Spectra for non-self-adjoint operators and integrable dynamics

Speaker: 

Michael Hitrik

Institution: 

UCLA

Time: 

Thursday, November 8, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Non-self-adjoint operators appear in many settings, from kinetic theory 
and quantum mechanics to linearizations of equations of mathematical 
physics. The spectral analysis of such operators, while often notoriously 
difficult, reveals a wealth of new phenomena, compared with their 
self-adjoint counterparts. Spectra for non-self-adjoint operators display 
fascinating features, such as lattices of eigenvalues for operators of 
Kramers-Fokker-Planck type, say, and eigenvalues for operators with 
analytic coefficients in dimension one, concentrated to unions of curves 
in the complex spectral plane. In this talk, after a general introduction, 
we shall discuss spectra for non-self-adjoint perturbations of 
self-adjoint operators in dimension two, under the assumption that the 
classical flow of the unperturbed part is completely integrable.
The role played by the flow-invariant Lagrangian tori of the completely 
integrable system, both Diophantine and rational, in the spectral analysis 
of the non-self-adjoint operators will be described. In particular, we 
shall discuss the spectral contributions of rational tori, leading to 
eigenvalues having the form of the "legs in a spectral centipede". This 
talk is based on joint work with Johannes Sj\"ostrand.

Nonlinear detection of connections

Speaker: 

Gabriel Paternain

Institution: 

Cambridge, visiting U Washington

Time: 

Thursday, May 16, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

A connection is a geometric object that allows to parallel transport vectors along a curve in a domain. A natural question that often arises is whether one can recover a connection inside a domain from the knowledge of the parallel transport along a set of special curves running between boundary points of the domain. In this talk I will discuss this geometric inverse problem in various settings including Riemannian manifolds with boundary and Minkowski space. This problem is related to other inverse problems and is tackled with a range of techniques that I will explore during the talk.

Applied random matrix theory

Speaker: 

Joel Tropp

Institution: 

Caltech

Time: 

Thursday, January 31, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications. This talk is designed for a general audience in mathematics and related fields.

Gromov-Hausdorff limits of Kahler manifolds

Speaker: 

Gabor Szekelyhidi

Institution: 

University of Notre Dame

Time: 

Thursday, May 9, 2019 - 4:00pm to 5:00pm

Location: 

RH 306

Through the work of Cheeger, Colding, Naber and others we have a deep understanding of the structure of Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature lower bounds. For polarized Kahler manifolds, this was taken further by Donaldson-Sun, who showed that under two-sided Ricci curvature bounds, non-collapsed limit spaces are projective varieties, leading to major progress in Kahler geometry. I will discuss joint work with Gang Liu giving an extension of this result to the case when the Ricci curvature is only bounded from below.

The Shape of Associativity

Speaker: 

Satyan Devadoss

Institution: 

University of San Diego

Time: 

Thursday, May 31, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Associativity is ubiquitous in mathematics.  Unlike commutativity, its more popular cousin, associativity has for the most part taken a backseat in importance.  But over the past few decades, associativity has blossomed and matured, appearing in theories of particle collisions, elliptic curves, and enumerative geometry.  We start with a brief look at this history, and then explore the visualization of associativity in the forms of polytopes, manifolds, and complexes. This talk is heavily infused with imagery and concrete examples.

A geometer’s view of the simplex algorithm for linear optimization

Speaker: 

Jesus de Loera

Institution: 

UC Davis

Time: 

Thursday, May 3, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

Linear programs (LPs) are, without any doubt, at the core of both the theory and the practice of mondern applied and computational Optimization (e.g., in discrete optimization LPs are used in practical computations using branch-and-bound, and in approximation algorithms, e.g., in rounding schemes). At the same time  Dantzig’s Simplex method is one of the most famous algorithms to solve LPs and SIAM elected it as one of the top 10 most influential algorithms of the 20th Century.

But despite its key importance, many simple easy-to-state mathematical properties of the Simplex method and its geometry remain unknown. The geometry of the simplex method is very much the convex-combinatorial geometry of polyhedra (e.g., cubes, simplices, etc). Perhaps the most famous geometric-combinatorial challenge is to determine a worst-case upper bound for the graph diameter of polyhedra.  Although a lot of progress has been made, today even for the most elementary textbook linear programs we remain ignorant as to what the exact diameter upper bounds are. In this talk, I will discuss this  geometric problem and present the key ideas for proving that the diameter of graphs of all network-flow polytopes satisfy the Hirsch linear bound. This is joint work with S. Borgwardt (Univ of Colorado) and E. Finhold (Fraunhofer Institut).

Braids, Polynomials and Hilbert's 13th Problem

Speaker: 

Jesse Wolfson

Institution: 

UC Irvine

Time: 

Thursday, March 1, 2018 - 4:00pm to 5:00pm

Location: 

RH 306

There are still completely open fundamental questions about polynomials in one variable. One example is Hilbert's 13th Problem, a conjecture going back long before Hilbert.  Indeed, the invention of algebraic topology grew out of an effort to understand how the roots of a polynomial depend on the coefficients. The goal of this talk is to explain part of the circle of ideas surrounding these questions. 

Along the way, we will encounter some beautiful classical objects - the space of monic, degree d square-free polynomials, algebraic functions, lines on cubic surfaces, level structures on Jacobians, braid groups, Galois groups, and configuration spaces - all intimately related to each other, all with mysteries still to reveal.

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