Algebra Seminar
(Mon 3:30-4:30pm, Kap 245)

Organizer: Cris Negron

Topics include (but are not limited to):

Algebra, representation theory, algebraic and arithmetic geometry, homotopy theory, and mathematical physics.

Date Location Speaker Abstract
January colloquia
Feb 5 Sean Eberhard (Queen's Univ Belfast) Diameter bounds for finite classical groups generated by transvections

Abstract: The diameter of a group G with respect to a symmetric generating set X is the smallest integer d such that every element of G is the product of at most d elements of X. A well-known conjecture of Babai predicts that every nonabelian finite simple group G has diameter (log |G|)^O(1) with respect to any generating set. This is known to be true for bounded-rank groups of Lie type (Helfgott; Pyber--Szabo; Breuillard--Green--Tao), but the conjecture is wide open for high-rank groups. There has bee a good deal of progress recently for generating sets containing either special elements or random elements. In this talk I will outline the proof that the conjecture holds for the classical groups SL_n(q), Sp_{2n}(q), SU_n(q) and any generating set containing a transvection. The proof is based essentially on (a) the positive resolution of Babai's conjecture in bounded rank and (b) a result of Kantor classifying finite irreducible linear groups generated by transvections.
Feb 12 Claire Levaillant (USC) Reducibility criterion for the Cohen-Wales representation of the Artin group of type E6

Abstract: We introduce tangles of type En and relations on these tangles. We use this novel diagrammatic algebra to build a representation of the Birman-Murakami-Wenzl algebra of type E6. As a representation of the Artin group of type E6, this representation is equivalent to the faithful representation of Cohen and Wales introduced by them in 2000 as a generalization to the Artin groups of the faithful Lawrence-Krammer representation of the braid group. The latter representation became famous as it is the only representation of the braid group that is known to be faithful. We use our newly built representation to find a reducibility criterion for the Cohen-Wales representation of the Artin group of type E6. Our method generalizes to types E7 and E8. This talk is given to honor the memory of my late Ph.D. advisor David Wales (1939-2023).
Feb 19 off
Feb 26 Ben Davison (Edinburgh) Okounkov's conjecture via BPS Lie algebras

Abstract: Given an arbitrary finite quiver Q, Maulik and Okounkov defined a new Yangian-style quantum group. It is built from the FRT formalism and their construction of R matrices on the cohomology of Nakajima quiver varieties, via the stable envelopes that they also defined. Just as in the case of ordinary Yangians, there is a Lie algebra g_Q inside their new algebra, and the Yangian is a deformation of the current algebra of this Lie algebra.
     Outside of extended ADE type, numerous basic features of g_Q have remained mysterious since the outset of the subject, for example, the dimensions of the graded pieces. A conjecture of Okounkov predicts that these dimensions are given by the coefficients of Kac's polynomials, which count isomorphism classes of absolutely indecomposable Q-representations over finite fields. I will explain a recent proof, with Botta, of this conjecture. By proving that the Maulik-Okounkov Lie algebra g_Q is isomorphic to certain BPS Lie algebras, we prove Okounkov's conjecture, as well as essentially determining the isomorphism class of g_Q, thanks to recent joint work of myself with Hennecart and Schlegel Mejia.
March 4 Fabien Morel (München) Some properties and computations of cellular A^1-homology of smooth projective rational k-schemes

Abstract: In this talk I will discuss some computations related to the study A^1-connected smooth projective schemes X over a fixed field k (mostly perfect). For a smooth projective k-scheme, A^1-connected means A^1-chain connected, and rational smooth projective k-schemes are A^1-connected on nice field (char 0 for instance) as we observed some time ago with A. Asok. After briefly introducing the cellular A^1-homology sheaves (with Anand Sawant) I will explain why these are good for, for instance by understanding the top dimensional one. I will explain some of the basic facts and difficulties involved concerning strictly A^1-invariant sheaves, and give some examples and computations of those. In particular I will explain the case of smooth projective rational surfaces, where Poincare' duality holds (over a perfect field).
March 11 off
March 18 Matthew Harper (UC Riverside) The Heisenberg and Weyl categories

Abstract: Building on work of Quinn, the Weyl category is an abstract monoidal category whose generating objects Q_+ and Q_- act by induction and restriction on the sum (over $n$) of the categories of Temperley-Lieb modules. In analogy with the Heisenberg category of Khovanov, Licata, and Savage, the Grothendeick group of our category is closely related to the Weyl algebra. In describing the Weyl category, we will point out some key structural differences between it and the Heisenberg category, specifically the need for 2-categorical structure and for idempotent completion. The full description of the Weyl category is based on joint work with Peter Samuelson.
March 25 Jason Fulman (USC) Cycle index methods in algebra and number theory

Abstract: It has been said that most mathematicians have just a few tricks. In this talk, I reveal my favorite trick: cycle index generating functions. We discuss this for the symmetric group and for finite groups of Lie type. We point out connections with algebra and number theory.
April 1 Nick Rozenblyum (Toronto) Hamiltonian flows in (relative) Calabi-Yau categories

Abstract: I will describe a general categorical approach to constructing Hamiltonian actions on moduli spaces from categorical data. In particular cases, this specializes to give a "universal" Hitchin integrable system, the Calogero-Moser system, and the Hamiltonian action of necklace Lie algebras on Nakajima quiver varieties. A key input is a description of deformation of Calabi-Yau structures and its relation to a cyclic version of the Deligne conjecture, which is of independent interest. This is joint work with Brav.
April 8 Tom Gannon (UCLA) Quantization of the Ngô Morphism

Abstract: We will discuss work, joint with Victor Ginzburg, on a quantization (non-commutative deformation) of the Ngô morphism, a morphism of group schemes which plays a key role in Ngô’s proof of the fundamental lemma in the Langlands program. We will also discuss how the tools used to construct this morphism can be used to prove conjectures of Ben-Zvi—Gunningham, which predict that this morphism gives “spectral decomposition” of DG categories with an action of a reductive group over the coarse quotient of a maximal Cartan subalgebra by the affine Weyl group.
April 15 Dan Halpern-Leistern (Cornell) The noncommutative minimal model program

Abstract: There are many situations in which the derived category of coherent sheaves on a smooth projective variety admits a semiorthogonal decomposition that reflects something interesting about its geometry. I will present a new unifying framework for studying these semiorthogonal decompositions using Bridgeland stability conditions. Then, I will formulate some conjectures about canonical flows on the space of stability conditions that imply several important conjectures on the structure of derived categories, such as the D-equivalence conjecture and Dubrovin's conjecture.
April 17 KAP 414 [Note the unusual date and location] Yuri Bakhturin (Memorial U) Generalizing classical correspondences between groups and Lie algebras

Abstract: There are different ways of building a Lie algebra of a group. One approach, due to A. I. Malcev, allows to introduce the structure of a Lie algebra on the (divisible nilpotent) group itself. Conversely, one can build a group on a (nilpotent) Lie algebra, using Baker-Campbell-Hausdorf formula. As a result, we have "hybrid" object, one author called them "groupalgebras" or "algebragroups". We observe that this approach works in a far wide setting of arbitrary nilpotent algebras. We also have new applications in the case of the classical Malcev correspondence. (This is joint work with A. Olshanskii.)
April 22 Dan Rogalski (UCSD) Homological integrals for weak Hopf algebras

Abstract: The integral is an important structure in a finite-dimensional Hopf algebra. Lu, Wu, and Zhang generalized this to define a homological integral for any Artin-Schelter Gorenstein Hopf algebra. This homological integral has many applications in the study of Hopf algebras of small GK-dimension. A weak Hopf algebra is a generalization of a Hopf algebra in which the comultiplication does not necessarily preserve the unit. Weak Hopf algebras arise naturally in the study of tensor categories, for example. In this work we show how to define a homological integral for an AS Gorenstein weak Hopf algebra, and that it has good properties. This is joint work in progress with Rob Won and James Zhang.
April 29 Ilya Dumanski (MIT)

Previous semester

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See our friends at the Geometry seminar (M 2:00 pm) and the Combinatorics seminar (W 2:00 pm).