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In this talk, I would like to discuss Deligne’s version of Geometric Class Field theory, with special emphasis on the correspondence between rigidified 1-dimensional l-adic local systems on a curve and 1-dimensional l-adic local systems on Pic with certain compatibilities. We should like to give a sense of how this relates to the OG class field theory, and how Deligne demonstrates this correspondence via the geometry of the Abel-Jacobi Map. If time permits, we would also like to discuss the correspondence between continuous 1-dimensional l-adic representations of the etale fundamental group of a curve and local systems.

The string 2-group is a fundamental object in string geometry, which is a refinement of spin geometry required to describe the spinning string. While many models for the string 2-group exist, the construction of a representation for it is new. In this talk, I will recall the notion of strict 2-group, and then give two examples: the automorphism 2-group of a von Neumann algebra, and the string 2-group. I will then describe the representation of the string 2-group on the hyperfinite III_1 factor, which is a functor from the string 2-group to the automorphism 2-group of the hyperfinite III_1 factor.

Since the completion of the Elliott classification programme it is an important question to ask which C*-algebras satisfy the assumptions of the classification theorem. We will ask this question for the case of crossed-product C*-algebras associated to actions of nonamenable groups and focus on two extreme cases: Actions on commutative C*-algebras and actions on simple C*-algebras. It turns out that for a large class of nonamenable groups, classifiability of the crossed product is automatic under the minimal assumptions on the action. This is joint work with E. Gardella, S. Geffen, P. Naryshkin and A. Vaccaro. 

One of the most important constructions in operator algebras is the tracial ultrapower for a tracial state on a C*-algebra. This tracial ultrapower is a finite von Neumann algebra, and it appears in seminal work of McDuff, Connes, and more recently by Matui-Sato and many others for studying the structure and classification of nuclear C*-algebras. I will talk about how to generalise this to unbounded traces (such as the standard trace on B(H)). Here the induced tracial ultrapower is not a finite von Neumann algebra, but its multiplier algebra is a semifinite von Neumann algebra.

 I will discuss ongoing work with Toke Carlsen and Aidan Sims on ideal structure of C*-algebras of commuting local homeomorphisms. This is one aspect of a general attempt to bridge C*-algebras with multidimensional (symbolic) dynamics.

Hirschman-Widder densities may be viewed as the probability density functions of positive linear combinations of independent and identically distributed exponential random variables. They also arise naturally in the study of Pólya frequency functions, which are integrable functions that give rise to totally positive Toeplitz kernels. This talk will introduce the class of Hirschman-Widder densities and discuss some of its properties. We will demonstrate connections to Schur polynomials and to orbital integrals. We will conclude by describing the rigidity of this class under composition with polynomial functions.

 This is joint work with Dominique Guillot (University of Delaware), Apoorva Khare (Indian Institute of Science, Bangalore) and Mihai Putinar (University of California at Santa Barbara and Newcastle University).

A few years back, Smale spaces were shown to exhibit noncommutative Poincaré duality (with Jerry Kaminker and Ian Putnam). The fundamental class was represented as an extension by the compacts. In current work we describe a Fredholm module representation of the fundamental class. The proof uses delicate approximations of the Smale space arising from a refining sequence of (open) Markov partition covers. I hope to explain all these notions in an elementary manner. This is joint work with Dimitris Gerontogiannis and Joachim Zacharias.

Groupoid C*-algebras and twisted groupoid C*-algebras are introduced by Renault in the late ’70. Twisted groupoid C*-algebras have since proved extremely important in the study of structural properties for large classes of C*-algebras. On the other hand, Steinberg algebras are introduced independently by Steinberg and Clark, Farthing, Sims and Tomforde around 2010 which are a purely algebraic analogue of groupoid C*-algebras. Steinberg algebras provide useful insight into the analytic theory of groupoid C*-algebras and give rise to interesting examples of *-algebras. In this talk, I will first recall some relevant background on topological groupoids and twisted groupoid C*-algebras, then I will introduce twisted Steinberg algebras which generalise the Steinberg algebras and provide a purely algebraic analogue of twisted groupoid C*-algebras. If I have enough time, I will further introduce pair of algebras which consist of a Steinberg algebra and an algebra of locally constant functions on the unit space, it is an algebraic analogue of Cartan pairs

Measures of discrete curvature are a recent addition to the toolkit of network analysts and data scientists. At the basis lies the idea that networks and other discrete objects exhibit distinct geometric properties that resemble those of smooth objects like surfaces and manifolds, and that we can thus find inspiration in the tools of differential geometry to study these discrete objects. In this talk, I will introduce how this has lead to the development of notions of discrete curvature, and what they are good for. Furthermore, I will discuss our latest results on a new notion of curvature on graphs, based on the effective resistance. These new "resistance curvatures" are related to other well-known notions of discrete curvature (Ollivier, Forman, combinatorial curvature), we find evidence for convergence to continuous curvature in the case of Euclidean random graphs and there is a naturally associated discrete Ricci flow.

A preprint on this work is available on arXiv: https://arxiv.org/abs/2201.06385