RUTGERS ALGEBRA SEMINAR - Spring 2025
Wednesdays at 2:00-3:00 PM in H705

no seminar January 19 as we start in remote mode
26 Jan Weihong Xu (Rutgers) "Quantum $K$-theory of Incidence Varieties" (remote)
 2 Feb Rudradip Biswas (Manchester) "Cofibrant objects in representation theory" (remote) 
 9 Feb Chuck Weibel (RU) "An introduction to monoid schemes"
16 Feb Ian Coley  (RU)  "Hochster's description of Spec(R)"
23 Feb no   seminar           --------------------------------------------
 2 Mar Alexei Entin (Tel Aviv)  "The minimal ramification problem in inverse Galois theory" (remote)
 9 Mar Pham Tiep (RU)   "Representations and tensor product growth"
16 Mar   no seminar      ------------------- Spring Break ----------
30 Mar Tim Burness (Bristol, UK) "Fixed point ratios for primitive groups and applications"
 6 Apr Eugen Rogozinnikov (Strasbourg) "Hermitian Lie groups of tube type as symplectic groups over noncommutative algebras"
13 Apr Eilidh McKemmie (RU) "A survey of various random generation problems for finite groups"
20 Apr Yael Davidov (RU)     "Exploring the admissibility of Groups and an Application of Field Patching"
27 Apr Daniel Douglas (Yale) "Skein algebras and quantum trace maps" 
 4 May

15 Sept Yom Kippur
29 Sept Yoav Segev   "A characterization of the quaternions using commutators"
 6 Oct Lev Borisov (RU) "Explicit equations for fake projective planes"
13 Oct Ian Coley   (RU) "Introduction to topoi"
20 OctAnders Buch (RU) POSTPONED to 12/8 ("Tevelev Degrees")
27 Oct Eilidh McKemmie (RU) "The probability of generating invariably a finite simple group"
 3 Nov Ian Coley and Chuck Weibel  "Localization, and the K-theory of monoid schemes" 
10 Nov Max Peroux (Penn) "Equivariant variations of topological Hochschild homology"
17 Nov Shira Gilat (RU)    "The infinite Brauer group"
24 Nov --- no seminar ---   Thanksgiving is Nov. 25
29 Nov (Monday) Wednesday class schedule 
 1 Dec --- no seminar ---   
 8 Dec Anders Buch (RU) "Tevelev Degrees"

27 Jan Ian Coley (Rutgers) "Tensor Triangulated Geometry?"
 3 Feb Aline Zanardini (U.Penn) "Stability of pencils of plane curves"
10 Feb Svetlana Makarova (U.Penn) "Moduli spaces of stable sheaves over quasipolarized K3 surfaces, and Strange Duality"
17 Feb no seminar
24 Feb Patrick McFaddin (Fordham) "Separable algebras and rationality of arithmetic toric varieties"
 3 Mar Christian Klevdal (U.Utah) "Integrality of G-local systems"
10 Mar Justin Lacini   (U.Kansas) "On log del Pezzo surfaces in positive characteristic"
17 Mar  no seminar      ------------------- Spring Break ----------
24 Mar John Kopper (Penn State)  "Ample stable vector bundles on rational surfaces"
14 Apr Allechar Serrano Lopez (U.Utah) "Counting elliptic curves with prescribed torsion over imaginary quadratic fields"
21 Apr Franco Rota (Rutgers)   "Motivic semiorthogonal decompositions for abelian varieties"
28 Apr Ben Wormleighton (Washington U./St.Louis) "Geometry of mutations: mirrors and McKay"
 5 May Morgan Opie (Harvard)      "Complex rank 3 vector bundles on CP5"
12 May Avery Wilson (N. Carolina) "Compactifications of moduli of G-bundles and conformal blocks"

Abstracts of seminar talks

Posets of partial decompositions (Kevin Piterman, April 14, 2025)
Given a finite vector space V, let D(V) be the poset of (unordered) proper direct sum decompositions of V, ordered by refinement. That is, elements of D(V) are sets {S_1,...,S_r} where $S_1\oplus...\oplus S_r = = V$, and S_i\neq 0,V for all i. The topology of this poset is closely related to the poset of p-subgroups of GL(V), where p is a prime dividing the order of the unit group of the underlying finite field, and to posets of Levi subgroups. Moreover, results by Volkmar Welker show that D(V) has the homotopy type of a wedge of spheres of the maximal possible dimension (namely, dim(V)-2). In particular, its homology is concentrated in a single degree, and we can try to understand its properties as a GL(V)-module.
In this talk, I'll present results of an ongoing research with John Shareshian and Volkmar Welker on the topology of posets related to D(V), and the computation of the homology module. We will mainly focus on the poset of partial direct sum decompositions PD(V), obtained by taking subsets of elements of D(V) with ordering given by refinement, and their ordered versions (where sets of subspaces are now tuples), denoted by OD(V) and OPD(V) respectively. We will see that OPD(V) is spherical and describe its representation module in terms of the Steinberg module. Analogous models for other types of classical groups will also be explored during the talk.

Jets of Local Complete Intersection Morphisms (Andrew Stout, April 9, 2025)
I will give the natural conditions for which the enriched Hom functor (the jet operator) induces a local complete intersection (LCI) morphism given a LCI morphism of schemes. This subsumes my previous work on induced flatness. In addition, it turns out that this is the relativized version of well-known results of Mustata which were used by Eisenbud and Frenkel to prove an important generalization of a theorem of Kostant concerning the nilpotent cone of a Lie algebra. This talk will be accessible to a general mathematical audience.

My Mathematics with I.M. Isaacs: Some open questions left (Gabriel Navarro, April 2, 2025)
In the 28 papers that we coauthored, 3 of them with P.H. Tiep, there are some open questions left. I will survey some of these as a tribute to I.M. Isaacs.

A dual basis for the equivariant quantum K-theory of cominuscule varieties (Kevin Summers, March 26, 2025)
The equivariant quantum K-theory ring of a flag variety is a Frobenius algebra equipped with a perfect pairing called the quantum K-metric. It is known that in the classical K-theory ring for a given flag variety the ideal sheaf basis is dual to the Schubert basis with regard to the sheaf Euler characteristic.
We define a quantization of the ideal sheaf basis for the equivariant quantum K-theory of cominuscule flag varieties. These quantized ideal sheaves are then dual to the Schubert basis with regard to the quantum K-metric. We provide explicit type-uniform combinatorial formulae for the quantized ideal sheaves in terms of the Schubert basis for any cominuscule flag variety.
Time permitting, we will also discuss an application utilizing the quantized ideal sheaves to calculate the Schubert structure constants associated to multiplication by the top exterior power of the tautological quotient bundle in QK_T(Gr(k,n)).

Exponential bound for finite classical groups (Yuqiao Huang, March 12, 2025)
In "Character levels and character bounds for finite classical groups" by Robert M. Guralnick, Michael Larsen, Pham Huu Tiep (GLT), the authors proved for finite symplectic groups and orthogonal groups explicit bounds on the character values that depend on the sizes of centralizers. More precisely, they proved that, for γ between 0.8 and 1, if the centralizer of element g has order bounded by q^(n^2δ), then the length |χ(g)| is bounded by 4χ(1)^γ. However, these bounds degrade sharply as γ approaches 0.8, with δ nearing 0.
We improve these results in two directions: (1) we extend the admissible range of γ to (0.5, 1), and (2) we establish sharper bounds for δ. Crucially, our results prevent δ from collapsing near 0 as γ approaches 0.8. We also derive δ as explicit linear or quadratic functions of γ, enabling direct computation for applications.

Inductive Feit condition for small-rank simple groups of Lie type (Carlos Tapp-Monfort, March 5, 2025)
Some very deep questions in group representation theory are reduced by the Classification of Finite Simple Groups to proving a so-called ‘Inductive condition’ for all finite simple groups. This is the case of the Feit's conjecture or the Galois-McKay conjecture. In this talk I will discuss these conjectures with an emphasis on how we can prove the inductive conditions for some low rank finite groups of Lie type.

Cycles on moduli spaces of curves and abelian varieties (Samir Canning, Feb. 26, 2025)
Moduli spaces of curves and abelian varieties are central objects in algebraic geometry, number theory, and topology. Despite their importance, many basic aspects of the topology of these moduli spaces are not well understood. I will explain surprising new results about the cohomology of moduli spaces of curves proven by studying the cohomology of moduli spaces of abelian varieties, and vice versa.

A tale of two spaces: Hilbert schemes and Branch stacks (Ritvik Ramkumar, 3:30 on Feb. 12, 2025)
Varieties are the solution sets of polynomial equations. A distinctive feature of algebraic geometry is that certain collections of varieties are often themselves a variety. A fundamental example is the Hilbert scheme, constructed by Grothendieck in the 1960s. When the varieties are collections of points on a surface, the associated Hilbert scheme is smooth and has found significant applications beyond algebraic geometry. However, when the surface is replaced by a threefold, the Hilbert scheme becomes singular, and its structure remains largely unexplored. I will outline my research program for studying these singularities introducing methods from algebra, geometry, and combinatorics. I will also highlight how the tools developed can be applied to other problems.
To study higher-dimensional varieties, such as collections of curves, Alexeev and Knutson proposed a better alternative to the Hilbert scheme, known as the Branch stack. They conjectured its projectivity, a property that would make it as versatile as the Hilbert scheme. I will describe my work on proving this conjecture, thereby enabling the Branch stack to be used in various applications.

A local-global principle in group representation theory (Damiano Rossi, Feb. 12, 2025)
In its broader definition, representation theory is the attempt to linearize algebraic structures by studying their actions on vector spaces. The general hope is to recover information about the algebraic structure by answering a (typically) easier question. In the case of (finite) group representation theory, one could tackle this problem one prime at a time. On one hand, we can describe the structure of a group by looking at its p-local structure (given by the set of p-subgroups and their embedding) for each prime p. On the other hand, we can describe the representations of such a group by analyzing their properties at each prime p. The local-global principle in group representation theory asserts that, for each fixed prime p, the p-local structure of a group is directly and intimately linked to the representation theory of the group looked at through the prime p. I will present several fundamental conjectures in the area and explain how these can all be recovered from a unifying statement known as Dade's Conjecture. I will then describe a research program I designed to prove Dade's Conjecture and explain its connections to algebraic topology, homotopy theory, and algebraic geometry.

Links between character triples (Nariel Monterro, 2:00 on Feb. 12, 2025)
We will introduce the notion of a link between character triples. The motivation to consider this notion has multiple reasons.
Firstly, links between character triples induce special character triple isomorphisms. This provides a new perspective for isomorphisms between character triples and a different conceptual understanding.
Secondly, links between character triples provide equivalent reformulations of complicated conditions (involving projective representation) that play a fundamental role in the reductions of the McKay conjecture and the Feit conjecture to finite simple groups. This is joint work with Robert Boltje and John Revere McHugh.

K-regularity and normality (Chuck Weibel, Feb. 5, 2025)
Algebraic K-theory (and K-regularity) can detect geometric properties of a singular variety like X=Spec(A). We give two theorems to this effect, that K₂-regularity implies normality, and a result for local complete intersections that improves previous bounds by almost a factor of 2. This is joint work with C. Haesemeyer.