# Algebraic Geometry/Commutative Algebra Seminar, 2022–2023

To volunteer to give a talk, or for anything else regarding the seminar, contact Claudiu Raicu or Eric Riedl.

Abstracts can be found below.

## Fall Schedule

The seminar will meet on Tuesdays, 2:30–3:30pm, either on Zoom or in 258 Hurley, unless otherwise noted. Related events are also listed below.

Date Speaker Title
Tuesday, Sep. 6 Steven Karp (Notre Dame) Wronskians, total positivity, and real Schubert calculus
Tuesday, Sep. 13 Joe Waldron (Michigan State) Purely inseparable Galois theory
Tuesday, Sep. 20 Mark Skandera (Lehigh University) Type-BC character evaluations and P-tableaux
Tuesday, Sep. 27 Joe Cummings (Notre Dame) A Fano compactification of the free group character variety
Tuesday, Oct. 4 Joshua Mundinger (Chicago) Quantization of restricted Lagrangian subvarieties in positive characteristic
Tuesday, Oct. 11, 2:30–3:30pm Bernd Ulrich (Purdue) TBA
Tuesday, Oct. 11, 3:30–4:30pm
127 Hayes-Healy
Note special time/location
Hunter Simper (Purdue) TBA
Tuesday, Oct. 18 No seminar (Fall break)
Tuesday, Oct. 25 Cheng Meng (Purdue) TBA
Tuesday, Nov. 1
Tuesday, Nov. 8 Brandon Alberts (Eastern Michigan) TBA
Tuesday, Nov. 15 Nitin Chidambaram (Bonn) TBA
Tuesday, Nov. 22 Mihai Fulger (Connecticut) TBA
Tuesday, Nov. 29 Martha Precup (Washington University in St. Louis) TBA
Tuesday, Dec. 6 Lena Ji (Michigan) TBA

## Abstracts

### Sep. 6, 2022

Speaker
Steven Karp (Notre Dame)
Title
Wronskians, total positivity, and real Schubert calculus
Abstract
The totally positive flag variety is the subset of the complete flag variety Fl(n) where all Plücker coordinates are positive. By viewing a complete flag as a sequence of subspaces of polynomials of degree at most n-1, we can associate a sequence of Wronskian polynomials to it. I will present a new characterization of the totally positive flag variety in terms of Wronskians, and explain how it sheds light on conjectures in the real Schubert calculus of Grassmannians. In particular, a conjecture of Eremenko (2015) is equivalent to the following conjecture: if V is a finite-dimensional subspace of polynomials such that all complex zeros of the Wronskian of V are real and negative, then all Plücker coordinates of V are positive. This conjecture is a totally positive strengthening of a result of Mukhin, Tarasov, and Varchenko (2009), and can be reformulated as saying that all complex solutions to a certain family of Schubert problems in the Grassmannian are real and totally positive.

### Sep. 13, 2022

Speaker
Joe Waldron (Michigan State)
Title
Purely inseparable Galois theory
Abstract
Given a field K of characteristic p, a classical result of Jacobson provides a Galois correspondence between finite purely inseparable subfields of K of exponent one (i.e. those which contain K^p), and sub-restricted Lie algebras of Der(K). I will discuss joint work with Lukas Brantner in which we extend this Galois correspondence to subfields of arbitrary exponent using methods from derived algebraic geometry.

### Sep. 20, 2022

Speaker
Mark Skandera (Lehigh University)
Title
Type-BC character evaluations and P-tableaux
Abstract
For w in S_n corresponding to a smooth type-A Schubert variety X(w), there exists a poset P = P(w) which provides a combinatorial interpretation of the Kazhdan-Lusztig basis element C_w(1) in Z[S_n]. Evaluations of characters at C_w(1) may be computed by counting special P-tableaux. We state a type-BC analog of these results. Namely, for w in the hyperoctahedral group B_n such that the type-B and type-C Schubert varieties are simultaneously smooth, there exists a type-BC poset P = P(w) which provides a combinatorial interpretation of the Kazhdan-Lusztig basis element C_w(1) in Z[B_n]. Again, evaluations of characters at the Kazhdan-Lusztig basis element C_w(1) of Z[B_n] may be computed by counting special P-tableaux.

### Sep. 27, 2022

Speaker
Joe Cummings (Notre Dame)
Title
A Fano compactification of the free group character variety
Abstract
We show that a certain compactification of the free group character variety $\mathcal{X}(F_g, Sl_2(\mathbb{C}))$ is Fano. This compactification has been studied previously by Manon, and separately by Biswas, Lawton, and Ramras. Part of the proof involves the construction of a large family of integral reflexive polytopes arising from trivalent graphs of genus $g$. This project is joint with Chris Manon.

### Oct. 4, 2022

Speaker
Joshua Mundinger (Chicago)
Title
Quantization of restricted Lagrangian subvarieties in positive characteristic
Abstract
In order to answer basic questions in modular and geometric representation theory, Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties in positive characteristic. These are certain noncommutative algebras over a field of positive characteristic which have a large center. I will discuss recent work describing how to construct an important class of modules over such algebras.