Upcoming Seminars & Events

Subscribe to Seminars & Events
April 27, 2015
3:15pm - 4:30pm
Geometric convolutions and Fourier restriction beyond curves and hypersurfaces

I will present recent results relating to two problems in Fourier analysis, L^p-improving properties of convolutions with singular measures and the Fourier restriction problem, both of which deal with the analysis of operators associated to submanifolds of Euclidean space. In both cases the theory is much more well-developed for curves and hypersurfaces than it is for submanifolds of intermediate dimension. This relative lack of
positive results is due in part to the problem that the Phong-Stein nonvanishing rotational curvature condition is frequently impossible to satisfy for surprisingly deep algebraic reasons. I will focus primarily on the case of 2-surfaces in R^5, which does not fit nicely into previously-existing combinatorial strategies, and will present a new approach with the potential to apply to a broad range of new cases. 

Speaker: Phil Gressman, UPenn
Location:
Fine Hall 314
April 27, 2015
4:30pm - 5:30pm
Fast algorithms for electronic structure analysis

Kohn-Sham density functional theory (KSDFT) is the most widely used electronic structure theory for molecules and condensed matter systems. For a system with N electrons, the standard method for solving KSDFT requires solving N eigenvectors for an O(N) * O(N) Kohn-Sham  Hamiltonian matrix.  The computational cost for such procedure is expensive and scales as O(N^3), and limits routine KSDFT calculations to hundreds of atoms.  In recent years, we have developed an alternative procedure called the  pole expansion and selected inversion (PEXSI) method [1-2].  The PEXSI method solves KSDFT without solving any eigenvalue and eigenvector, and directly evaluates physical quantities including electron density, energy, atomic force, density of states, and local density of states. The overall algorithm scales as at most O(N^2) for all materials including insulators, semiconductors and the difficult metallic systems.  The PEXSI method can be efficiently parallelized over 10,000 - 100,000 processors on high performance machines.  It has been integrated into standard electronic structure software packages such as SIESTA for ab initio materials simulation over 20,000 atoms [3].  Recently we have been able to use PEXSI to study electronic structure of large scale graphene nanoflakes [4] and  phosphorene nanoribbons [5] to unprecedented scale (more than 10,000 atoms).

[1] L. Lin, J. Lu, L. Ying, R. Car and W. E, Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems, Commun. Math. Sci. 7, 755, 2009

[2] L. Lin, M. Chen, C. Yang and L. He, Accelerating atomic orbital-based electronic structure calculation via pole Expansion and selected inversion, J. Phys. Condens. Matter 25, 295501, 2013

[3] L. Lin, A. Garcia, G. Huhs and C. Yang, SIESTA-PEXSI: Massively parallel method for efficient and accurate ab initio materials simulation without matrix diagonalization, J. Phys. Condens. Matter 26, 305503, 2014

[4] W. Hu, L. Lin, C. Yang and J. Yang, Electronic structure of large-scale graphene nanoflakes, J. Chem. Phys. 141, 214704, 2014

[5] W. Hu, L. Lin and C. Yang, Edge reconstruction in armchair phosphorene nanoribbons revealed by discontinuous Galerkin density functional theory, submitted

Speaker: Lin Lin, UC-Berkeley
Location:
Fine Hall 214
April 28, 2015
4:30pm - 5:30pm
A class of gapped Hamiltonians on quantum spin chains and its classification

The MPS (matrix product state) formalism gives a recipe to construct Hamiltonians in quantum spin chains from $n$-tuples of $k\times k$- matrices. This $n$-tuple defines a completely positive map and the existence of the uniform spectral gap of the Hamiltonian is related to the spectral property of the associated CP map. I would like to talk about a classification problem of this class of Hamiltonians. Through the relation between Hamiltonians and CP maps, the problem is reduced to the question of path connectedness of a class of CP maps.

Speaker: Yoshiko Ogata, University of Tokyo
Location:
Jadwin Hall 343
April 29, 2015
1:30pm - 2:30pm
Another (q,t) world

A well studied (q,t)-analogue of symmetric functions are the Macdonald polynomials. In this talk I will survey another (q,t)-analogue, where q is a prime power from a finite field and t is an indeterminate. Analogues of facts about the symmetric group S_n are given for GL_n(F_q), including (1) counting factorizations of certain elements into reflections, (2) combinatorial properties of appropriate (q,t)-binomial coefficients, (3) Hilbert series for invariants on polynomial rings. Some new conjectured explicit Hilbert series of rings of invariants over finite fields are given. This is joint work with Joel Lewis and Vic Reiner. 

Speaker: Dennis Stanton , University of Minnesota
Location:
Fine Hall 214
April 29, 2015
3:00pm - 4:00pm
Tails of Random Projections and the Atypicality of Cramer’s Theorem

In recent years, there has been much interest in the interplay between geometry and probability in high-dimensional spaces.  One striking result that has been established is a central limit theorem for random projections of random variables that are uniformly distributed in high-dimensional convex sets.   It is therefore natural to ask if such random projections also exhibit other properties satisfied by sums of iid random variables, such as large deviation principles.   As a step in this direction, we establish  (quenched and annealed) large deviation principles for random projections of sequences of random vectors that are uniformly distributed on $l^p$ balls in $n$-dimensional Euclidean space.   We also prove several interesting consequences of these principles, including the perhaps surprising fact that the well known Cramer’s theorem, which describes the large deviations of sums of iid random variables, is atypical.  Such questions, besides being of intrinsic interest, are also of relevance to statistics and data analysis.  This talk is based on joint work with Nina
Gantert and Steven Kim.

Speaker: Kavita Ramanan, Brown
Location:
Fine Hall 214
April 29, 2015
4:30pm - 5:30pm
Zeta(3) in arithmetic and geometry

Euler proved in 1735 that zeta(2) = $\pi^2 /6$, and also computed the special values of zeta(n) at all positive even integers. Yet it took almost another 250 years for Apery proved that zeta(3) was irrational. In this talk, we shall talk about zeta(3) as well as its p-adic version, and the connection of these numbers to both arithmetic and geometry.

Speaker: Frank Calgary, Northwestern
Location:
Fine Hall 314
April 30, 2015
2:00pm - 3:30pm
Higher Rank Orbit Closures in Genus 3

The moduli space of translation surfaces is stratified by the orders of the zeros of Abelian differentials. We classify GL^+(2,R) orbit closures in the strata of translation surfaces in genus 3 with at most two zeros, with the property that they have rank 2 (in the sense of Alex Wright).   This is joint work with Duc-Manh Nguyen and Alex Wright. 

Speaker: David Aulicino, University of Chicago
Location:
Fine Hall 601
April 30, 2015
3:00pm - 4:00pm
TBA - Browder

Please note special time.

Speaker: William Browder, Princeton University
Location:
Fine Hall 314
April 30, 2015
3:00pm - 4:00pm
Stability results in additive combinatorics and graph theory

We will talk about stability results in extremal combinatorics. Along the way, we solve a question of Erdos and Sarkozy on sumsets of integers avoiding perfect squares, and reprove the Posa-Seymour conjecture on hamiltonian cycles for large graphs. Joint work with A.Jamshed, A. Khalfalah, and E. Szemeredi.

Speaker: Simao Herdade, Rutgers University
Location:
Fine Hall 224
April 30, 2015
4:30pm - 6:00pm
Schauder estimates for a class of non-local parabolic equations and applications

Regularity theory for non-local elliptic and parabolic equations has undergone rapid development in recent years and reached applications in several models of fluid dynamics. In this talk we will present new Schauder estimates for parabolic equations with general kernels (lacking any traditional symmetry of evenness assumptions) and show their use in studying well-posedness of an Euler-like scalar model.

Speaker: Roman Shvydkoy, University of Illinois at Chicago
Location:
Fine Hall 322
April 30, 2015
4:30pm - 5:30pm
TBA - Lihe Wang

Please note different day (Thursday) and time (4:30). 

Speaker: Lihe Wang, University of Iowa
Location:
TBD
April 30, 2015
4:30pm - 5:30pm
The total surgery obstruction

The 1960's Browder-Novikov-Sullivan-Wall high-dimensional surgery theory for deciding if an n-dimensional  Poincare duality space X is homotopy equivalent to an n-dimensional topological manifold has two obstructions. There is a primary topological K-theory obstruction to the existence of a topological bundle structure on the Spivak spherical fibration \nu{X \subset S^{n+k}}. There is also a secondary algebraic L-theory surgery obstruction in the Wall group L_n(Z[\pi_1(X)]) of quadratic forms over the fundamental group ring Z[\pi_1(X)], which depends on the resolution of the primary obstruction. In 1979 (in Princeton) I united these two obstructions in a single "total surgery obstruction" s(X)  \in S_n(X). This homotopy invariant lives  in an even more generalized Witt group S_n(X) defined for any space X, such that s(X)=0 if (and for n>4 only if) X is homotopy equivalent to a manifold. The object of the talk is to describe the construction of s(X) from the stable homotopy class \rho \in \pi_{n+k}(T(\nu_X)) of the Pontrjagin-Thom map \rho:S^{n+k} \to T(\nu_X) to the Thom space. The description involves the algebraic surgery theory analogue of the homotopy groups of a Thom space, for any spherical fibration. 

Speaker: Andrew Ranicki, University of Edinburgh
Location:
Fine Hall 314
April 30, 2015
4:30pm - 5:30pm
The total surgery obstruction

Please note special time.   The 1960's Browder-Novikov-Sullivan-Wall high-dimensional surgery theory for deciding if an n-dimensional  Poincare duality space X is homotopy equivalent to an n-dimensional topological manifold has two obstructions. There is a primary topological K-theory obstruction to the existence of a topological bundle structure on the Spivak spherical fibration \nu{X \subset S^{n+k}}. There is also a secondary algebraic L-theory surgery obstruction in the Wall group L_n(Z[\pi_1(X)]) of quadratic forms over the fundamental group ring Z[\pi_1(X)], which depends on the resolution of the primary obstruction. In 1979 (in Princeton) I united these two obstructions in a single "total surgery obstruction" s(X)  \in S_n(X). This homotopy invariant lives  in an even more generalized Witt group S_n(X) defined for any space X, such that s(X)=0 if (and for n>4 only if) X is homotopy equivalent to a manifold. The object of the talk is to describe the construction of s(X) from the stable homotopy class \rho \in \pi_{n+k}(T(\nu_X)) of the Pontrjagin-Thom map \rho:S^{n+k} \to T(\nu_X) to the Thom space. The description involves the algebraic surgery theory analogue of the homotopy groups of a Thom space, for any spherical fibration. 

Speaker: Andrew Ranicki, University of Edinburgh
Location:
Fine Hall 314
April 30, 2015
4:30pm - 5:30pm
Uniform bounds for the number of rational points on curves of small Mordell-Weil rank

We show that there is a bound N(d,g,r) for the number of K-rational points on hyperelliptic curves C of genus g when the degree of the number field K is d and the Mordell-Weil rank r of the Jacobian of C is at most g-3. The proof uses an extension of the method of Chabauty-Coleman, based on p-adic integration on (p-adic) disks and annuli covering the p-adic points of the curve. We also deduce a uniform version of the result (due to Poonen and the speaker) that "most" hyperelliptic curves of odd degree over Q have only one rational point, where "uniform" refers to families of curves defined by congruence conditions. 

Speaker: Michael Stoll, Bayreuth University
Location:
Fine Hall 214
May 1, 2015
1:30pm - 2:30pm
Periodic Symplectic Cohomologies

Periodic cyclic homology group associated to a mixed complex was introduced by Goodwillie. In this talk, I will explain how to apply this construction to the symplectic cochain complex of a Liouville domain and obtain two periodic symplectic cohomology theories, which are called periodic symplectic cohomology and finitely supported periodic symplectic cohomology, respectively. The main result is that there is a localization theorem for the finitely supported periodic symplectic cohomology.

Speaker: Jingyu Zhao, Columbia University
Location:
IAS Room S-101
May 1, 2015
2:00pm - 3:30pm
Randomization of initial data and global well-posedness

Please note special day (Friday) and location.  In this talk, we will present techniques of randomization to get global well-posedness on sets of initial data with non zero measure, the measure being related to the randomization. We will in particular present almost sure Strichartz estimates and explain how an invariant measure can be used as an invariant of the equation such as the mass or the energy. We will apply these techniques to the radial cubic nonlinear wave equation. 

Speaker: Anne-Sophie de Suzzoni, Universit´e Paris 13, CNRS
Location:
TBD
May 1, 2015
3:00pm - 4:00pm
TBA - Jeff Viaclovsky
Speaker: Jeff Viaclovsky, University of Wisconsin
Location:
Fine Hall 314
May 4, 2015
3:15pm - 4:30pm
TBA - Michael I. Weinstein
Speaker: Michael I. Weinstein, Columbia University
Location:
Fine Hall 314
May 7, 2015
2:00pm - 3:30pm
Outer billiards and the plaid model

Outer billiards is a billiards-like dynamical system which is defined on the outside of a convex shape in the plane. Even for simple shapes, like kites (bilaterally symmetric quadrilaterals), the orbits have an intricate fractal-like structure. I'll explain a combinatorial model, which I call the plaid model, which gives a precise picture of that the outer billiards orbits look like on kites. This model explains, among other things, why the orbits have a (coarsely) self-similar structure in case the parameter associated to the kite is a quadratic irrational. 

Speaker: Richard Schwartz, Brown University
Location:
Fine Hall 601
May 7, 2015
3:00pm - 4:00pm
Moduli space actions and cyclic operads

I will describe a combinatorial dg model for the homology of the open moduli spaces of punctured Riemann spheres. This model acts (in the operadic sense) on chain complexes computing eg string topology, cyclic cohomology of a Frobenius algebra, or equivariant cohomology of an S^1 space.  We thus find homology operations old and new, as well as a homotopy invariant description of the chain level structures which produce these operations. 

Speaker: Ben Ward , Simons Center, Stony Brook
Location:
Fine Hall 314

Pages