Syracuse Algebra Seminar

Fall 2025 Schedule:

Next Talk:

• 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.

Upcoming Talks:

• October 3: Ben Kaufman (Syracuse University), TBA

• October 10: No talk (Fall Break weekend)

• October 24: Mahrud Sayrafi (McMaster University), TBA

• 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.