Syracuse Algebra Seminar

Fall 2026 Schedule:

• September 4: Kory Pollicove, Syracuse University, Systems of Higher Homotopies and Koszul Morphisms

  • Abstract: Systems of higher homotopies have been an indispensable tool in the development of the homological theory of complete intersections. For example, the Eisenbud-Shamash resolution provides a way to transfer resolutions over complete intersections using  data of the higher homotopies. In joint work with Dorian Kalir and Ryan Watson, we generalize the definition of systems of higher homotopies in a way that recovers the Eisenbud-Shamash resolution, Bar resolution, and the Priddy resolution for Koszul algebras. In the talk, we will introduce this definition and see several of its applications to Koszul morphisms.

• September 11: Josh Pollitz, Syracuse University, Derived differential operators 

  • Abstract: A classical construction of Grothendieck associates to a map of commutative rings its corresponding ring of differential operators. There have been a wealth of applications of these operators in a wide variety of areas, especially when the target is smooth over a field of characteristic zero. In this talk, I will discuss joint work with Jeffries, Mallory, Miller, and Quinlan-Gallego where we provide a dg enhancement of the derived functors of Smith and Van den Bergh that replaces the ordinary ring of differential operators with a dg algebra of derived differential operators. I will discuss how this dg algebra fits into a framework of Koszul duality, and, in the setting of F-finite rings, I will explain a derived analog of a classical interpretation of the ordinary ring of differential operators in terms of the Frobenius endomorphism.

• September 18: No seminar, campus holiday

• September 25: Kent Vashaw, SUNY-Albany, Quantum-symmetric equivalence via Manin's universal quantum groups

  • Abstract: Quantum-symmetric equivalence was introduced in joint work with Huang--Nguyen--Ure--Veerapen--Wang as a tool to lift ring-theoretical and homological properties in noncommutative algebra to the context of Hopf algebras and tensor categories. Two N-graded algebras are called quantum-symmetrically equivalent if there is a monoidal equivalence between the categories of comodules for their associated universal quantum groups (in the sense of Manin) which sends one algebra to the other. We prove that if two N-homogeneous algebras have equivalent graded module categories, then they are quantum-symmetrically equivalent; that (building on work of Raedschelders—Van den Bergh) the class of Koszul Artin—Schelter regular algebras of a fixed global dimension form a single quantum-symmetric equivalence class; and that quantum-symmetric equivalence classes are controlled by a cogroupoid that can be defined via generators and relations. This talk focuses on work that is joint with Hongdi Huang, Van Nguyen, Charlotte Ure, Padmini Veerapen, and Xingting Wang. 

• October 2: Aryaman Maithani, University of Utah, Homological properties of invariant rings of permutation groups

  • Abstract: Consider the action of a finite group G on the polynomial ring S := k[x₁, ..., xₙ] by degree-preserving automorphisms, and consider the ring of invariants S^G. It is well-known that the functor of invariants is not exact in positive characteristic and several key isomorphisms break down in this case. Nevertheless, we show that if G acts by permuting variables and the characteristic of S is not two, then the following are independent of k, and may thus be computed using characteristic zero techniques: the a-invariant of S^G, the property of S^G being quasi-Gorenstein, and the Hilbert functions of the local cohomology modules Hⁿ(S)^G and Hⁿ(S^G); moreover, these Hilbert functions coincide. Additionally, we determine the ring of k-linear differential operators on S^G, and show that each differential operator lifts to one over ℤ.

• October 16: Timothy Tarter, James Madison University, Poncelet's Closure Theorem & Certain Generalizations

  • Abstract: In 1812, Poncelet posited the question: "given two generic plane curves, one properly contained inside the other, is it always possible to construct an in-and-circumscribed (non-convex) polygon on the two curves?" A decade later, in 1822, Poncelet proved that, independent of the starting point of the construction, there were either infinitely many such polygons, or zero. Later, in 1828, Jacobi presented an analytic proof of this theorem as an application of the theory of elliptic functions, and in the 1850's, Cayley found explicit conditions to find such "Poncelet" curves. More recently, in 1977, Griffiths and Harris generalized Poncelet's theorem to quadric surfaces in space expanding on Jacobi's application of the theory of elliptic functions. In this talk, we will discuss the proof of both the classical Poncelet theorem and its generalization as well as some potential directions of future research in arithmetic geometry following the work of Chipalkatti (2017) and Wang (2025).

• October 23: Maria Akter, University of Alabama, m-adic Continuity of Frobenius Splitting Ratio

  • Abstract: The study of singularities under perturbation is classical, with origins dating back to the 50's and 60's through the work of Samuel and Hironaka. In this talk, we introduce the theory of F-singularities in prime characteristic rings and examine continuity problems in relation to perturbation theory. Our main result characterizes the continuity of the F-splitting ratio.

• November 6: Joe Waldron, Michigan State University

  • Abstract: TBA

• November 13: Uli Walther, Purdue University

  • Abstract: TBA

• November 20: TBA

  • Abstract: TBA

• December 4: TBA

  • Abstract: TBA