Title: Spectral Optimization and Free Boundary Problems

Date: 02/12/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

A classic subject in analysis is the relationship between the spectrum of the Laplacian on a domain and that domain's geometry. One approach to understanding this relationship is to study domains which extremize some function of their spectrum under geometric constraints. I will explain how to attack these problems using tools from the calculus of variations to find solutions. A key difficulty with this method is showing that the optimizers (which are a priori very weak) are actually smooth domains, and I address this issue in some recent work with Fanghua Lin. Our results are based on relating spectral optimization problems to certain vector-valued free boundary problems of Bernoulli type.

We're hoping to share our experiences from different projects related to supporting student learning outside of the classroom to generate ideas about how the MLC can shift to facilitate more productive learning for our students.

We will begin by outlining the construction of a cluster algebra associated to any surface with boundary (and marked points). Then we will discuss a formula, due to Schiffler, which explicitly gives an arbitrary cluster variable as a Laurent monomial in the initial variables, using the perfect matchings of an associated graph, called a "snake graph".

This talk will be a brief introduction to the Turaev-Viro Invariant. The Turaev-Viro Invariant is a 3-manifold invariant defined on a triangulation of a manifold. Using skein-theoretic methods, I will demonstrate a proof of its invariance with a technique known as chain-mail. This technique illustrates a close relationship between the Turaev-Viro Invariant and the surgery-presentation invariants originally defined by Reshetikhin and Turaev.

Title: Quantization of conductance in gapped interacting systems

Date: 02/15/2018

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

I will present two closely connected results. The first is the linear response theory in gapped interacting systems, and a proof of the associated Kubo formula. The second is a short proof of the quantization of the Hall conductance for gapped interacting quantum lattice systems on the two-dimensional torus.

Title: Auction Dynamics for Semi-Supervised Data Classification

Date: 02/16/2018

Time: 4:10 PM - 5:00 PM

Place: B117 Wells Hall

We reinterpret the semi-supervised data classification problem using an auction dynamics framework (inspired by real life auctions) in which elements of the data set make bids to the class of their choice. This leads to a novel forward and reverse auction method for data classification that readily incorporates volume/class-size constraints into an accurate and efficient algorithm requiring remarkably little training/labeled data. We prove that the algorithm is unconditionally stable, and state its average and worst case time complexity.

Von Neumann algebras are certain *-subalgebras of bounded operators acting on a Hilbert space. They are generally thought of as non-commutative measure spaces and offer connections to many fields of mathematics (e.g. group theory, low-dimensional topology, logic, ergodic theory, and random matrix theory to name a few). In some instances an analogy with probability spaces is more appropriate, and indeed this is precisely what informs the field of free probability, wherein one uses non-commutative analogs of probabilistic notions to study the structure of von Neumann algebras. One particular example of this is free transport. In probability theory, transport refers to a measurable map between probability spaces that pushes one measure onto the other. Following work of Brenier in 1991, transportation theory has known great success. Free transport, the non-commutative analog that was introduced by Guionnet and Shlyakhtenko in 2014, offers methods for proving isomorphisms between von Neumann algebras. In this talk, I will discuss these ideas as well my work, which used free transport to prove isomorphisms between certain so-called "non-tracial" von Neumann algebras.