# Seminars & Events for 2011-2012

##### Rigidity and Origami

Cauchy's famous rigidity theorem for 3D convex polyhedra has been extended in various directions by Dehn, Weyl, A.D.Alexandrov, Gluck and Connelly. These results imply that a disk-like polyhedral surface with simplicial faces is, generically, flexible, if the boundary has at least 4 vertices. What about surfaces with rigid but not necessarily simplicial faces?

##### Applications of Multilinear Restriction and Restriction Estimates

The restriction problem is an important open problem in harmonic analysis. The 2-dimensional case was proven in the 1970's. The case of three (or more) dimensions remains open, with interesting partial results. In 2005, Bennett, Carbery, and Tao proved a ``multilinear" restriction estimate.

##### A Random Walk on Image Patches

Algorithms that analyze patches extracted from time series or images have led to state-of-the art techniques for classification, denoising, and the study of nonlinear dynamics. In the first part of the talk we describe two examples of such algorithms: a novel method to estimate the arrival-times of seismic waves from a seismogram, and a new patch-based method to denoise images.

##### Geometrically Characterizing Representation Type of Finite-dimensional Algebras

Given a finite-dimensional algebra $A$, the set of $A$-modules of a fixed dimension d can be viewed as a variety. This variety carries a group action whose orbits correspond to isomorphism classes of $A$-modules. A natural problem is to characterize various properties of an algebra $A$ in terms of its module varieties.

##### On Measures Invariant Under Diagonalizable Groups on Quotients of Semi-simple Groups

Actions of diagonalizable algebraic groups (which are referred to as tori in the theory of algebraic groups, though in cases of interest to us are not compact) on arithmetic quotient spaces play an important role in many number theoretic and other applications. I will present a joint result with Einsiedler (extending earlier work with A.

##### Effect of Emergent Distinguishability of Particles in a non-Equilibrium Chaotic System

We consider the behavior of classical particles which evolution consists of free motion interrupted by binary collisions. The fluid of hard balls and the dilute gas with arbitrary short-range interactions are treated, where the total number of particles is moderate (say, five particles).

##### Polyhedral Products, Toric Manifolds, and Twisted Cohomology

I will discuss the cohomology with coefficients in rank one local systems for various polyhedral products, including real Davis-Januszkiewicz spaces and toric complexes. As one application (joint work with Alvise Trevisan), I will show how to determine the Betti numbers and the cup products of real, quasi-toric manifolds.

##### Intersections of Quadrics: 25 years later

Consider $F:\R^n\rightarrow\mathbb{R}^2$ given by two quadratic forms and let $V=F^{-1}(0)$ and $Z=V\cap S^{n-1}$. In January 1984 I began to study the topology of the generic $Z$ when the quadratic forms are simultaneously diagonalizable. By the end of the year I had an answer, but only around 1986-87 I wrote the details of a proof that left out some cases.

##### Modularity Lifting in Non-Regular Weight

Modularity lifting theorems were introduced by Taylor and Wiles and formed a key part of the proof of Fermat's Last Theorem. Their method has been generalized successfully by a number of authors but always with the restriction that the Galois representations and automorphic representations in question have regular weight. I will describe a method to overcome this restriction in certain cases.

##### Loss of Compactness and Bubbling in the Space of Complete Minimal Surfaces in Hyperbloc Space

We consider the space of complete minimal surfaces in $H^3$ with a (free) boundary at infinity. We explain how the Willmore energy is a natural functional on this space. We study the possible loss of compactness in the space of such surfaces with energy bounded above.

##### Homotopy Theory and Toric Spaces

**Suyoung Choi** of Ajou University, Korea - Toric rigidity of simple polytopes and moment-angle manifolds

##### Well-Posedness and Finite-Time Blowup for the Zakharov System on Two-Dimensional Torus

We consider the Zakharov system on two-dimensional torus. First, we show the local well-posedness of the Cauchy problem in the energy space by a standard iteration argument using the $X^{s,b}$ norms. Our result does not depend on the period of torus.

##### Dimension Reduction, Coarse-Graining and Data Assimilation in High-Dimensional Dynamical Systems

Modern computing technologies, such as massively parallel simulation, special-purpose high-performance computers, and high-performance GPUs permit to simulate complex high-dimensional dynamical systems and generate time-series in amounts too large to be grasped by traditional “look and see” analyses.

##### Low Density Limit of BCS Theory and Bose-Einstein Condensation of Fermion Pairs

We consider the low density limit of a Fermi gas in the BCS approximation. We show that if the interaction potential allows for a two-particle bound state, the system at zero temperature is well approximated by the Gross-Pitaevskii functional, describing a Bose-Einstein condensate of fermion pairs. This is joint work with Robert Seiringer.

##### Braid Group Techniques for fundamental groups of surfaces, The K3 example

##### On a Hidden Symmetry of Simple Harmonic Oscillators

Since the original 1926 Schroedinger's paper, there was a misconception that the “simple” harmonic oscillator can be solved only by the separation of variables, which results in a traditional “static” electron density distribution.

##### On the L2 Bounded Curvature Conjecture in General Relativity

In order to control locally a space-time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The L2 bounded curvature conjecture roughly asserts that one should only need L2 bound on the curvature tensor on a given space-like hypersuface.

##### Homology of Clique Complexes and an Application to Algebraic Geometry

To any graph $G$ one can associate a simplicial complex $C(G)$, called the clique complex of $G$, whose simplices are in one-to-one correspondence with the complete subgraphs of $G$. It is thus natural to study the homology groups of $C(G)$ in connection to properties of the graph $G$.

##### Stable and Unstable Properties of Real Johnson-Wilson Spectra

I will try to describe the properties of certain spectra known as real Johnson-Wilson spectra, which are obtained as fixed points of involutions on the usual Johnson Wilson spectra. These spectra, that go by the symbol $ER(n)$, have several intriguing properties. For example, they are periodic and they support a self map whose cofiber is the Johnson Wilson spectrum $E(n)$.

##### The Tamagawa Number Formula Via Chiral Homology (joint with J. Lurie)

Let $X$ a curve over $F_q$ and $G$ a semi-simple simply-connected group. The initial observation is that the conjecture of Weil's which says that the volume of the adelic quotient of $G$ with respect to the Tamagawa measure equals 1, is equivalent to the Atiyah-Bott formula for the cohomology of the moduli space $Bun_G(X)$ of principal G-bundles on $X$.