and probabilistic graphical modeling) to achieve efficient classification, use it for recognition and solve inverse problems for recovery of network wiring. Furthermore, their combination enables predictive modeling of dynamic networks. I will describe the methodology and provide examples of real neurobiological systems for which the developed tools were applied. These include olfaction in moths, C Elegans worm nervous system, and sun-compass navigation in Monarch butterflies. The speaker is a candidate for the joint Math/Neuro position.

This is a continuation of last week's talk on super Riemann surfaces. I'll introduce NS and R punctures, describe the obstruction to splitting of super moduli space, and (if there's time) introduce integration on supermanifolds.

We study the long-time stability of shock-free solutions of hyperbolic systems of conservations laws, under arbitrarily large initial disturbance in . We use the relative entropy method, a robust tool which allows us to consider rough and large disturbances. We display practical examples in several space dimensions, for scalar equations as well as isentropic gas dynamics. For full gas dynamics, we use a trick due to G.-Q. Chen and coll., in which the estimate is made in terms of the relative mechanical energy instead of the relative mathematical entropy.

If M is a 2-manifold and f: M is a simple closed curve then Im(f) is the centerline of N, an annulus or Mbius band, imbedded in M. Sewing disks, D, along their boundary to (M - Int(N)) decomposes M into another 2-manifold, allowing a two-person game to be played.

Classical ergodic theory is well-developed for probability-preserving actions of amenable groups. However, for non-amenable groups many of the main objects of the theory behave differently, or simply have no obvious definition. One such is the classical Kolmogorov-Sinai entropy of an action, but a few years ago Bowen introduced a useful substitute: the sofic entropy, defined (roughly) as the exponential growth rate of certain spaces of `good models' for the original action. These spaces of models carry a natural Hamming metric, and in general this aspect of their structure is important for the theory. My talk will give a brief overview of these metric spaces of models, and discuss some more recent work which defines another new invariant of systems using a different geometric feature of these metric spaces.

Many first courses in number theory introduce the idea of quadratic reciprocity and prove it in a somewhat obscure way. This talk aims to show reciprocity in it's most natural setting and talk about how its extensions lead to important questions in modern number theory.

TBA: This will give a gentle introduction (assuming no background) into equivariant cohomology and more generally homotopy theory, leading to some understanding of constructions in Noel's Topology seminar.

I will explain my proof of some conjectures of Gross,Hacking,Keel and Kontsevich which imply the Full Fock Goncharov conjecture for the flag variety of SL_n, and in particular produce a canonical basis for each irreducible representation of SL_n, and a canonical procedure for producing explicit equations for (infinitely many differnt) bounded lattice polytopes whose integer points parameterize the basis.

Many important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shear layers in solutions of transport dominated equations. While the ability to reliably capture and sparsely represent anisotropic structures is obviously the more important the higher the number of spatial variables is, principal difficulties arise already in two spatial dimensions. Since it was shown that the well-known (isotropic) wavelet systems are not capable of efficiently approximating such anisotropic features, the need arose to introduce appropriate anisotropic representation systems. Among various suggestions, shearlets are the most widely used today. Main reasons for this are their optimal sparse approximation properties within a model situation in combination with their unified treatment of the continuum and digital realm, leading to faithful implementations. An additional advantage is the availability of stable compactly supported systems for high spatial localization. In this talk, we will provide an introduction to the anisotropic representation system of shearlets, in particular, compactly supported shearlets, present the main theoretical results, and discuss applications to imaging science and adaptive numerical solution of partial differential equations. Finally, we will present a general framework for systems with optimal sparse approximation properties with respect to anisotropic features, coined parabolic molecules, which in particular includes compactly supported shearlet systems.

This talk will survey a variety of algorithms related to the classical gossip algorithm, including some nonlinear variants.

This event is part of the Statistics Seminar Series. It cross-listed here as maybe of interest to attendees of the Random Structures seminar. Event details, including a schedule for lunch with the speaker, can be found here: http://piratepad.nl/ZxgxmquX9j

"Classical" invariant theory, as described by Hermann Weyl in his book, The Classical Groups, has seen very substantial progress since Weyl wrote. This talk will survey four stages of development in the theory, of which the fourth stage is currently unfolding. A key role is played by the Weyl algebra, which was introduced by Weyl in his discussions of quantum mechanics, but not used by him in invariant theory. The main ideas used are from representation theory. The talk will include a summary of relevant notions from representation theory.