• Title:  Low-temperature localization of directed polymers
• Date: 09/07/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Erik Bates, Stanford University

On the d-dimensional integer lattice, directed polymers can be seen as paths of a random walk in random environment, except that the environment updates at each time step. The result is a statistical mechanical system, whose qualitative behavior is governed by a temperature parameter and the law of the environment. Historically, the phase transitions of this system have been best understood by whether or not the path’s endpoint localizes. While the endpoint is no longer a Markov process as in a random walk, its quenched distribution is. The key difficulty is that the space of measures is too large for one to expect convergence results. By adapting methods recently used by Mukherjee and Varadhan, we develop a compactification theory to resolve the issue. In this talk, we will discuss this intriguing abstraction, as well as new concrete theorems it allows us to prove for directed polymers constructed from SRW or any other walk. (This talk is based on joint work with Sourav Chatterjee.)

• Title: Stability of superselection sectors in infinite quantum spin systems
• Date: 09/14/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Matthew Cha, MSU

Superselection sectors are equivalence classes of unitarily equivalent representations and can be used to label charges in a quantum system.  We consider a family of superselection sectors for infinite quantum spin systems corresponding to almost localized endomorphisms. If the vacuum state is pure and satisfies certain locality conditions, we show how to recover the charge statistics. In particular, the superselection structure is that of a braided tensor category, and further, is stable against deformations by a quasi-local dynamics. We apply our results to prove stability of anyons in Kitaev's quantum double.  Braided tensor categories naturally appear as the algebraic theory of anyons in topological phases of matter.  Our results provide evidence that the anyonic structure is an invariant of topologically ordered states.  This is work is joint with Pieter Naaijkens and Bruno Nachtergaele.

• Title: On the quantum and PDE models of Polaron
• Date: 09/21/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Gang Zhou, SUNY Binghamton

Polaron theory is a model of an electron in a crystal lattice. It is studied in the framework of nonequilibrium statistic mechanics. There are two different mathematical models: H. Frohlich proposed a quantum model in 1937; L. Landau and S. I. Pekar proposed a system of nonlinear PDEs in 1948. In this talk I will present a proof that these two models are equivalent to certain orders, and present some other related works. These are joint works with Rupert Frank.

• Title: BPS monopoles with non-maximal symmetry breaking and the Nahm transform
• Date: 10/03/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Akos Nagy, Fields Institute/University of Waterloo

The notion of “broken symmetry” is central in gauge theories. For BPS monopoles, symmetry breaking can be defined in terms of the eigenvalues of the Higgs-field at infinity. The symmetry breaking is maximal if the eigenvalues are distinct. Monopoles with maximal symmetry breaking have been studied extensively by both mathematicians and physicists for decades now, but little is known about the general case. In this talk, I will show how to produce monopoles with arbitrary symmetry breaking using the Nahm transform, and I will also outline the construction of its inverse. The inversion heavily uses first order elliptic PDE's on non-compact spaces, more concretely, the theory of Callias-type operators in 3D. This is a joint project with Benoit Charbonneau.

• Title: TBA
• Date: 11/02/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Marius Lemm, IAS

No abstract available.