Seminars and Colloquia by Series

Implicit interface boundary integral methods

Series
Applied and Computational Mathematics Seminar
Time
Friday, March 13, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Richard Tsai – University of Texas at Austin
I will present a new approach for computing boundary integrals that are defined on implicit interfaces, without the need of explicit parameterization. A key component of this approach is a volume integral which is identical to the integral over the interface. I will show results applying this approach to simulate interfaces that evolve according to Mullins-Sekerka dynamics used in certain phase transition problems. I will also discuss our latest results in generalization of this approach to summation of unstructured point clouds and regularization of hyper-singular integrals.

Birational Models of Moduli of Sheaves on Surfaces via the Derived Category

Series
Algebra Seminar
Time
Wednesday, March 11, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Aaron Bertram – University of Utah
Jacobians aren't particularly interesting from the point of view of the minimal model program, and neither are the moduli spaces of vector bundles on curves. But once we pass to vector bundles of higher rank (or torsion-free sheaves) on surfaces, then the birational geometry becomes very interesting. In this talk, I want to describe some recent results that rely on "tilting" the category of coherent sheaves on a surface to produce birational models of moduli that are themselves moduli spaces that come up naturally in the minimal model program.

Maximal Bounds on Cartesian Powers of Finite Graphs

Series
Analysis Seminar
Time
Wednesday, March 11, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jordan Greenblat – UCLA
In the course of their work on the Unique Games Conjecture, Harrow, Kolla, and Schulman proved that the spherical maximal averaging operator on the hypercube satisfies an L^2 bound independent of dimension, published in 2013. Later, Krause extended the bound to all L^p with p > 1 and, together with Kolla, we extended the dimension-free bounds to arbitrary finite cliques. I will discuss the dimension-independence proofs for clique powers/hypercubes, focusing on spectral and operator semigroup theory. Finally, I will demonstrate examples of graphs whose Cartesian powers' maximal bounds behave poorly and present the current state and future directions of the project of identifying analogous asymptotics from a graph's basic structure.

Onsager's Conjecture

Series
PDE Seminar
Time
Tuesday, March 10, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tristan Buckmaster – Courant Institute, NYU
In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy. The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Philip Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.

The structure of space curve arrangements with many incidences

Series
Combinatorics Seminar
Time
Tuesday, March 10, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joshua Zahl – MIT –
In 2010, Guth and Katz proved that if a collection of N lines in R^3 contained more than N^{3/2} 2-rich points, then many of these lines must lie on planes or reguli. I will discuss some generalizations of this result to space curves in three dimensional vector spaces. This is joint work with Larry Guth.

Inclusion of Spectrahedra, the Matrix Cube Problem and Beta Distributions.

Series
Algebra Seminar
Time
Monday, March 9, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Igor Klep – University of Auckland
Given a tuple A=(A_1,...,A_g) of symmetric matrices of the same size, the affine linear matrix polynomial L(x):=I-\sum A_j x_j is a monic linear pencil. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is called a spectrahedron. It is a convex basic closed semialgebraic subset of R^g. Given a spectrahedron S_L, the matrix cube problem of Nemirovskii asks for the biggest cube [-r,r]^g included in S_L. We solve a relaxation of this problem based on``matricial’’ spectrahedra and estimate the error inherent in this relaxation. The talk is based on joint work with B. Helton, S. McCullough and M. Schweighofer.

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