Students attending the seminar for credit will be asked to write in a few sentences what they will have learned from each talk.

Speaker: Arup Chattopadhyay (Indian Institute of Technology Guwahati and UNM)

Title: Submajorization and eigenvalue inequalities

Abstract: Let $\mathbb{M}_n$ be the $C^*$- algebra of $n\times n$ complex matrices and let $\varphi: \mathbb{M}_n \to \mathbb{M}_n$ be a completely positive map. Suppose $A \in \mathbb{M}_n$ is a self-adjoint matrix. In this lecture we will talk about a submajorization result concerning positive and negative parts of the spectrum of $\varphi (A)$. As an application of our main result, we will discuss some inequalities concerning the smallest and the largest eigenvalues of the Schur product of $A$ and $B$, where $A$ and $B$ are self-adjoint.

Speaker: Arup Chattopadhyay (Indian Institute of Technology Guwahati and UNM)

Title: Tensor product of quotient Hilbert modules

Abstract: In this lecture we will talk about a characterization result for doubly commuting quotient modules of a class of reproducing kernel Hilbert spaces. The class is large enough to contain the Hardy space and the weighted Bergman space over a polydisc. By considering the Hardy space over a polydisc or the weighted Bergman space $L^2_{a}(\mathbb{D}^n)$, with weight $a=(a_1,\dots,a_n)\in \mathbb{N}^n$, we will show that a quotient module $\mathcal{Q}$ of $L^2_{a}(\mathbb{D}^n)$ is doubly commuting if and only if $\mathcal{Q}=\mathcal{Q}_1\otimes\cdots\otimes \mathcal{Q}_n$ for some quotient module $\mathcal{Q}_i$ of $L^2_{a_i}(\mathbb{D})$, $i=1,\dots,n$. We will also discuss a characterization of co-doubly commuting and essentially doubly commuting submodules of $L^2_{a}(\mathbb{D}^n)$.

Speaker: Anna Skripka (UNM)

Title: Analysis of operator functions.

Abstract: I will make a brief introduction to analysis of matrix and, more generally, operator functions. These functions are defined at noncommuting points, which makes their analysis very different from the one in the classical commutative case. I will demonstrate several results in the noncommutative case whose commutative analogs are known from calculus and also show applications to perturbation theory. The talk should be accessible to students.

Speaker: Terry A. Loring (UNM)

Title: Emergent topology of insulators.

Abstract: Many insulators can be studied via finite matrix models. The joint Clifford spectrum, found using a non-commutative Dirac operator, is often an interesting hypersurface. Proving this uses one of ten KO or KU indices of finite-volume systems. Ongoing work with Schulz-Baldes compares these indices map to older indices for infinite volume systems. The mathematics here is closely related to recent work on emergent geometry in string theory.

Speaker: Abdel Mohamed (UNM)

Title: Spectral shift functions.

Abstract: The spectral theorem provides a method of defining complex-valued functions of Hermitian matrices as simple functions of an operator-valued measure of their spectrum. The spectral shift function then allows us to study some properties of these functions and is general enough to apply to operators on infinite dimensional spaces as well. Here we collect some results concerning spectral shift functions of finite order, and detail the computation of its sign numerically in the case of a special 2x2 Hermitian matrix.

Speaker: Maxim Zinchenko (UNM)

Title: Chebyshev polynomials on multi-component sets.

Abstract: Chebyshev polynomials are the unique monic polynomials that minimize the sup-norm on a given compact set. In this talk I will discuss classical and recent results on asymptotics of the Chebyshev polynomials on subsets of the real line.

Speaker: Bishnu Sedai (UNM)

Title: Trace formulas for perturbations of operators with Hilbert-Schmidt resolvents.

Abstract: We will discuss integral representations for the trace of Taylor remainders of operator functions corresponding to self-adjoint perturbations of self-adjoint operators with Hilbert-Schmidt resolvents.