Fall 2026

Title: Symplectic duality and bow varieties 

Abstract: Symplectic duality is the well-supported idea that symplectic algebraic varieties ought to come in pairs, with the properties of each variety in the pair deeply linked. One mechanism for constructing such pairs comes from quiver gauge theories. For gauge theories from type A quivers, the corresponding symplectic varieties are known as bow varieties and contain as examples flag varieties, Kleinian singularities, and certain moduli spaces of sheaves on surfaces. I will discuss some generalities about symplectic duality, with an aim toward explaining joint work with Tommaso Botta proving a certain enumerative algebro-geometric duality statement for bow varieties.

Title: Symplectic duality and bow varieties 

Abstract: Symplectic duality is the well-supported idea that symplectic algebraic varieties ought to come in pairs, with the properties of each variety in the pair deeply linked. One mechanism for constructing such pairs comes from quiver gauge theories. For gauge theories from type A quivers, the corresponding symplectic varieties are known as bow varieties and contain as examples flag varieties, Kleinian singularities, and certain moduli spaces of sheaves on surfaces. I will discuss some generalities about symplectic duality, with an aim toward explaining joint work with Tommaso Botta proving a certain enumerative algebro-geometric duality statement for bow varieties.

Abstract:  A Stanley-Reisner ring is a quotient of a polynomial ring by an ideal generated by square-free monomials; these rings are in bijective correspondence with simplicial complexes, and there is a profound relationship between the algebraic properties of the ring and the topology of the simplicial complex. Hochster's formula for local cohomology is one of many famous theorems built on this relationship. In this week's talk, we will examine the mechanisms underlying Hochster's formula. Next week, we will introduce the notion of a "t-Stanley-Reisner ring" and adapt Hochster's formula to this new setting (joint work with Mel Hochster). 

Abstract:  Let (V,t) be a discrete valuation ring with fraction field L and residue field K. A t-Stanley Reisner ring is the image of a Stanley-Reisner ring over V under the evaluation map sending x_0 to t; these are quotients of polynomial rings by ideals whose generators are square free "t-monomials," i.e. monomials in the variables and the uniformizing parameter t. There is a correspondence between t-Stanley Reisner rings and simplicial complexes with a distinguished vertex v_0. As is the case for classical Stanley-Reisner rings, this correspondence induces many deep connections between algebra and topology. The theory is particularly rich and nuanced when the simplicial complex has p-torsion, where (V,t) is a ring of mixed characteristic p. In this talk, we will discuss the adaptation of Hochster's formula to t-Stanley-Reisner rings. Instead of using simplicial cohomology over a fixed coefficient ring, we will need to consider "cellular sheaf cohomology." The cellular sheaf assigns one of L, V, or K to each face, depending on its proximity to the distinguished vertex v_0. 

Abstract:  In this talk, we will introduce an approach to the question of extendability of projective varieties via degeneration to ribbons. As an application of these methods, we will give a new proof of optimal results on the extendability of general K3 surfaces, classification of Fano threefolds and Mukai varieties, and the irreducibility of their Hilbert schemes. If time permits, we will also discuss some new results on the extendability of canonical surfaces and Calabi-Yau threefolds, and their impact on classification of Fano fourfolds.  

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