The goal of the math graduate seminar is to provide a place for graduate students to interact and to present new/interesting developments from our respective areas of research. The seminar is organized by graduate students, and is intended for a graduate student audience. The speaker must present interesting/important facts from any area of mathematics in a way that can be understood by students from different backgrounds. Having said that, anyone interested in math is welcome to join us!

We will meet every other Tuesday in Sloan 159 at 12:00PM unless otherwise announced. See below for schedule for the current academic year (2016-17). For past seminars, please visit past seminars.

FREE PIZZAS AND SODAS WILL BE SERVED!

If you would like to give a talk at the graduate seminar, please contact:

Seunghee Ye (syye@caltech.edu)

- Oct 11: Sofic Groups - Peter Burton

- Jan 10: Quantum Ramsey Numbers: a Probabilistic Method Approach to Operator Systems - Jalex Stark
Speaker: Jalex Stark

Abstract: A concrete operator system is a vector space of bounded linear operators on a (finite-dimensional) Hilbert space. Recently, an interpretation of these objects in terms of zero-error quantum information theory has spurred interest in a combinatorial approach, the so-called ``noncommutative graph theory''. In late 2015, Nik Weaver proved his quantum Ramsey theorem, which says that for any concrete operator system, there must be either a large subspace on which the action of the operator system is trivial or one on which the action is isomorphic to a full matrix subalgebra. In surprising contrast to the classical case, the bound on the subspace dimension is related polyonimally to the dimension of the hilbert space. We'll discuss how this can be used to give some tight control on the combinatorics of noncommutative graphs. Our main result is a probabilistic method argument showing a partial converse (a lower bound on the "quantum Ramsey number") which is asymptotically tight up to logarithmic factors in the off-diagonal regime. Along the way, we'll introduce tools from random matrix theory. This is joint work with Martino Lupini, Matthew Kennedy, Martin Argerami, and Marcin Sabok.

Hide - Jan 31: Sumsets of sequences of vectors and the Levy Steinitz Theorem - Josh Frisch
Speaker: Josh Frisch

Abstract: Given a conditionally, but not absolutely, you can rearrange it in order to sum to any real number. What about complex numbers or, more generally, finite dimensional vector spaces? An ingenious theorem of Levy and Steinitz says that the set of possible sums in this case, is always an affine subspace. We will prove this and, en route, a clever lemma about finite sums of vectors.

Hide - Feb 14: The translation flow on holomorphic maps out of the poly-plane - Dmitri Gekhtman
Speaker: Dmitri Gekhtman

Abstract: We study the family of holomorphic maps from the polydisk to the disk which restrict to the identity on the diagonal. In particular, we analyze the asymptotics of the orbit of such a map under the conjugation action of a parabolic subgroup of PSL_2(\R).

Hide - Feb 21: Connes' embedding conjecture and ergodic theory - Peter Burton
Speaker: Peter Burton

Abstract: Connes' embedding conjecture asserts that a wide class of von Neumann algebras have a certain finite approximation property. It has numerous implications in operator algebras, noncommutative geometry and quantum information theory. We will discuss emerging connections between Connes' embedding conjecture and the ergodic theory of direct products of free groups. These connections are interesting on an abstract level because they relate the 'static' embedding conjecture to dynamics, and on a more practical level because the ergodic theoretic reformulations of the embedding conjecture seem closer to current techniques than the operator algebraic statement.

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This page was created by Seunghee Ye, 2015