Syracuse Algebra Seminar

Fall 2025 Schedule:

Next Talk:

• October 24: Mahrud Sayrafi (McMaster University), Mutations of Exceptional Collections on Projective Varieties

  • Abstract: Utilizing recent joint work with Michael Brown, Souvik Dey, and Guanyu Li on computing extensions of bounded complexes of coherent sheaves on projective varieties, we present several applications involving computations in D^b(X), including manipulations of exceptional collections consisting of complexes and computing explicit monads based on them. We will also discuss related problems and conjectures from Helix theory in triangulated categories, particularly D^b(P^n).

Upcoming Talks:

• October 31: Mark E. Walker (University of Nebraska-Lincoln), TBA 

• November 7: Alexandra Seceleanu (University of Nebraska-Lincoln), TBA

• November 14: Antonia Kekkou (University of Utah), TBA

• November 21: Eamon Quinlan-Gallego (University of Illinois-Chicago), TBA

• November 28: No talk (Thanksgiving Break)

Past Talks: 

• September 5: Bhargavi Parthasarathy (Syracuse University), Homomorphisms of maximal Cohen-Macaulay modules over the cone of an elliptic curve

  • Abstract: Consider the ring R=k[[x,y,z]]/(f) where f=x3+y3+z3 with an algebraically closed field k and char(k) neq 3. In a 2002 paper, Laza, Popescu and Pfister used Atiyah's classification of vector bundles over elliptic curves to obtain a description of the maximal Cohen-Macaulay modules (MCM) over R. In particular, the matrix factorizations corresponding to rank one MCMs can be described using points in V(f). If M, N are rank one MCMs over R, then so is HomR(M,N). In this talk, I will discuss how the elliptic group law on f can be used to obtain the point in V(f) that describes the matrix factorization corresponding to HomR(M,N).

• September 12: Mengwei Hu (Yale University), On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities

  • Abstract: Kleinian singularities are quotients of C^2 by finite subgroups of SL_2(C). They are in bijection with the ADE Dynkin diagrams via the McKay correspondence. In this talk, I will introduce certain singular Lagrangian subvarieties in the minimal resolutions of Kleinian singularities that are related to the geometric classification of certain unipotent Harish-Chandra (g,K)-modules. The irreducible components of these singular Lagrangian subvarieties are P^1's and A^1's. I will describe how they intersect with each other through the realization of Kleinian singularities as Nakajima quiver varieties.

• September 26: Joe Beckmann (Syracuse University), What’s the most times that two curves in P^3 can intersect

  • Abstract: As is common in math, this easily-posed problem offers a wealth of possibilities. In spite of this, relatively few have attempted to answer it. Some recent work has been done by Hartshorne, Miró-Roig, and Ranestad with arithmetically Cohen-Macaulay curves over algebraically closed, characteristic zero fields. Less recently, Diaz and Giuffrida looked at reduced, irreducible, nondegenerate curves over algebraically closed fields. Fulton-MacPherson intersection theory is a powerful tool for solving enumerative geometry problems where the schemes of interest 'should' intersect; moreover, it works over any field. Less work has been done applying this theory to situations where the schemes generally shouldn't. In this talk, I will use Fulton-MacPherson intersection theory to show an upper bound for the number of times that a pair of reduced, irreducible, nondegenerate curves in P^3 over an arbitrary field can intersect. I will also discuss the generalizations of this question that my strategy can similarly answer.

• October 3: Ben Kaufman (Syracuse University), A characterization of Koszul algebras with coherent Koszul dual

  • Abstract: The extension conjecture asserts that for a finite dimensional algebra R and a simple module S, Ext^1(S,S)\neq 0 implies Ext^i(S,S)\neq0 for infinitely many i.  Using a result of Lenzing, we can show that the extension conjecture holds for Koszul algebras whose Koszul dual is coherent.  Unfortunately there are currently limited tools for determining which Koszul algebras have a coherent Koszul dual.  In this talk I will review the notions of Koszulness and coherence for graded (possibly non-connected) algebras and then discuss a recent result characterizing finite dimensional Koszul algebras with coherent Kozul dual in terms of linearity properties of syzygies.  

• October 10: No talk (Fall Break weekend)