Talk_id  Date  Speaker  Title 
27017

Monday 3/1 2:00 PM

Priyanga Ganesan, Texas A&M University

Quantum Graphs
 Priyanga Ganesan, Texas A&M University
 Quantum Graphs
 03/01/2021
 2:00 PM  2:50 PM
 Online (virtual meeting)
(Virtual Meeting Link)
 Brent Nelson (brent@math.msu.edu)
Quantum graphs are an operator space generalization of classical graphs. In this talk, I will motivate the idea of a quantum graph and its significance in quantum communication. We will look at the different notions of quantum graphs that arise in operator systems theory, noncommutative topology and quantum information theory. I will then introduce a nonlocal game with quantum inputs and classical outputs, that generalizes the non local graph coloring game. This is based on joint work with Michael Brannan and Samuel Harris.

27015

Tuesday 3/9 3:00 PM

Shiwen Zhang, U Minnesota

The landscape law for tight binding Hamiltonians
 Shiwen Zhang, U Minnesota
 The landscape law for tight binding Hamiltonians
 03/09/2021
 3:00 PM  4:00 PM

(Virtual Meeting Link)
 Jeffrey Hudson Schenker (schenke6@msu.edu)
The localization landscape theory, introduced in 2012 by Filoche and Mayboroda, considers the socalled the landscape function u solving Hu=1 for an operator H. The landscape theory has remarkable power in studying the eigenvalue problems of H and has led to numerous ``landscape baked’’ results in mathematics, as well as in theoretical and experimental physics. In this talk, we will discuss some recent results of the landscape theory for tightbinding Hamiltonians H=\Delta+V on Z^d. We introduce a box counting function, defined through the discrete landscape function of H. For any deterministic bounded potential, we give estimates for the integrated density of states from above and below by the landscape box counting function, which we call the landscape law. For the Anderson model, we get a refined lower bound for the IDS, throughout the spectrum. We will also discuss some numerical experiments in progress on the socalled practical landscape law for the continuous Anderson model. This talk is based on joint work with D. N. Arnold, M. Filoche, S. Mayboroda, and Wei Wang.

27014

Monday 3/15 2:00 PM

Christopher Shirley, ParisSaclay University

Stationary random Schrödinger operators at small disorder
 Christopher Shirley, ParisSaclay University
 Stationary random Schrödinger operators at small disorder
 03/15/2021
 2:00 PM  3:00 PM
 Online (virtual meeting)
(Virtual Meeting Link)
 Jeffrey Hudson Schenker (schenke6@msu.edu)
In this presentation, we will study Schrödinger operators with stationary potential and the existence of stationary Bloch waves for several types of stationarity and in particular in the random case. We will see that the Bloch waves of the unperturbed operator seem to vanish at weak disorder in the case of shortrange correlations. This phenomenon suggests a resonance problem, difficult to study due to the lack of compactness.
We therefore investigate this problem using Mourre theory.

28039

Thursday 3/18 5:00 PM

Brent Nelson, Michigan State University

Complex analysis applied to operator algebras
 Brent Nelson, Michigan State University
 Complex analysis applied to operator algebras
 03/18/2021
 5:00 PM  5:50 PM
 Online (virtual meeting)
(Virtual Meeting Link)
 Brent Nelson (banelson@msu.edu)
Given a positive definite matrix $D\in M_n(\mathbb{C})$ with $\text{Tr}(D)=1$, one can define a linear functional $\varphi\colon M_n(\mathbb{C})\to \mathbb{C}$ by $\varphi(x):=\text{Tr}(Dx)$ which we call a faithful state. This positive definite matrix also encodes a noncommutative dynamical system through $x\mapsto D^{it} x D^{it}$ for $t\in \mathbb{R}$. From the perspective of operator algebras, it is useful to encode this dynamical system as... well, an algebra of operators. More precisely, as a $*$algebra $\mathcal{M}$ containing $M_n(\mathbb{C})$ in a way that remembers the action of $\mathbb{R}$. In the general (infinite dimensional) setting, this is accomplished using crossed products and Tomita–Takesaki theory. In this talk, I will apply these methods to the more modest finite dimensional case, and show how a little bit of complex analysis allows one to find the analogue of $\text{Tr}$ on this larger $*$algebra $\mathcal{M}$. (This talk will assume some familiarity with linear algebra and complex analysis, but nothing further.)

27013

Monday 4/12 2:00 PM

Rui Han , Louisiana State University

Spectral gaps in graphene structures
 Rui Han , Louisiana State University
 Spectral gaps in graphene structures
 04/12/2021
 2:00 PM  3:00 PM
 Online (virtual meeting)
(Virtual Meeting Link)
 Jeffrey Hudson Schenker (schenke6@msu.edu)
We will present a full spectral analysis for a model of graphene in magnetic fields with constant flux through every hexagonal comb. In particular, we provide a rigorous foundation for selfsimilarity by showing that for any irrational flux, the spectrum of graphene is a zero measure Cantor set. I will also discuss the spectral decomposition, Hausdorff dimension of the spectrum and existence of Dirac cones. This talk is based on joint works with S. Becker and S. Jitomirskaya.

29049

Thursday 4/15 5:00 PM

Rolando de Santiago, Purdue University

Groups, Group Actions, and von Neumann Algebras
 Rolando de Santiago, Purdue University
 Groups, Group Actions, and von Neumann Algebras
 04/15/2021
 5:00 PM  5:50 PM
 Online (virtual meeting)
(Virtual Meeting Link)
 Brent Nelson (banelson@msu.edu)
Given a group $G$ acting on measure space $(X,\mu)$ Murray and von Neumann’s groupmeasure space construction describes a von Neumann algebra $L^\infty(X,\mu)\rtimes G $ which encodes both the group, the space and the action. The special case where the space is a singleton and the action is trivial produces the group von Neumann algebra $L(G) $.
In this talk, we will aim to describe properties of $L^\infty(X,\mu)\rtimes G $ in terms of the group, the space and the action; compute $L^\infty(X,\mu)\rtimes G $ in special cases; and describe how the groupmeasure space varies or the group von Neumann algebra varies with $G$. All this serves to illustrate the fundamental problem in this area: von Neumann algebras tend to have poor memory of their generating data.
This talk assumes a working knowledge of group theory and linear algebra, and while knowledge of measure theory may be helpful, it is not required.

27012

Monday 4/19 2:00 PM

Alexis Drouot, University of Washington

Mathematical aspects of topological insulators.
 Alexis Drouot, University of Washington
 Mathematical aspects of topological insulators.
 04/19/2021
 2:00 PM  3:00 PM
 Online (virtual meeting)
(Virtual Meeting Link)
 Jeffrey Hudson Schenker (schenke6@msu.edu)
Topological insulators are phases of matter that act like extraordinarily stable waveguides along their boundary. They have a rich mathematical structure that involves PDEs, spectral theory, and topology. I will survey some results and discuss some (semiclassical) directions of research.

27016

Monday 4/26 2:00 PM

Jacob Shapiro, Princeton University

Tightbinding limits in strong magnetic fields and the integer quantum Hall effect
 Jacob Shapiro, Princeton University
 Tightbinding limits in strong magnetic fields and the integer quantum Hall effect
 04/26/2021
 2:00 PM  3:00 PM
 Online (virtual meeting)
(Virtual Meeting Link)
 Jeffrey Hudson Schenker (schenke6@msu.edu)
Tightbinding models of noninteracting electrons in solids are commonly used when studying the integer quantum Hall effect or more generally topological insulators. However, up until recently there was no proof that the topological properties of these discrete Schrodinger operators agree with those of the continuum models of which they are the tightbinding limit. Before tending to this question, we first tackle the basic issue of the doublewell eigenvalue splitting in strong perpendicular constant magnetic fields in 2D. Once this is understood, we set up normresolvent convergence of a scaled continuum Schrodinger operator on L^2(R^2) (a magnetic Laplacian plus a lattice potential) to its tightbinding limit and finally show why the Chern numbers of these two models, discrete and continuum respectively, must agree. This talk is based on joint collaborations with C. L. Fefferman and M. I. Weinstein.
