## Welcome to the AGNT Seminar at Rice!

January 18, 2023 : [in-person]

Speaker: Chelsea Walton, Rice University

__Title__: Modernizing Modern Algebra: Category Theory is coming, whether we like it or not

__Abstract__: Inspired by my past stay at the University of Hamburg this Fall, I will chat about the development of the field, Modern Algebra, starting with the foundations set in Germany in the 1920s. Through the work of E. Artin and E. Noether, and through the writings of B. L. van der Waerden, Modern Algebra launched onto the mathematical scene as a sort of Haute Couture fashion in the 1930s. These days, this field is certainly presented as fashion for the masses, as is included in any standard undergraduate curriculum in mathematics. In fact, the contents of van der Waerden's landmark 1931 textbook "Moderne Algebra" is much in line with our syllabi for such courses today. Now at the 100 year mark since the emergence of Modern Algebra, one might wonder: What's next? I believe it's Category Theory, whether we like it or not. To support this belief, I'll spend most of the time in the talk presenting a case study for algebras in a few settings-- over a field, categorical, and 2-categorical. Applications of algebras in these settings will be emphasized, and the talk to be down-to-earth.

January 25, 2023 : [online]

Speaker: Be'eri Greenfeld, University of California, San Diego

__Title__: Growth of infinite-dimensional algebras, symbolic dynamics and amenability

__Abstract__: The growth of an infinite-dimensional algebra is a fundamental tool to measure its infinitude. Growth of noncommutative algebras plays an important role in noncommutative geometry, representation theory, differential algebraic geometry, symbolic dynamics and various recent homological stability results in number theory and arithmetic geometry.

We analyze the space of growth functions of algebras, answering a question of Zelmanov (2017) on the existence of certain 'holes' in this space, and prove a strong quantitative version of the Kurosh Problem on algebraic algebras. We use minimal subshifts with oscillating complexity to resolve a question posed by Krempa-Okninski (1987) and Krause- Lenagan (2000) on the GK-dimension of tensor products.

An important property implied by subexponential growth (both for groups and for algebras) is amenability. We show that minimal subshifts of positive entropy give rise to amenable graded algebras of exponential growth, answering a conjecture of Bartholdi (2007), naturally extending a wide open conjecture of Vershik on amenable group rings).

This talk is partially based on joint works with J. Bell and with E. Zelmanov.

February 1, 2023 : (open)

February 8, 2023 : (open)

February 15, 2023 : (open)

February 22, 2023 : (open)

March 1, 2023 : [in-person]

Speaker: Kent Vashaw, MIT

__Title__: An introduction to tensor-triangular geometry

__Abstract__: Since it was initiated in the early 2000s, tensor-triangular geometry has united disparate areas, including homotopy theory, algebraic topology, algebraic geometry, and representation theory, under a common general framework. I will give an introduction to the tools, goals, and methods of tensor-triangular geometry, with a particular emphasis on applications to representation theory, as well as discuss some recent results and conjectures on the noncommutative analogues of this geometry associated to finite tensor categories. This talk will include joint work with Dan Nakano and Milen Yakimov.

March 8, 2023 : (open)

March 22, 2023 :

Speaker: (CANCELLED) Allison Beemer, University of Wisconsin-Eau Claire

March 29, 2023 : (open)

April 5, 2023 : (open)

April 12, 2023 : [online]

Speaker: Soheil Memarian, University of Toronto

__Title__: Subvarieties of Complex Ball Quotients

__Abstract__: Many interesting moduli spaces admit complex ball uniformizations: moduli spaces of cubic threefolds, certain K3 surfaces, and symmetric cubic fourfolds with a specified group action, to name but a few. Motivated by these examples, we studied the hyperbolicity properties of complex ball quotients. In particular, we proved that all subvarieties of a complex ball quotient X are of general type provided that the cusps of X have sufficiently large depth.