Algebra Seminar
(Mon 3:304:30pm, 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 wellknown 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 boundedrank groups of Lie type (Helfgott; PyberSzabo; BreuillardGreenTao), but the conjecture is wide open for highrank 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 CohenWales 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 BirmanMurakamiWenzl 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 LawrenceKrammer 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 CohenWales 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 (19392023). 

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 Yangianstyle 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 Qrepresentations over finite fields. I will explain a recent proof, with Botta, of this conjecture. By proving that the MaulikOkounkov 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^1homology of smooth
projective rational kschemes Abstract: In this talk I will discuss some computations related to the study A^1connected smooth projective schemes X over a fixed field k (mostly perfect). For a smooth projective kscheme, A^1connected means A^1chain connected, and rational smooth projective kschemes are A^1connected on nice field (char 0 for instance) as we observed some time ago with A. Asok. After briefly introducing the cellular A^1homology 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^1invariant 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 TemperleyLieb 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 2categorical 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) CalabiYau 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 CalogeroMoser system, and the Hamiltonian action of necklace Lie algebras on Nakajima quiver varieties. A key input is a description of deformation of CalabiYau 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 (noncommutative 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 BenZvi—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 HalpernLeistern (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 Dequivalence 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 BakerCampbellHausdorf 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 finitedimensional Hopf algebra. Lu, Wu, and Zhang generalized this to define a homological integral for any ArtinSchelter Gorenstein Hopf algebra. This homological integral has many applications in the study of Hopf algebras of small GKdimension. 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) 
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See our friends at the Geometry seminar (M 2:00 pm) and the Combinatorics seminar (W 2:00 pm).