Student Operator Algebras Seminar
2024-04-18 00:00:00 +0000
Welcome!
The aim of this student seminar is to ignite interest in the fascinating field of operator algebras. We plan to introduce participants to a diverse range of topics within $C^*$-algebra and von Neumann algebra theory. Our goal is to foster a vibrant community where graduate students from various disciplines can attend, learn, and engage with significant results, concepts, and tools that intersect with their own research and coursework.
Topics include but are not limited to:
- Operator K-theory
- Index Theory
- Topological Dynamics
- $C^*$-algebra theory
- von Neumann algebra theory
- Noncommutative geometry and topology
If you are not familiar with these topics, not to worry, here are some references which go beyond Wikipedia:
An Introduction to $C^*$-Algebras and the Classification Program by Karen Strung
Von Neumann Algebras by Rolando de Santiago and Brent Nelson
An introduction to K-theory for $C^*$-algebras by Rørdam, Larsen, and Laustsen
An Introduction to $C^*$-Algebras and Noncommutative Geometry by Heath Emerson
The seminar is on Mondays at 4:30, and if you would like to give at talk, please email cano13 at purdue dot edu.
The schedule for fall 2024 is below:
August 29, Introductions and interests, location: MATH
- Speakers: Alejandro Cano, Hao Wan, Chrisil Ouseph
September 9, Direct integral decomposition of von Neumann algebras, location: SCHM 314
- Abstract: In this survey talk, we will explore the direct integral of von Neumann algebras and its applications to addressing problems related to the rigidity of group von Neumann algebras with diffuse center.
- Speaker: Adriana Fernández Quero, University of Iowa
September 16, Group $C^*$-Algebras, the Fourier transform and locally compact groups.
Abstract: We define two types of $C^*$-Algebras which we can associate to a large class of groups, and study the properties which each object can reflect from the other. In doing so, we state a number of theorems which tie the Fourier transform to a dual object of the group $C^*$-Algebra we construct. The tools along the way will include representation theory, abstract integration theory, and functional analysis.
- Speaker: Alejandro Cano, Purdue University
September 23, Hilbert Hotels on Hilbert Spaces. location: SCHM 314
Abstract: The talk will shine a light on the rich and underrated history of the interaction between set theory and operator algebras. We will warm up with a brisk jog through the necessary prerequisites in logic. A surprising survey of several results will follow, starting with an independence result in functional analysis. The remainder of the talk will be a showcase of some of the unexpected consequences in operator algebras of the Continuum Hypothesis and other independent axioms: a representation theorem for the double dual of $C[0,1]$, weak expectations on von Neumann algebras, ultrapowers, and automorphisms on the Caulkin Algebra.
- Speaker: Chrisil Ouseph, Purdue University
September 30, Introduction to finite free probability. location: SCHM 314
Abstract: Finite free additive and multiplicative convolutions are binary operations of polynomials that behave well with respect to the roots. These operations have gained interest in recent years due to its interpretation as expected characteristic polynomials of random matrix operations and their connection to free probability, geometry of polynomials, representation theory and combinatorics. We will study in detail the basic properties of these convolutions of polynomials, survey the results in the area, and mention some interesting open problems.
- Speaker: Daniel Perales, Texas A&M
October 21, The K-theoretic Atiyah-Singer Index Theorem. location: SCHM 314
Abstract: Index theorems relate global analytic information of a manifold (the number of solutions to an elliptic PDE on the manifold) to a sum of topological invariants over the manifold. Index theorems relate topology and analysis with applications ranging from mathematical physics to operator algebras. I will state and unpack the Atiyah-Singer Index theorem in the language of K-theory, which provides the most "big picture" overview of the result.
- Speaker: General Ozochiawaeze, Purdue University
October 28, The $C^*$-algebra version of the topological 2-torus. (Part 1) location: SCHM 314
Abstract: Irrational rotation algebras, also known as noncommutative tori, are noncommutative $C^*$-algebras that generalize the algebra of continuous functions on the two-dimensional torus. These algebras are defined as universal $C^*$-algebras associated with a specific set of generators and relations, in terms of a rotation angle $\theta\notin \mathbb{Q}.$ In particular, irrational rotation algebras have a unique tracial state and can be embedded into an AF-algebra, allowing us to explore their invariants via Elliott’s Theorem for AF-algebras. In this talk, we will discuss the construction of irrational rotation algebras and explore how the uniqueness of the tracial state facilitates the identification of complete invariants in K-theoretic terms. These invariants enable us to determine when two irrational rotation algebras are isomorphic based on their rotation parameters. If time permits, we will present details on the embedding technique used for the classification theorem. This talk will be mainly based on Chapter VI of the book $C^*$-algebras by Example by Ken Davidson.
- Speaker: Jose Manuel Barrientos Lopez, Purdue University
November 4, The $C^*$-algebra version of the topological 2-torus. (Part 2) location: SCHM 314
Abstract: Irrational rotation algebras, also known as noncommutative tori, are noncommutative $C^*$-algebras that generalize the algebra of continuous functions on the two-dimensional torus. These algebras are defined as universal $C^*$-algebras associated with a specific set of generators and relations, in terms of a rotation angle $\theta\notin \mathbb{Q}.$ In particular, irrational rotation algebras have a unique tracial state and can be embedded into an AF-algebra, allowing us to explore their invariants via Elliott’s Theorem for AF-algebras. In this talk, we will discuss the construction of irrational rotation algebras and explore how the uniqueness of the tracial state facilitates the identification of complete invariants in K-theoretic terms. These invariants enable us to determine when two irrational rotation algebras are isomorphic based on their rotation parameters. If time permits, we will present details on the embedding technique used for the classification theorem. This talk will be mainly based on Chapter VI of the book $C^*$-algebras by Example by Ken Davidson.
- Speaker: Jose Manuel Barrientos Lopez, Purdue University
November 11 , Rigidity for von Neumann Algebras Arising from Groups. location: SCHM 314
Abstract: I will survey some rigidity results for II$_1$ factors arising from countable groups, e.g. $L(G)$ and $L^{\infty}(X, \mu) \rtimes G$. Time permitting, I will either go into more detail about the philosophy of deformation/rigidity or emphasize the geometric properties of groups and their actions necessary for rigidity results.
- Speaker: Patrick Debonis, Purdue University