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.
Spring 2024
| 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. |
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| 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). |
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| 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. |
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| 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). |
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| 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. |
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| 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. |
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| 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. |
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| 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. |
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| 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. |
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| 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.) |
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| 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. |
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| April 29 | Ilya Dumanski (MIT) | Coherent Satake category, Q-systems, and Feigin-Loktev fusion product Abstract: Feigin--Loktev fusion product is an operation between graded cyclic modules over the current Lie algebra. We explain how it is related to monoidal operations in two other categories: modules over the quantum loop group and the category of perverse coherent sheaves on the affine Grassmannian. This relation may explain similarities between these two categories (most notably the cluster patterns). Moreover, this allows us to tranfer some results from one of these categories into another. Namely, we establish the existence of short exact sequences in the coherent Satake category, which are analogs of Q-systems, and which conjecturally stand for cluster mutations in this category. Based on arXiv:2308.05268. |
Fall 2023
| Date | Location | Speaker | Abstract |
|---|---|---|---|
| Aug 28 | Harold Williams (USC) | Canonical Bases for Coulomb Branches
Abstract: Following work of Kapustin-Saulina and Gaiotto-Moore-Neitzke, one expects half-BPS line defects in certain supersymmetric 4d quantum field theories to form monoidal categories with rich structures (for example, a monoidal cluster structure in many cases). This talk explains an algebro-geometric approach to defining this category in the case of gauge theories with polarizable matter. This definition takes the form of a nonstandard t-structure on the dg category of coherent sheaves on the derived Braverman-Finkelberg-Nakajima space of triples. We refer to objects in this category as Koszul-perverse coherent sheaves, as this t-structure interpolates between the perverse coherent t-structure and certain t-structures appearing in the theory of Koszul duality (specializing to these in the case of a pure gauge theory and an abelian gauge theory, respectively). As a byproduct, this defines a canonical basis in the associated quantized Coulomb branch by passing to classes of irreducible objects. This is joint work with Sabin Cautis. |
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| holiday | |||
| Sept 11 | Joaquín Morage (UCLA) | Coregularity of Fano varieties
Abstract: Fano varieties are one of the three important blocks of algebraic varieties. In this talk we will review the history of Fano varieties Then, we will discuss some recent invariant, called the coregularity of the Fano variety that measures certain combinatorial complexity of the variety. We will explain how this invariant reflects in the geometry of the Fano variety and how it can help for the classification of such varieties. |
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| Sept 18 | |||
| Sept 25 | Daniele Garzoni (USC) | Hilbert's irreducibility theorem via random walks
Abstract: Let G be a linear algebraic group over a number field K (e.g., SLn), and let f: V → G be a cover of finite degree. Perform a long random walk on a Cayley graph of a fixed finitely generated subgroup of G(K) (e.g., SLn(Z)). Should you expect to hit elements g whose fibre f-1(g) is K-irreducible? After giving motivation, we will see that the answer is Yes, under some conditions on G and f. This represents a quantitative version of Hilbert's irreducibility theorem for linear algebraic groups. Joint work with Lior Bary-Soroker. |
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| Oct 2 | |||
| Oct 9 | Ben Elias (U Oregon) | Demazure operators for G(m,m,n) and quantum geometric Satake at a root of unity
Abstract: The reflection representation of the affine Weyl group in type A admits a q-deformation, which plays a role in quantum geometric Satake. Specializing q to a primitive 2m-th root of unity, one obtains a reflection representation of the complex reflection group G(m,m,n). In this way, a category of "singular Soergel bimodules" for G(m,m,n) is related to representations of quantum groups at roots of unity. To properly study these bimodules, additional algebraic background is needed. Associated to a reflection representation one has an action of a reflection group W on a polynomial ring R. A classic result of Demazure for Coxeter groups states that the inclusion of invariant polynomials into all polynomials is a Frobenius extension. A key tool in Demazure's proof is the nilCoxeter algebra, which acts on R by certain operators now called Demazure operators. Demazure proves that the Frobenius trace agrees with the Demazure operator associated to the longest element. These facts are what make the theory of Soergel bimodules tick. In joint work with Juteau and Young, we study the affine reflection representation of G(m,m,n). We prove the corresponding Frobenius extension result. We investigate an exotic nilCoxeter algebra with Demazure operators associated to elements of the affine Weyl group, acting on the ring R. When n=3, we determine which elements of the affine Weyl group are "longest elements" for G(m,m,n). |
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| Oct 16 | |||
| Oct 23 | Agustina Czenky (U Oregon) | Low rank symmetric fusion categories in positive characteristic
Abstract: In this talk, we look at the classification problem for symmetric fusion categories in positive characteristic. We recall the second Adams operation on the Grothendieck ring and use its properties to obtain some classification results. In particular, we show that the Adams operation is not the identity for any non-trivial symmetric fusion category. We also give lower bounds for the rank of a (non-super-Tannakian) symmetric fusion category in terms of the characteristic of the field. As an application of these results, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self-dual simple objects. |
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| Oct 30 | Pablo S. Ocal (UCLA) | Quantum symmetries via twisted tensor products and their Balmer spectrum
Abstract: In this talk I will present an approach to understand the representation theory of Hopf algebras. The ideas are inspired by the Balmer spectrum of a symmetric tensor triangulated category, a topological tool analogous to the usual spectrum of a commutative ring. Our main tool will be twisted tensor products, a noncommutative generalization of the coordinate ring of a variety. After introducing them, I will give some properties of their module categories, and set up how to understand their tensor-triangulated geometry using cohomology. |
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| Nov 6 | Peter Haine (UC Berkeley) | On Voevodsky’s reconstruction theorem
Abstract: In Grothendieck’s 1983 letter to Faltings, he made a number of conjectures about reconstructing schemes—initiating the field of anabelian geometry. The first conjecture is about reconstructing schemes from their étale fundamental groups. The second is about reconstructing schemes from their étale topoi. In 1990, Voevodsky proved that normal schemes of finite type over a finitely generated field of characteristic zero can be reconstructed from their étale topoi. However, this is not the optimal result; one expects to be able to reconstruct seminormal schemes from their étale topoi. In this talk, we’ll discuss joint work with Sebastian Wolf that generalizes Voevodsky’s theorem. In particular, in characteristic zero, we prove the optimal result. |
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| Nov 11 (Saturday!) |
Multiple Speakers | Southern California Algebraic Geometry Seminar
sites.google.com/site/socalags/fall-2023 |
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| Nov 13 | Tony Feng (UC Berkeley) | Mirror symmetry and the Breuil-Mezard Conjecture
Abstract: I will introduce the Breuil-Mezard Conjecture, which predicts the existence of hypothetical “Breuil-Mezard cycles” in the moduli space of representations of Galois groups of p-adic numbers. I will talk about a new approach to the Breuil-Mezard Conjecture, joint with Bao Le Hung, which is based on the intuition that it is analogous to homological mirror symmetry. In particular, we construct Breuil-Mezard cycles by combining certain instances of homological mirror symmetry with Bezrukavnikov-Mirkovic-Vilonen localization and a bit of the theory of quantum groups. This talk will be aimed at a general audience of algebraists. |
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| Nov 20 | early holiday | ||
| Nov 27 | Monica Vazirani (UC Davis) | Skeins on Tori
Abstract: We study skeins on the 2-torus and 3-torus via the representation theory of the double affine Hecke algebra of type A and its connection to quantum D-modules. As an application we can compute the dimension of the generic SLN- and GLN-skein module of the 3-torus for arbitrary N. This is joint work with Sam Gunningham and David Jordan. |
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See our friends at the Geometry seminar (M 2:00 pm) and the Combinatorics seminar (W 2:00 pm).