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.

Hide

- April 11: An Introduction to the Kapustin-Witten Equations and Witten's Program - Siqi He
Speaker: Siqi He

Abstract: In this talk, we will introduce a Witten's Program on the SL(2,C) Casson invariant, Jones polynomials and the Kapustin-Witten equations. Over manifold with boundary, Witten suggest to study a non elliptic boundary condition of Kapustin-Witten equations which will leads to study a non-linear uniformly degenerate elliptic PDE. We will discuss some unsolved math problems in this program.

Hide - April 18: The Theory of Pseudo-differential Operators on the Noncommutative n-Torus - Jim Tao
Speaker: Jim Tao

Abstract: The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudo-differential operators. In a foundational paper, Connes showed that, by direct analogy with the theory of pseudo-differential operators on $\mathbb R^n$, one may derive a similar pseudo-differential calculus on noncommutative $n$ tori $\mathbb T_{\theta}^n$, and with the development of this calculus came many results concerning the local differential geometry of noncommutative tori for $n=2,4$, as shown in the groundbreaking paper in which the Gauss--Bonnet theorem on $\mathbb T_{\theta}^2$ is proved and later papers. Certain details of the proofs in the original derivation of the calculus were omitted, such as the evaluation of oscillatory integrals, so we make it the objective of this paper to fill in all the details. After reproving in more detail the formula for the symbol of the adjoint of a pseudo-differential operator and the formula for the symbol of a product of two pseudo-differential operators, we define the corresponding analog of Sobolev spaces for which we prove the Sobolev and Rellich lemmas. We then extend these results to finitely generated projective right modules over the noncommutative $n$ torus.

Hide - April 25: Reconstruction in Gauge Gromov-Witten Theory - Seunghee Ye
Speaker: Seunghee Ye

Abstract: Gauge GW invariants are K-theoretic invariants associated to the moduli space of Gieseker bundles. The moduli spaces are complete but not compact, making it nontrivial to prove well-definedness of the invariants, let alone compute them. We will show that n-pointed gauge GW invariants can be reconstructed from 3-pointed invariants, providing a concrete way to compute these invariants.

Hide - May 2: Determinacy - Connor Meehan
Speaker: Connor Meehan

Abstract: Hello, reader. I want to play a game. Here is what happens if you lose: I had a winning strategy all along. We will fix a set A of infinite strings of natural numbers. Then you pick a number, then I choose one, then you pick again, and so on. Your challenge: ensure the final string we create is in A.

We discuss the thrilling topological games that people play, the cunning strategies involved, and the shocking consequences one may derive in descriptive set theory. Aptness of adjectives in the previous sentence not guaranteed.

Hide

This page was created by Seunghee Ye, 2015