• Speaker: Shiwen Zhang, U Minnesota
• Title: The landscape law for tight binding Hamiltonians
• Date: 03/09/2021
• Time: 3:00 PM - 4:00 PM
• Place:  (Virtual Meeting Link)
• Contact: Jeffrey Hudson Schenker (schenke6@msu.edu)

The localization landscape theory, introduced in 2012 by Filoche and Mayboroda, considers the so-called 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 tight-binding 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 so-called 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.

• Speaker: Brent Nelson, Michigan State University
• Title: Complex analysis applied to operator algebras
• Date: 03/18/2021
• Time: 5:00 PM - 5:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: 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.)

• Speaker: Rolando de Santiago, Purdue University
• Title: Groups, Group Actions, and von Neumann Algebras
• Date: 04/15/2021
• Time: 5:00 PM - 5:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Brent Nelson (banelson@msu.edu)

Given a group $G$ acting on measure space $(X,\mu)$ Murray and von Neumann’s group-measure 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 group-measure 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.