Algebra & Discrete Math and Algebraic Geometry seminars at UC Davis
This is an informal webpage for Algebra & Discrete Math and
Algebraic Geometry seminars at UC Davis.
You can access the official semiar webpages by clicking at the links above.
Due to the coronavirus pandemic, the seminars run on Zoom at 1-2pm (Pacific time) on Mondays and Wednesdays.
For Zoom link and password please contact
Jose Simental Rodriguez or
Eugene Gorsky .
Program
(AG) 4/1 : Speaker: Niklas Garner (UC Davis)
Title: Coulomb branches and plane curve singularities
Abstract: Recent constructions inspired by the physics of 3d N=4 gauge theories provide a
plethora of interesting objects in geometric representation theory. In this talk I will discuss an
application of one such construction to the HOMFLY-PT homology of links coming from plane curve singularities.
Notes: notes
(ADM) 4/6 : Speaker: Amzi Jeffs (University of Washington)
Title: Sunflower Theorems and Convex Codes
Abstract: In the 1970s neuroscientists O'Keefe and Dostrovsky made a groundbreaking experimental observation: neurons called "place cells" in a rat's hippocampus were active in a convex subset of the animal's environment, and thus encoded a cognitive map of the environment. In 2013, Curto et al. introduced a mathematical model of place cells and began the task of classifying the combinatorial "convex codes" that arise in this model. A key question in this field is the following: given a convex code on n neurons, what is the smallest dimension of Euclidean space in which one can find a realization using convex open sets? We will introduce some new discrete geometry theorems in the spirit of Helly and Tverberg. We will use these theorems to prove that, surprisingly, the smallest dimension mentioned above may be exponentially large in terms of the number of neurons n.
Notes: notes
(AG) 4/8 : Speaker: Vasily Krylov (MIT)
Title: Generalized slices in the affine Grassmanian for minuscule cocharacters and applications
Abstract: In the paper "Coulomb branches of
3d N=4 quiver gauge theories and slices in the affine Grassmannian" authors defined so-called generalized transversal slices in the affine Grassmannian of a reductive group
G. These varieties depend on a pair of cocharacters \(\lambda,\mu\) of G such that lambda is dominant and \(\mu\leq \lambda\)
. In types ADE these varieties are isomorphic to Coulomb branches of the corresponding framed quiver gauge theories.
Symplectic duality predicts that these varieties should be isomorphic to affine spaces in the case when
\(\lambda\) is minuscule and \(\mu\) lies in the W-orbit of \(\lambda\). Hiraku Nakajima proved this fact in type A using the
identification of generalized slices with bow varieties (not published). We will give the proof for any reductive group
G and will also describe the standard Poisson structure on slices. Time permitting we will then discuss applications
of this result including coverings of some convolution diagrams of generalized slices by affine spaces
(trying to answer a question of Hiraku Nakajima and Michael Finkelberg). This is a work in progress with Ivan Perumov.
Notes: notes
(ADM) 4/13 : Speaker: Siddharth Venkatesh, UCLA
Title: Algebraic Geometry and Representation Theory in the Verlinde category in positive characteristic
Abstract: In this talk I will give an overview of symmetric tensor categories, with a focus on the Verlinde category, a universal base for semisimple symmetric tensor categories in positive characteristic. I will give an explicit construction of the Verlinde category, describe its fusion rules and motivate its study via some representation theoretic constructions. In particular, I will describe an elementary construction in the Verlinde category that gives us examples of exceptional Lie superalgebras in charactetistic p.
Subsequently, I will describe the algebro-geometric properties of finitely generated commutative rings in the Verlinde category, with an emphasis on describing the Frobenius images of simple summands of these rings, and then use this to construct a correspondence between affine group schemes in this category and Harish-Chandra pairs, i.e., pairs of ordinary affine group schemes and Lie algebras in the Verlinde category with compatible adjoint actions.
Notes: notes
(AG) 4/15 : No seminar
(ADM) 4/20 : Speaker: Noah Snyder, Indiana University
Title: The exceptional knot polynomial
Abstract: Many Lie algebras fit into discrete families like \(GL_n, O_n, Sp_n\). By work of Brauer, Deligne and others, the corresponding
planar algebras fit into continuous familes \(GL_t\) and \(OSp_t\). A similar story holds for quantum groups, so we
can speak of two parameter families \((GL_t)_q\) and \((OSp_t)_q.\) These planar algebras are the ones attached to
the HOMFLY and Kauffman polynomials. There are a few remaining Lie algebras which don't fit into any of the classical
families: \(G_2, F_4, E_6, E_7,\) and \(E_8.\) By work of Deligne, Vogel, and Cvitanovic, there is a conjectural
1-parameter continuous family of planar algebras which interpolates between these exceptional Lie algebras.
Similarly to the classical families, there ought to be a 2-paramter family of planar algebras which introduces a
variable \(q\), and yields a new exceptional knotpolynomial. In joint work with Scott Morrison and Dylan Thurston,
we give a skein theoretic description of what this knot polynomial would have to look like. In particular, we
show that any braided tensor category whose box spaces have the appropriate dimension and which satisfies some mild
assumptions must satisfy these exceptional skein relations.
Notes: notes
(AG) 4/22 : Speaker: Bernd Sturmfels, UC Berkeley and MPI Leipzig
Title: Theta surfaces
Abstract: A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincare showed that theta surfaces are precisely the surfaces of double translation, i.e. obtained as the Minkowski sum of two space curves in two different ways. These curves are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.
Notes: notes
(ADM) 4/27 : Speaker: Victor Ostrik, University of Oregon
Title: Incompressible tensor categories
Abstract: This talk is based on joint work with Benson and Etingof. We say that a symmetric tensor category is incompressible if there is no symmetric tensor functor from this category to a smaller tensor category. Our main result is a construction of new examples of incompressible tensor categories in positive characteristic.
Notes: notes
(AG) 4/29 : Speaker: Peter Crooks, Northeastern University
Title: Poisson slices and Hessenberg varieties
Abstract: Hessenberg varieties constitute a rich and well-studied class of closed subvarieties in the flag variety. Prominent examples include Grothendieck-Springer fibres, the Peterson variety, and the projective toric variety associated to the Weyl chambers. These last two examples belong to the family of standard Hessenberg varieties, whose total space is known to be a log symplectic variety. I will exhibit this total space as a Poisson slice in the log cotangent bundle of the wonderful compactification, thereby building on Balibanu's recent results. This will yield a canonical closed embedding of each standard Hessenberg variety into the wonderful compactification.
This represents joint work with Markus Roeser.
Notes: notes
(ADM) 5/4 : Speaker: Federico Castillo, University of Kansas
Title: Todd Class of Permutohedral Variety
Abstract: Berline and Vergne described a precise relation between the number of integer points of a polytope and the volumes of its faces. This relation can be seen as a higher dimensional analogue of Pick's theorem. We study the specific case of the permutohedron via the connection with toric varieties. This is joint work with Fu Liu.
Notes: notes
(AG) 5/6 : Speaker: Maria Angelica Cueto, Ohio State University
Title: Combinatorics and real lifts of bitangents to tropical quartic curves
Abstract: Smooth algebraic plane quartics over algebraically closed fields have 28 bitangent lines. By contrast, their tropical counterparts have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system associated to an effective tropical theta characteristic on the tropical quartic curve.
In this talk, I will discuss recent work joint with Hannah Markwig ( arxiv:2004.10891 ) on the combinatorics of these bitangent classes and its connection to the number of real bitangents to real smooth quartic curves characterized by Pluecker. We will see that they are tropically convex sets and they come in 39 symmetry classes. The classical bitangents map to specific vertices of these polyhedral complexes, and each tropical bitangent class captures four of the 28 bitangents. We will discuss the situation over the reals and show that each tropical bitangent class has either zero or four lifts to classical bitangent defined over the reals, in agreement with Pluecker's classification.
Notes: notes
(ADM) 5/11 : Speaker: Dimitar Grantcharov, University of Texas
Title: Bounded modules of direct limit Lie algebras
Abstract: In this talk we will discuss recent results on the category of weight modules with
bounded sets of weight multiplicities of the direct limit Lie algebras
sl(infty), o(infty) and sp(infty). Classification of the simple objects and properties of the
category will be provided. This is a joint work with I. Penkov.
Notes: notes
(ADM) 5/6 : Speaker: Nicolle Gonzalez, UCLA
Title: sl(n)-homology theories obstruct ribbon concordance
Abstract: In a recent result, Zemke showed that a ribbon concordance between two knots induces an
injective map between their corresponding knot Floer homology. Shortly after, Levine and Zemke proved the
analogous result for ribbon concordances between links and their Khovanov homology. In this talk I will explain
joint work with Caprau-Lee-Lowrance-Sazdanovic and Zhang where we generalize this construction further to show that a
link ribbon concordance induces injective maps between sl(n)-homology theories for all n\ge 2.
Notes: notes
(ADM) 5/18 : Speaker: Arseniy Sheydvasser, Graduate Center at CUNY
Title: Algebraic Invariants of Hyperbolic 4-Orbifolds
Abstract: Given an arithmetic subgroup
G of the isometry group of hyperbolic n-space H^n, one can consider the orbifold H^n/G.
Hyperbolic 2- and 3-orbifolds are reasonably well-understood; for example, hyperbolic 3-orbifolds
correspond to orders of split quaternion algebras and there are algorithms that make use of this structure to
compute geometric invariants of the orbifolds such as their volume, numbers of cusps, and fundamental groups.
However, already hyperbolic 4-orbifolds belong to untamed wilds. We shall examine this frontier by introducing a
class of arithmetic groups that have many of the same properties as the Bianchi groups and for which we can compute
some geometric invariants of the orbifolds via algebraic invariants of rings with involution.
Notes: notes
(AG) 5/20 : Speaker: Sean Griffin, University of Washington
Title: Springer fibers, rank varieties, and generalized coinvariant rings
Abstract: Springer fibers are a family of varieties with the remarkable property that their cohomology rings
R_lambda have the structure of a symmetric group module, even though there is no
S_naction on the varieties themselves. This is one of the first examples of a geometric representation. In the 80s, De Concini and Procesi proved that
R_lambda has another geometric description as the coordinate ring of the scheme-theoretic intersection of a nilpotent orbit closure with diagonal matrices. This led them to an explicit presentation for
R_lambda in terms of generators and relations, which was further simplified by Tanisaki. In this talk, we present a
generalization of this work to the coordinate ring of a scheme-theoretic intersection of Eisenbud-Saltman rank
varieties. We then connect these coordinate rings to the generalized coinvariant rings recently introduced by Haglund,
Rhoades, and Shimozono in their work on the Delta Conjecture from Algebraic Combinatorics. We then give combinatorial
formulas for the Hilbert series and graded Frobenius series of our coordinate rings generalizing those of
Haglund-Rhoades-Shimozono and Garsia-Procesi.
Notes: notes
(ADM) 5/26 : Speaker: Digjoy Paul, IMSC Chennai
Title: New approaches to the restriction problem
Abstract: Given an irreducible polynomial representation
\(W_n\) of the general linear group \(GL_n\), we can restrict it to the representations of the symmetric group
\(S_n\) that seats inside \(GL_n\) as a subgroup. The restriction problem is to find a combinatorial interpretation
of the restriction coefficient: the multiplicity of an irreducible
\(S_n\) modules in such restriction of \(W_n\). This is an open problem (see OPAC 2021!) in algebraic combinatorics.
In Polynomial Induction and the Restriction Problem, we construct the polynomial induction functor, which is the
right adjoint to the restriction functor from the category of polynomial representations of
\(GL_n\) to the category of representations of \(S_n\). This construction leads to a representation-theoretic proof of
Littlewood's Plethystic formula for the restriction coefficient.
Character polynomials have been used to study characters of families of representations of symmetric groups
(see Garsia and Goupil), also used in the context of FI-modules by Church, Ellenberg, and Farb (see FI-modules and
stability for representations of symmetric groups).
In Character Polynomials and the Restriction Problem, we compute character polynomial for the family of restrictions
of \(W_n\) as \(n\) varies. We give an interpretation of the restriction coefficient as a moment of a certain character
polynomial. To characterize partitions for which the corresponding Weyl module has non zero
\(S_n\)-invariant vectors is quite hard. We solve this problem for partition with two rows, two columns, and for
hook-partitions.
Notes: notes
(AG) 5/27 : Speaker: Gurbir Dhillon, Stanford University
Title: Steinberg-Whittaker localization and affine Harish--Chandra bimodules
Abstract: A fundamental result in representation theory is Beilinson--Bernstein localization, which identifies the representations of a reductive Lie algebra with fixed central character with D-modules on (partial) flag varieties. We will discuss a localization theorem which identifies the same representations instead with (partial) Whittaker D-modules on the group. In this perspective, representations with a fixed central character are equivalent to the parabolic induction of a `Steinberg' category of D-modules for a Levi.
Time permitting, we will explain how these methods can be used to identify a subcategory of Harish--Chandra bimodules for an affine Lie algebra and prove that it behaves analogously to Harish--Chandra bimodules with fixed central characters for a reductive Lie algebra. In particular, it contains candidate principal series representations for loop groups. This a report on work with Justin Campbell.
Notes: notes
(ADM) 6/1 : Speaker: Iva Halacheva, Northeastern University
Title: Self-dual puzzles in Schubert calculus branching
Abstract: In classical Schubert calculus, Knutson and Tao’s puzzles are a
combinatorial tool that gives a positive rule for expanding the product of two Schubert classes
in equivariant cohomology of the (type A) Grassmannian. I will describe a positive rule that uses
self-dual puzzles to compute the restriction of a Grassmannian (type A) Schubert class to the symplectic
(type C) Grassmannian in equivariant cohomology. The proof uses the machinery of quantum integrable systems.
I will also discuss a generalization in which the Grassmannians are upgraded to their cotangent bundles and Schubert
classes—to Segre-Schwartz-MacPherson classes. The resulting construction involves Lagrangian correspondences and
produces a generalized puzzle rule with a geometric interpretation. This is joint work with Allen Knutson and
Paul Zinn-Justin.
Notes: notes
(AG) 6/3 : Speaker: Pablo Boixeda Alvarez, MIT
Title: On the center of the small quantum group
Abstract: We compute the
\(G\)-invariant part of the center of the small quantum group at a regular block in terms of the cohomology of an equivalued affine Springer fiber.
Notes: notes