Mathematics Colloquium

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.

  1. 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.
  2. In the framework of pseudo-time integration, we have constructed an analytical treatment to suppress the nonlinear instability.
  3. 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.

Tanglegrams and tanglegram crossing numbers

When

January 25, 2019 | 2:30 - 3.30 p.m.

Where

Campbell Hall 443

Speaker

Dr. Laszlo Szekely, University of South Carolina

Abstract

A tanglegram is a graph, consisting of two rooted binary trees of the same size and a perfect matching joining their leaves. A tanglegram layout is a special straight line drawing of this graph, where the leaves of the trees are placed on two parallel lines, the two trees are drawn as plane trees on either side of the strip created by these lines, and the matching is drawn inside the strip. It is desirable to draw a tanglegram with the least possible number of pairs of crossing edges in the strip. This is the Tanglegram Layout Problem. The minimum number of the crossings obtained in this way is the tanglegram crossing number of the tanglegram. The tanglegram is planar, if it has a layout without crossings. The Tanglegram Layout Problem is NP-hard and is well studied in computer science.

Tanglegrams play a major role in phylogenetics, especially in the theory of coevolution. The first binary tree is the phylogenetic tree of hosts, while the second binary tree is the phylogenetic tree of their parasites, e.g. gopher and louse. The matching connects the host with its parasite. The tanglegram crossing number has been related to the number of times parasites switched hosts.

I present a Kuratowski type characterization of planar tanglegrams and a conjecture for a similar finite characterization of some more general tanglegram crossing number problems. This is a joint work with Eva Czabarka and Stephan Wagner.


Subintervals of (Random Subsets of) Intervals and Entanglement in Quantum Spin Chains

When

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

Where

Campbell Hall 443

Speaker

Dr. Gunter Stolz, UAB

Abstract

The entanglement entropy of eigenstates provides a measure which can be used to study if an interacting quantum system is localized. In particular, guided by the well known phenomenon of Anderson localization for non-interacting systems, one expects that the exposure of the system to disorder will increase its tendency towards localization. We discuss this in the example of the quantum Ising chain. This simple model allows for a very explicit description of how eigenstate entanglement is reduced by the introduction of disorder. This is part of a joint project with Houssam Abdul-Rahman and Christoph Fischbacher.