Spring 2022

Welcome to the AGNT Seminar at Rice!

Default Research Talk Format: Research Talk (50 minutes) + Questions (10 minutes)

Learning Group Information:
AG Group, HBH 227 3:00pm - 4:00pm, Details [here]
QA Group, HBH 442 3:00pm - 4:00pm, Details [here]

January 19, 2022 : QA Speaker [online]
Speaker: Marcelo Aguiar, Cornell University
Title: Lie theory relative to a hyperplane arrangement
Abstract: A result due to Joyal, Klyachko, and Stanley relates free Lie algebras to partition lattices. We will discuss the precise relationship and interpret the result in terms of the braid hyperplane arrangement. We will then extend this result to arbitrary (finite, real, and central) hyperplane arrangements, and do the same with several additional aspects of classical Hopf-Lie theory. The Tits monoid of an arrangement, and the notion of lune, play central roles in the discussion. This is part of joint work with Swapneel Mahajan.

January 26, 2022 : Learning Groups - see above

February 2, 2022 : AG Speaker [online]
Speaker: Sachi Hashimoto, Boston University
Title: Integral points on elliptic curves using p-adic Gross--Zagier
Abstract: Faltings' theorem is a non-explicit theorem that states that there are finitely many rational points on nice projective curves of genus at least 2 (and also implies there are finitely many integral points on genus 1 affine curves). The quadratic Chabauty method makes explicit some cases of Faltings' theorem. By studying p-adic heights of points of the Jacobian, we obtain locally analytic functions that cut out a finite set of p-adic points containing these rational or integral points on the curve. In this talk, I will explain how we can leverage information from p-adic Gross--Zagier formulas to compute these locally analytic functions without directly knowing any points on the Jacobian in the simplest case of rank 1 elliptic curves. These Gross--Zagier formulas relate analytic quantities, special values of p-adic L-functions, to invariants of algebraic cycles, the p-adic height and logarithm of Heegner points.

February 9, 2022 : Learning Groups - see above

February 16, 2022 : QA Speaker [online]
Speaker: Ryan Kinser, University of Iowa
Title: Quantum symmetries of algebras related to quivers [talk notes]
Abstract: The term "quantum symmetries" in this talk refers to Hopf actions of quantum groups, and related Hopf algebras, on other algebras. While quantum symmetries of field extensions, polynomial rings, and their deformation have been explored for several decades, only recently have quantum symmetries of algebras related to quivers received close attention. This talk will survey results on this topic spanning several works by combinations of authors including Berrizbeitia, Etingof, Oswald, Walton, and the speaker.

February 23, 2022 : Learning Groups - see above

March 2, 2022 : AG Speaker [online]
Speaker: Rosa Winter, King's College London
Title: Density of rational points on del Pezzo surfaces of degree 1
Abstract: Let X be an algebraic variety over an infinite field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. For example, is X(k) empty or not? And if it is not empty, is X(k) dense in X with respect to the Zariski topology? Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d ≥ 3, these are the smooth surfaces of degree d in P^d). For del Pezzo surfaces of degree at least 2 over a field k, we know that the set of k-rational points is Zariski dense provided that the surface has one k-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field k, even though we know that they always contain at least one k-rational point, we do not know if the set of k-rational points is Zariski dense in general.
      I will talk about density of rational points on del Pezzo surfaces, state what is known so far, and show a result that is joint work with Julie Desjardins, in which we give sufficient and necessary conditions for the set of k-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where k is finitely generated over Q.

March 9, 2022 : Learning Groups - see above

March 16, 2022 : QA Speaker [online]
Francesca Gandini, Kalamazoo College
Title: Constructive Invariant Theory in the exterior algebra (and beyond?)
Abstract: When we consider the action of a finite group on a polynomial ring, an invariant is a polynomial unchanged by the action. Noether's Degree Bound states that in characteristic zero the maximal degree of a minimal generating invariant is bounded above by the order of the group. In (commutative) invariant theory, Derksen showed that the generators of the Hilbert ideal can be found via elimination theory from the vanishing ideal of a subspace arrangement. We show that the same approach works over the exterior algebra and prove Noether's Degree Bound in this context. Our methods rely on a bound on the Castelnuovo-Mumford regularity of intersections of linear ideals in the exterior algebra, which we proved in previous work using tools from combinatorial representation theory. We also show a transference of bounds from the symmetric algebra to the exterior algebra using these tools. A bound on invariant skew polynomials in the exterior algebra also bounds some square-free invariants in the (-1)-skew algebra and motivates future investigations in the theory of skew polarization.

March 23, 2022 : Learning Groups - see above

March 30, 2022 : AG Speaker [in-person]
Speaker: Renee Bell, University of Pennsylvania
Title: Monodromy of Tamely Ramified Covers of Curves
Abstract: The étale fundamental group \pi_1^{et} in algebraic geometry formalizes an analogy between Galois theory and topology, extending our intuition to spaces in which loops, as defined traditionally, do not yield meaningful information. For a curve X over an algebraically closed field of characteristic 0, finite quotients of $\pi_1^{et}$ can be described solely in topological terms, but in characteristic p, dramatic differences and new phenomena have inspired many conjectures, including Abhyankar's conjectures. Let k be an algebraically closed field of characteristic p and let X be the projective line over k with three points removed. In joint work with Booher, Chen, and Liu, we show that for each prime p ≥ 5, there are families of tamely ramified covers with monodromy the symmetric group S_n or alternating group A_n for infinitely many n, producing these covers from moduli spaces of elliptic curves, and relating the fiber of these covers to the Markoff surface.

April 6, 2022 : Learning Groups - see above

April 13, 2022 : QA Speaker [in-person]
Speaker: Chelsea Walton, Rice University
Title: Modular categories: What are they, why care, and what have I done with them?
Abstract: In this talk, I will answer the questions in the title; the last question pertains to recent work with Robert Laugwitz (ArXiv: 2010.11872, 2202.08644).

April 20, 2022 : Colloquium Speaker (still at 3pm) [online]
Stephan Ramon Garcia, Pomona College
Title: Factorization lengths in numerical semigroups
Abstract: Numerical semigroups are simple combinatorial objects that lead to deep and subtle questions. We answer in one fell swoop virtually all asymptotic questions about factorization lengths in numerical semigroups. Surprisingly, this uses tools from complex, harmonic, and functional analysis, probability theory, algebraic combinatorics, and computer-aided design! Our results yield uncannily accurate predictions that agree with numerical computations, along with some totally unexpected byproducts. Partially supported by NSF Grants DMS-1800123 and DMS-2054002. Joint work with A. Böttcher, M. Omar, C. O'Neill, and undergraduate students G. Udell ('21), T. Wesley ('21), S. Yih ('18).

April 27, 2022 : AG Speaker [online]
Speaker: Carlos Rivera, University of Washington
Title: Persistence of the Brauer-Manin obstruction for cubic surfaces
Abstract: Let X be a smooth cubic hypersurface over a field k. Cassels and Swinnerton-Dyer have conjectured that X has a k-rational point as soon as it has a 0-cycle of degree 1 or, equivalently, as soon as X has a closed point of degree prime to 3. In 1974, D. Coray showed several results in this direction including, in the case of cubic surfaces, that the existence of a closed point of degree prime to 3 implies the existence of a closed point of degree 1, 4 or 10. In this talk, for k a global field and X a cubic surface, we will show that a Brauer-Manin obstruction to the existence of k-points on X will persist over every extension L/k with degree prime to 3. Therefore proving that the conjecture of Colliot-Thélène and Sansuc on the sufficiency of the Brauer-Manin obstruction for cubic surfaces implies the conjecture of Cassels and Swinnerton-Dyer in this case. This is joint work with Bianca Viray.

Fall 2021

Learning Group Information:
AG Group, 423 HBH, 3:00pm - 4:00pm, Details [here]
QA Group, 427 HBH, 3:00pm - 4:00pm, Details [here]

August 25, 2021 : Learning Groups [in-person]- see above

September 1, 2021 : AG Speaker [in-person]
Tony Várilly-Alvarado, Rice University
Title: Rational surfaces and locally recoverable codes
Abstract: Motivated by large-scale storage problems around data loss, a budding branch of coding theory has surfaced in the last decade or so, centered around locally recoverable codes. These codes have the property that individual symbols in a codeword are functions of other symbols in the same word. If a symbol is lost (as opposed to corrupted), it can be recomputed, and hence a code word can be repaired. Algebraic geometry has a role to play in the design of codes with locality properties. In this talk I will explain how to use algebraic surfaces birational to the projective plane to both reinterpret constructions of optimal codes already found in the literature, and to find new locally recoverable codes, many of which are optimal (in a suitable sense). This is joint work with Cecília Salgado and Felipe Voloch.

September 8, 2021 : Learning Groups - see above

September 15, 2021 : QA Speaker [in-person]
Sean Sanford, Indiana University
Title: Fusion Categories over Non–Algebraically Closed Fields
Abstract: Much of the early work on Fusion Categories was inspired by physicists desire for rigorous foundations of topological quantum field theory. One effect of this was that base fields other than the complex numbers were rarely considered, if at all. The relevant features of $\mathbb C$ that make the theory work are the fact that it is characteristic zero, and algebraically closed. This talk will focus on the interesting things that can be found when the algebraically closed requirement is removed. The content will be introductory, with lots of examples.

September 22, 2021 : Learning Groups - see above

September 29, 2021 : AG Speaker [in-person]
Alex Barrios, Carleton College
Title: Local data of elliptic curves with prescribed isogeny graph
Abstract: Given a rational elliptic curve, one can use Tate's algorithm to compute the following local data at each rational prime: the local conductor exponent, the Néron-Kodaira type, and the local Tamagawa number. This talk will investigate how the local data changes between two isogenous elliptic curves. We will focus on how the Néron-Kodaira type and the local Tamagawa number change under isogeny. This has been well-studied in the literature but remains open for special cases corresponding to a 2- or 3-isogeny. We aim to answer the remaining cases by explicitly classifying the local data of elliptic curves with a prescribed non-trivial isogeny graph. This is joint work with Darwin Chimarro, Manami Roy, Nandita Sahajpal, Bella Tobin, and Hanneke Wiersema.

October 6, 2021 : Learning Groups - see above at 3pm
Xingting Wang, Howard University (Special time at 4pm)
Title: Can a Zhang twist be a cocycle twist?
Abstract: In this talk, we will discuss two different twists of algebra structures in the literature, namely Zhang twist and 2-cocycle twist. We will provide sufficient conditions for them to coincide by introducing the notion of a twisting pair. We will explore twisting pairs for many well-known examples of Hopf algebras including Manin’s universal quantum group from quantum symmetry and FRT construction from a solution of the quantum Yang-Baxter equation. This is joint work with Hongdi Huang, Van C. Nguyen, Charlotte Ure and Kent B. Vashaw.

October 13, 2021 : QA Speaker [online]
Zachary Dell, Ohio State University
Title: A characterization of braided enriched monoidal categories
Abstract: Monoidal categories enriched in symmetric monoidal categories are well studied in the literature. In recent years attention has been given to the case where the enriching category is merely braided. In this talk I will give an overview of the existing results characterizing such categories in terms of braided oplax monoidal functors into the Drinfeld centers of ordinary monoidal categories and describe how this construction extends to an equivalence of 2-categories.

October 20, 2021 : Learning Groups - see above

October 27, 2021 : AG Speaker [online]
Elisa Bellah, University of Oregon
Title: Bounding Lifts of Markoff Triples mod p
Abstract: In 2016, Bourgain, Gamburd, and Sarnak proved that Strong Approximation holds for the Markoff surface in most cases. That is, the modulo p solutions to the equation x^2+y^2+z^2=3xyz are covered by the integer points for most primes p. In this talk, we will discuss how the algorithm given in the paper of Bourgain, Gamburd, and Sarnak can be used to obtain upper bounds on lifts of Markoff triples modulo p. We will also discuss ongoing work to improve these bounds on average by assuming the Markoff mod p graphs form an expander family. This is joint work with Elena Fuchs and Lynnelle Ye.

November 3, 2021 : Learning Groups - see above

November 10, 2021 : QA Speaker [online]
Agustina Czensky, University of Oregon
Title: On low rank odd-dimensional MTCs
Abstract: Classification of fusion categories by rank is an open (and seemingly out of reach) problem. For modular tensor categories (MTCs), Bruillard, Ng, Rowell and Wang showed in 2016 that there are finitely many of a fixed rank. Bruillard and Rowell also showed that odd-dimensional MTCs are pointed up to rank 11, which means that they can be classified by group-theoretical data. In this talk we will describe joint work with Julia Plavnik were we contribute to this classification problem. We will discuss some useful results for the classification of low-rank odd-dimensional MTCs, and expand on their classification for ranks 15 to 23.

November 17, 2021 : Learning Groups - see above

November 24, 2021 : (no activity)

December 1, 2021 : AG Speaker [online]
Humberto Diaz, Washington University
Title: The Hodge Conjecture for varieties with small Chow group
Abstract: The integral Hodge conjecture asserts that every integral Hodge class is the class of an algebraic cycle. It has been known to be false since the counterexamples of Atiyah and Hirzebruch. Since then, many other types of counterexamples have been produced. In this talk, I will report on recent work that gives yet another type of counterexample, this time arising from a variety whose Chow group of algebraic cycles is as small as possible.

December 8, 2021 : QA Speaker [online]
Florencia Orosz, University of Denver
Title: Tensor categories arising from the Virasoro algebra
Abstract: In this talk we will discuss the tensor structure associated with certain representations of the Virasoro algebra. In particular, we will show that there is a braided tensor category structure on the category of C1-cofinite modules for the Virasoro vertex operator algebras of arbitrary central charge. This talk is based on joint work with Thomas Creutzig, Cuibo Jiang, David Ridout and Jinwei Yang.