Seminar Information

Main Talk: Wednesdays at 11:00am, duration of 50min.

Pre-Talk: Wednesday at 10:30am, duration of 25min.

Zoom Link: 959 3082 9877 (Zoom ID). Recordings available for members of UC Davis.

Password: In numerics, the number of lines in any smooth cubic surface in projective 3-space.

Department Website: The seminar list in the department website.

Organizers: R. Casals, E. Carlsson and E. Gorsky.

Talks in Fall 2020

1. December 2: Morsifications and mutations by Pavlo Pylyavskyy (University of Minnesota).
   
   Slides of the Talk (Available Soon) And Abstract:
   
   We describe and investigate a connection between the topology of isolated singularities of plane curves and the mutation equivalence, in the sense of cluster algebra theory, of the quivers associated with their morsifications. This is joint work with Sergey Fomin, Eugenii Shustin, and Dylan Thurston.
   
   
   
   
2. November 18: Augmentations, Fillings, and Clusters by Daping Weng (Michigan State University).
   
   Slides of the Talk And Abstract:
   
   A Legendrian link is a 1-dimensional closed manifold that is embedded in real Euclidean 3-space and satisfies certain tangent conditions. Rainbow closures of positive braids are natural examples of Legendrian links. In the study of Legendrian links, one important task is to distinguish different exact Lagrangian fillings of a Legendrian link, up to Hamiltonian isotopy, in the symplectization. We introduce a cluster K2 structure on the augmentation variety of the Chekanov-Eliashberg dga for the rainbow closure of any positive braid. Using the Eckholm-Honda-Kalman functor from the cobordism category of Legendrian links to the category of dga’s, we prove that a big family of fillings give rise to cluster seeds on the augmentation variety of a positive braid closure, and these cluster seeds can in turn be used to distinguish non-Hamiltonian isotopic fillings. Moreover, by relating a cyclic rotation concordance on a positive braid closure with the Donaldson-Thomas transformation on the corresponding augmentation variety, we prove that other than a family of positive braids that are associated with finite type quivers, the rainbow closure of all other positive braids admit infinitely many non-Hamiltonian isotopic fillings. This is joint work with H. Gao and L. Shen.
   
   
   
   
3. November 4: Schubert calculus and Lagrangian correspondences by Iva Halacheva (Northeastern University).
   
   Slides of the Talk And Abstract:
   
   For a reductive algebraic group G, a natural question is to consider the inclusions of partial flag varieties H/Q into G/P and their pullbacks in equivariant cohomology, in terms of Schubert classes. We will look at the case of the symplectic and usual Grassmannian, and describe the pullback map combinatorially using puzzles. A generalization of this construction involves Maulik-Okounkov classes and cotangent bundles of the Grassmannians, with Lagrangian correspondences playing a key role. This is joint work with Allen Knutson and Paul Zinn-Justin.
   
   
   
   
4. October 28: Symplectic embeddings via algebraic positivity by Ben Wormleighton (Washington University in St. Louis).
   
   Slides of the Talk And Abstract:
   
   A fundamental and remarkably subtle question in symplectic geometry is “when does one symplectic manifold embed in another?”. There are two paths to approaching such problems: constructing embeddings, and obstructing embeddings; I will focus on the latter. Connections with algebraic geometry emerged from work of Biran and McDuff-Polterovich relating embeddings of disjoint unions of balls (i.e. ball packing problems) and the algebraic geometry of blowups of P², and this talk will describe work over the last few years continuing in the vein of employing algebraic techniques to study symplectic embedding problems. We describe a sequence of invariants of a polarised algebraic surface that obstruct symplectic embeddings, in many interesting cases sharply. Using this perspective we prove a combinatorial bound on the Gromov width of toric surfaces conjectured by Averkov-Nill-Hofscheier, and discuss related phenomena in algebraic positivity inspired by these symplectic findings.
   
   
   
   
5. October 21: Wall-crossing phenomena for Newton-Okounkov bodies by Laura Escobar (Washington University in St. Louis).
   
   Slides of the Talk And Abstract:
   
   A Newton-Okounkov body is a convex set associated to a projective variety, equipped with a valuation. These bodies generalize the theory of Newton polytopes. Work of Kaveh-Manon gives an explicit link between tropical geometry and Newton-Okounkov bodies. We use this link to describe a wall-crossing phenomenon for Newton-Okounkov bodies. This is joint work with Megumi Harada.
   
   
   
   
6. October 14: Parabolic Hilbert schemes and representation theory by José Simental Rodríguez (UC Davis).
   
   Slides of the Talk And Abstract:
   
   We explicitly construct an action of type A rational Cherednik algebras and, more generally, quantized Gieseker varieties, on the equivariant homology of the parabolic Hilbert scheme of points on the plane curve singularity given by an (n,m)-cusp, where m and n are coprime positive integers. We show that the representation we get is a highest weight irreducible representation and explicitly identify its highest weight. We will also place these results in the recent context of Coulomb branches and BFN Springer theory. This is joint work with Eugene Gorsky and Monica Vazirani.
   
   
   
   
7. October 7: Polytopes, wall crossings, and cluster varieties by Man-Wai Mandy Cheung (Harvard).
   
   Slides of the Talk And Abstract:
   
   Cluster varieties are log Calabi-Yau varieties which are a union of algebraic tori glued by birational "mutation" maps. Partial compactifications of the varieties, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties. However, it is not clear from the definitions how to characterize the polytopes giving compactifications of cluster varieties. We will show how to describe the compactifications easily by broken line convexity. As an application, we will see the non-integral vertex in the Newton Okounkov body of Gr(3,6) comes from broken line convexity. Further, we will also see certain positive polytopes will give us hints about the Batyrev mirror in the cluster setting. The mutations of the polytopes will be related to the almost toric fibration from the symplectic point of view. Finally, we can see how to extend the idea of gluing of tori in Floer theory which then ended up with the Family Floer Mirror for the del Pezzo surfaces of degree 5 and 6. The talk will be based on a series of joint works with Bossinger, Lin, Magee, Najera-Chavez, and Vienna.