Upcoming Seminars & Events

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April 17, 2015
1:30pm - 2:30pm
Unlinked fixed points of Hamiltonian diffeomorphisms and a dynamical construction of spectral invariants

Hamiltonian spectral invariants have had many interesting and important applications in symplectic geometry. Inspired by Le Calvez’ theory of transverse foliations for dynamical systems of surfaces, we introduce a new dynamical invariant, denoted by N, for Hamiltonians on surfaces (except the sphere). We prove that, on the set of autonomous Hamiltonians, this invariant coincides with the classical spectral invariant. This is joint work with Vincent Humilière and Frédéric Le Roux.

Speaker: Sobhan Seyfaddini, MIT
Location:
IAS Room S-101
April 17, 2015
3:00pm - 4:00pm
On the Morse index in the min-max theory of minimal surfaces

The min-max theory, developed by F. Almgren and J. Pitts, is a variational theory aimed to construct unstable minimal surfaces in a closed manifold. Recently, the min-max theory has had seminal achievements lead by
F. Marques and A. Neves. However, the understanding of this theory is still very limited. In this talk, I will focus on the geometric properties of the min-max surface. In particular, I will discuss several recent progresses on
the Morse index of the min-max surfaces. 

Speaker: Xin Zhou, MIT
Location:
Fine Hall 314
April 17, 2015
4:15pm - 5:15pm
Direct methods of moving planes, moving spheres, and blowing-ups for the fractional Laplacian

Many conventional approaches on partial differential operators do not work on the nonlocal fractional operator. To overcome this difficulty arising from non-localness, Caffarelli and Silvestre introduce the extension method to reduced the problem into a local one in one higher dimensions, which has become a powerful tool in studying such nonlocal problems and has yielded  a series of fruitful results. However, due to technical restrictions, sometimes one needs to impose extra conditions when studying the extended problems in higher dimensions, and these conditions may not be necessary if we investigate the original nonlocal problems directly. In this talk, we will introduce direct methods of *moving planes*, *moving spheres*, and *blowing-up and re-scaling arguments* for the fractional Laplacian.  By an elementary approach, we will first show the key ingredients needed in the {\em method of moving planes} either in a bounded domain or in the whole space, such as {\em strong maximum principles for anti-symmetric functions}, {\em narrow region principles}, and {\em decay at infinity}. Then, using simple examples, semi-linear equations involving the fractional Laplacian,  we will illustrate how this new {\em method of moving planes} can be conveniently employed to obtain symmetry and non-existence of positive solutions, under much weaker conditions than in the previous literatures. We firmly believe that these ideas and approaches can be effectively applied to a wide range of nonlinear problems involving fractional Laplacians or other nonlocal operators. 

Speaker: Wenxiong Chen, Yeshiva University
Location:
Fine Hall 314
April 20, 2015
3:15pm - 4:30pm
Asymptotics of Chebyshev polynomials for general subsets of the real line

Given a compact subset, E in R, the Chebyshev polynomials are the unique minmizers of the sup norm over E among all degree n monic polynomials. I will describe some recent results of Christensen, Simon and inchenko on this classical subject. We settle a 45 year old conjecture of Widom on the large n pointwise asymptotics for the case of finite gap sets. We also extend upper bounds on the norm that Totik and Widom obtained for the finite gap case to positive measure Cantor sets. Our proof in this case is new and even in the finite gap case is simpler and more explicit bounds than the earlier work. 

Speaker: Barry Simon, Caltech
Location:
Fine Hall 314
April 20, 2015
4:30pm - 5:30pm
Efficient use of semidefinite programming for selection of rotamers in protein conformations and other applications

In this paper we study a semidefinite programming relaxation of the (NP-hard) side chain positioning problem. We show that the Slater constraint qualification (strict feasibility) fails for the SDP relaxation. We then show the advantages of using facial reduction to regularize the SDP. In fact, after applying facial reduction, we have a smaller problem that is more stable both in theory and in practice. We include a discussion of the background of SDP relaxations and exploiting pecial structure in various applications such as sensor network localization, SNL.  Work with: Forbes Burkowski (University of Waterloo) Yuen-Lam Cheung Voronin (University of Colorado, Boulder)

Speaker: Henry Wolkowicz, University of Waterloo
Location:
Fine Hall 214
April 21, 2015
4:30pm - 5:30pm
From molecular dynamics to kinetic theory and hydrodynamics

Please note that this colloquium talk is on Tuesday, not Wednesday.

Speaker: Laure Saint-Raymond, École Normale Supérieure
Location:
Fine Hall 314
April 21, 2015
4:30pm - 5:50pm
Singular Eigenvalue Perturbation Theory

Eigenvalue Perturbation Theory is central to the theory of nonrelativistic quantum mechanics going back to Schrodinger's first papers. This lecture will review what is known about the eigenvalues in physical situations where one doesn't have simple convergence to a new isolated eigenvalue. Included are the anharmonic oscillator and Zeeman effect (divergent series and summability), autoionizing states in atoms (complex scaling and resonances), Stark effect (exponentially small resonances) and double wells (instantons).

Speaker: Barry Simon, Caltech
Location:
Jadwin Hall 343
April 22, 2015
12:15pm - 5:30pm
Celebrating the work of Prof. Edward Nelson (1932-2014)

In place of the regular departmental colloquium and pre-colloquium presentations, we shall have three talks focused on different aspects of Prof. Edward Nelson's work:

12:15 - 1:15 (Fine Hall Common Room, 3rd fl.)
"Ed Nelson's Work in Analysis, Especially Related to Quantum Field Theory"
Barry Simon, California Institute of Technology

4:30 - 5:30 pm (Fine Hall 314)

"Questioning the Infinite"
Greg Lawler, University of Chicago
"Ed Nelson's stochatic mechanics"
Eric Carlen, Rutgers University
Presentations moderated by professor emeritus Simon Kochen.

Reminiscences will be held in the Common Room after the talks.
The family has arranged for a eucharist, with special intention for the memory of Edward Nelson, which will be celebrated on April 21 at 5:15 pm in the Marquand Transept of the University Chapel.  All are welcome to attend.

Speaker: ,
April 23, 2015
2:00pm - 3:30pm
Rigidity phenomena in random point sets and applications

In several naturally occurring (infinite) point processes, the number the points inside a finite domain can be determined, almost surely, by the point configuration outside the domain. There are also other processes where such ''rigidity'' extends also to a number of moments of the mass distribution. The talk will focus on point processes with such curious "rigidity" phenomena, and their implications. We will also talk about applications to stochastic geometry and some questions in harmonic analysis. 

Speaker: Subhro Ghosh, Princeton University
Location:
Fine Hall 601
April 23, 2015
3:00pm - 4:00pm
Incidences between points and lines and extremal configuration of lines in Euclidean spaces

In 1983, Szemer\'edi and Trotter proved a tight bound for the number of incidences between points and lines in the plane. Ever since then, incidence geometry has been a very active research area, bridging between computer science and mathematics, with many connections to diverse topics, from range searching algorithms to the Kakeya problem. Over 25 years later, Guth and Katz proved a tight incidence bound for points and lines in three dimensions. Their proof introduced methods from advanced algebra and especially from algebraic geometry which were not used in combinatorics before. This enabled Guth and Katz to (almost) settle the Erd\"os distinct distances problem - a problem which stubbornly stood open for over 60 years, despite very brave attempts to solve it. The work of Guth and Katz has given significant added momentum to incidence geometry, making many problems, deemed hopeless before the breakthrough, amenable to the new techniques. In this talk I will present the area of incidence geometry, before and after, highlighting the basics of the new ``algebraic'' approach, and will also present very recent results, joint with Micha Sharir, among which we studied incidences between points and lines in four dimensions, incidences between points and lines that lie on special surfaces and other related questions. We will also give a variety of interesting open related questions.

Speaker: Noam Solomon, Hebrew University
Location:
Fine Hall 224
April 23, 2015
4:30pm - 5:30pm
TBA - Jonathan Hanselman
Speaker: Jonathan Hanselman, Columbia University
Location:
Fine Hall 314
April 23, 2015
4:30pm - 5:30pm
Extensions of the Gross-Zagier formula

I will first discuss the general conjectural picture relating algebraic cycles to L-functions and some extensions of the Gross-Zagier formula involving p-adic L-functions. This leads naturally to the question of constructing algebraic cycles corresponding to the vanishing of certain Rankin-Selberg L-functions at the center of symmetry. Finally, I will outline some new constructions of such cycles, based on work in progress with A. Ichino.

Speaker: Kartik Prasanna, University of Michigan
Location:
IAS Room S-101
April 24, 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
April 24, 2015
3:00pm - 4:00pm
A priori estimates for semistable solutions of semilinear elliptic equations

In this talk we will discuss semistable solutions of the boundary value problem $Lu+f(u)=0$ in $\Omega$ and $u=0$ on $\partial\Omega$, where $Lu:=\partial_i(a^{ij}u_j)$ is uniformly elliptic. By semistability we mean that the lowest Dirichlet eigenvalue of the linearized operator at u is nonnegative. The basic problem (which has a long history) is to obtain a priori $L^{\infty}$ bounds on a solution under minimal assumptions on $f(t)$.  A basic and standard assumption is that $u>0$ in $\Omega$  and $f\in C^2$ is positive, nondecreasing, and superlinear at infinity, i.e. $f(0)>0$, $f' \geq 0$ and $f(t)/t$ tends to infinity as $t$ tends to infinity. For radially symmetric solutions,  an $L^{\infty}$ bound for $u$ is known for $n\leq 9$. On the other hand there exists unbounded semistable solutions when $n\geq 10$ for $f(u)=e^u$. This problem, like many other semilinear elliptic problems studied in recent years, seems to be related to minimal surface stability but this still remains mysterious. 

Speaker: Joel Spruck, Johns Hopkins University
Location:
Fine Hall 314
April 27, 2015
3:15pm - 4:30pm
TBA - Phil Gressman
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
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

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