Mathematics

Particle-vortex statistics and the nature of dense quark matter

Aleksey Cherman (INT Washington)

Abstract: Dense nuclear matter is expected to be a superfluid. Quark matter, which is expected to appear at higher densities, is also a superfluid. But quark matter turns out to be sharply distinct from a standard superfluid, because it supports Z3-valued particle-vortex braiding phases, and its effective action includes a coupling to a topological quantum field theory. Physically, our results imply that certain mesonic and baryonic excitations have orbital angular momentum quantized in units of ħ/3 in the presence of a superfluid vortex. If Z3 braiding phases and angular momentum fractionalization are absent in lower density hadronic matter, as is widely expected, then the quark matter and hadronic matter regimes of dense QCD must be separated by at least one phase transition. Since the low-density regime is a `confined' phase, while the high-density regime is a `Higgs' phase due to color superconductivity, our results also have interesting implications for Higgs-confinement complementarity.

The Fifth Arithmetic Operation

Eric Wawerczyk (University of Notre Dame)

Abstract: Martin Eichler is attributed to saying: “There are five elementary operations in Number Theory: addition, subtraction, multiplication, division, and modular forms.” The point of this talk is to demonstrate a variety of amazing arithmetic formulas which can be derived using these five “basic” operations. We will be presenting amazing proofs by Euler, Riemann, and Ramanujan.

The $Q$-system of type $A_2^{(2)}$

Minyan Simon Lin (UIUC)

Abstract: In this talk, I will introduce the $Q$-system of type $A_2^{(2)}$, which is a recursion relation that is satisfied by the characters of certain finite-dimensional $\mathfrak{sl}_2$-modules. I will derive some properties of this system, such as its integrability and Laurent properties, and discuss how these properties have a natural extension to the noncommutative case. If time permits, I will discuss some of the consequences of the properties of the noncommutative $Q$-system of type $A_2^{(2)}$.

Resonance rigidity for Schrödinger operators

Tanya Christiansen (University of Missouri)

Abstract: From a mathematical point of view, resonances may provide a replacement for discrete spectral data for a class of operators with continuous spectrum. Physically, resonances may correspond to decaying waves. This talk will introduce the notion of resonances for Schrödinger operators. We discuss results, both by the speaker and others, related to the rigidity of the set of resonances of a Schrödinger operator on ${\mathbb R}^d$ with potential $V\in L^\infty_c({\mathbb R}^d)$. For example, within this class of operators, is the Schrödinger operator with $0$ potential determined by its resonances? What can we say about other sets of isoresonant potentials?