# Mathematics Colloquium

## Probability bounds in classical and quantum statistical mechanics

### When

March 22, 2019 | 2:30 - 3:30 p.m.

### Where

Campbell Hall 443

### Speaker

Dr. Shannon Star, UAB

### Abstract

In the first half we will show that spectral gap bounds lead to concentration of measure bounds. Methods for the former were initiated by Lu and Yau, and further developed for quantum spin systems. This is joint with Michael Froehlich (UAB) a medical researcher pursuing a biostats PhD. In the second half of the talk, we will describe how probability bounds for the classical dimer model and six vertex model in 2d give bounds for quantum systems in 1d: general XY models, the XXZ model and the Hubbard model. This is joint work with Scott Williams (UAB) a former math major. The talk is accessible to students, and is particularly relevant to those interested in Markov chains.

## Existence results for some classes of integrodifferential equations of Gurtin-Pipkin type

### When

March 1, 2019 | 2:30 - 3:30 p.m.

### Where

Campbell Hall 443

### Speaker

Prof. Toka Diagana, The University of Alabama in Huntsville

### Abstract

Integro-differential equations of Gurtin-Pipkin type play an important in studying various practical problems. In particular, they have been used to study the heat conduction in materials with memory, the sound propagation in viscoelastic media, or in homogenization problems in perforated media (Darcy’s Law).

To read the full abstract, view the PDF version.

## Multiplicity One Conjecture in Min-max theory

### When

February 18, 2019 | 2:30 - 3.30 p.m.

### Where

Campbell Hall 443

### Speaker

Dr. Xin Zhou, Institute for Advance Study (Princeton)/UC Santa Barbara

### Abstract

I will present a recent proof of the Multiplicity One Conjecture in Min-max theory. This conjecture was raised by Marques and Neves as the key step to establish a Morse theory for the area functional. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one. As direct corollaries, it implies the generalized Yau's conjecture for such manifolds with positive Ricci curvature, which says that there exist a sequence of minimal hypersurfaces with area tending to infinity, and the Weighted Morse Index Bound Conjecture by Marques and Neves. The talk will be for general audience.

## Reducibility of the Fermi surface for periodic quantum-graph operators

### When

February 15, 2019 | 2:30 - 3.30 p.m.

### Where

Campbell Hall 443

### Speaker

Dr. Stephen Shipman, LSU

### Abstract

The Fermi, or Floquet, surface for a periodic operator at a given energy level is an algebraic variety that describes all complex wave vectors admissible by the periodic operator at that energy. Its reducibility is intimately related to the construction of embedded eigenvalues supported by local defects. The rarity of reducibility is reflected in the fact that a generic polynomial in several variables cannot be factored. The "easy" mechanism for reducibility is symmetry. However, reducibility ensues in much more general and interesting situations. This work constructs a class of non-symmetric periodic Schrödinger operators on metric graphs (quantum graphs) whose Floquet surface is reducible. The graphs in this study are obtained by coupling two identical copies of a periodic quantum graph by edges to form a bilayer graph. Reducibility of the Floquet surface for all energies ensues when the coupling edges have potentials belonging to the same asymmetry class, that is, when their "spectral A-functions" are identical. If the potentials of the connecting edges belong to different asymmetry classes, then typically the Floquet surface is not reducible. Bilayer graphene is a notable exception—its Floquet surface is always reducible.

## Supergeometry: introduction and some applications

### When

February 8, 2019 | 2:30 - 3.30 p.m.

### Where

Campbell Hall 443

### Speaker

Dr. Theodore Voronov, University of Manchester

### Abstract

Supergeometry owes its name to supersymmetry, which is a (theoretical) symmetry mixing bosons and fermions in physics. Supersymmetric models appeared in 1970s, but "super" ideas have much deeper roots both in mathematics and physics e.g. in the works of Hermann Graßmann and Élie Cartan. The pioneer of supermathematics was Felix Alexandrovich Berezin. His discovery of what we now call "Berezin integral" was used by Faddeev and Popov for quantization of gauge fields even before the official birth of supermathematics.

Supergeometry provides powerful tools for "ordinary" mathematics and mathematical physics, and has led to spectacular applications such as analytic proof of the Atiyah-Singer index theorem based on supermanifold quantization. It possesses a unifying power, making it possible to see e.g. Clifford algebra and differential operators as basically the same thing. Recent applications of supergeometry are related with symmetry structures "up to homotopy." This includes the proof of the existence of deformation quantization of Poisson manifolds due to Kontsevich.

I will give an introduction to the main ideas of supergeometry with examples (including my own work). No prior knowledge will be assumed. I hope to touch on some of my recent works, such as volumes of classical supermanifolds (originated from my counterexample to a conjecture by Witten), differential operators on the superline, and (time permitting) an entirely new idea of "microformal geometry."

## An overview of numerical algorithms for the Poisson-Boltzmann equation in biomolecular electrostatics

### When

February 1, 2019 | 2:30 - 3.30 p.m.

### Where

Campbell Hall 443

### Speaker

Dr. Shan Zhao, University of Alabama

### Abstract

The Poisson-Boltzmann Equation (PBE) is a widely used implicit solvent model for the electrostatic analysis of solvated biomolecules. The numerical solution of the PBE is known to be challenging, due to the consideration of discontinuous coefficients, complex geometry of protein structures, singular source terms, and strong nonlinearity. In this talk, I will offer a brief overview of recent studies in the literature as well as new developments in our group for resolving the PB numerical difficulties.

- For treating dielectric interface and complex geometry, both finite element methods and Cartesian grid finite difference methods have been developed for delivering a second order accuracy in space.
- In the framework of pseudo-time integration, we have constructed an analytical treatment to suppress the nonlinear instability.
- For treating charge singularity in solvated biomolecules, we have introduced a new regularization approach, which combines the efficiency of two-component schemes with the accuracy of the three-component methods. Finally, numerical experiments of several benchmark examples and free energy calculations of protein systems are presented to demonstrate the stability, accuracy, and efficiency of the new algorithms.