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Events for February 9, 2018 from General and Seminar calendars

Newton-Okounkov Bodies

Time: 1:00PM - 2:30PM

Location: BLOC 624

Speaker: Frank Sottile, Texas A&M University

Title: Mixed volumes and an extension of intersection theory of divisors

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: James Kennedy, University of Lisbon

Title: Asymptotically optimal Laplacian eigenvalues and Polya's conjecture

Abstract: A longstanding problem in spectral geometry is to determine the domain(s) which minimise a given eigenvalue of a differential operator such as the Laplacian with Dirichlet boundary conditions, among all domains of given volume. For example, the Theorem of (Rayleigh--) Faber--Krahn states that the smallest eigenvalue is minimal when the domain is a ball. Very little to nothing is known about domains minimising the higher eigenvalues, but the Weyl asymptotics suggest that the ball should in a certain sense be asymptotically optimal.

In the first part of this talk, we will sketch a new approach to this problem initiated by a paper of Colbois and El Soufi in 2014, which asks not after the minimising domains themselves but properties of the corresponding sequence of minimal values. This serendipitously also yields a new Ansatz for tackling the more than 50 year old conjecture of Polya that the k-th eigenvalue of the Dirichlet Laplacian on any domain always lies above the corresponding first term in the Weyl asymptotics for that eigenvalue. Along the way, we will additionally meet variants of Gauss circle problem.

In a second part, we will present some recent analogous results for the Laplacian with Robin boundary conditions, which are ongoing joint work with Pedro Freitas.

Mathematical Physics and Harmonic Analysis Seminar

Time: 2:50PM - 3:50PM

Location: Blocker 605AX

Speaker: Yaiza Canzani, UNC Chapel Hill

Title: On the growth of eigenfunctions averages

Abstract: In this talk we discuss the behavior of Laplace eigenfunctions when restricted to a fixed submanifold by studying the averages given by the integral of the eigenfunctions over the submanifold. In particular, we show that the averages decay to zero when working on a surface with Anosov geodesic flow regardless of the submanifold (curve) that one picks. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages. This is based on joint work with Jeffrey Galkowski.

Algebra and Combinatorics Seminar

Time: 3:00PM - 3:50PM

Location: BLOC 628

Speaker: Nathan Williams, UT Dallas

Title: Fixed Points of Parking Functions

Abstract: We define an action of words in [m]^n on R^m to give a new characterization of rational parking functions. We use this viewpoint to give a simple definition of Gorsky, Mazin, and Vazirani's zeta map on rational parking functions when m and n are coprime, and prove that this zeta map is invertible. A specialization recovers Loehr and Warrington's sweep map on rational Dyck paths. This is joint work with Jon McCammond and Hugh Thomas.

Geometry Seminar

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Tri Lai, University of Nebraska - Lincoln

Title: Tilings and More

Abstract: The field of enumeration of tilings dates back to the early 1900s when MacMahon proved his classical theorem on plane partitions. The enumeration of tilings has since taken on a life of its own as a subfield of combinatorics with connections and applications to diverse areas of mathematics, including representation theory, linear algebra, group theory, mathematical physics, graph theory, probability, and cluster algebra, just to name a few. In this talk, we focus on an interesting connection between tilings, linear algebra, and a mathematical model of electrical networks. In particular, we will go over the proof of a conjecture of Kenyon and Wilson on `tiling-representation' of semi-contiguous minors.

Linear Analysis Seminar

Time: 4:00PM - 5:00PM

Location: BLOC 220

Speaker: Mateusz Wasilewski, IMPAN

Title: Non-conjugacy of generator MASAs in q-Gaussian algebras

Abstract: In the study of group von Neumann algebras, sometimes maximal abelian subgroups give rise to maximal abelian subalgebras (MASAs); it happens so for free groups and abelian subgroups generated by a single free generator. These subalgebras are conjugate by an automorphism (coming from the automorphism of a free group), but not by an inner automorphism. Free group factors also admit a different representation, using Voiculescu's free Gaussian functor; in this setting one can generate a MASA using a vector from a real Hilbert space. Bo┼╝ejko and Speicher introduced a deformation of free group factor, called the q-Gaussian algebras for which analogues of generator MASAs can be defined. I will show that these MASAs are never conjugate by an inner automorphism, using Popa's intertwining techniques (joint work with Martijn Caspers and Adam Skalski).