Kalina's personal website

Algebra and Combinatorics Seminar

We meet weekly on Wednesday at 3:00 PM (Central time), GI-126

Organizers: Alessandra Constantini and Kalina Mincheva.

January 14

Trung Chau, Chennai Mathematical Institute.
Frobenius singularities of permanental varieties. abstract.

A permanent of a square matrix is exactly its determinant with all minus signs becoming plus. Despite the similarities, the computation of a determinant can be done in polynomial time, while that of a permanent is an NP-hard problem. In 2002, Laubenbacher and Swanson defined $P_t(X)$ to be the ideal generated by all t-by-t subpermanents of $X$, and called it a permanental ideal. This is a counterpart of determinantal ideals, the center of many areas in Algebra and Geometry. We will discuss properties of $P_2(X)$, including their Frobenius singularities over a field of prime characteristic, and related open questions.

January 21

Mahir Can, Tulane University.
When Schubert Varieties Miss Being Toric by One. abstract.

Schubert and Richardson varieties in flag varieties provide a rich testing ground for various group actions. In this talk I will discuss two “borderline toric” phenomena. First, I will introduce nearly toric Schubert varieties. They are spherical Schubert varieties for which the smallest codimension of a torus orbit is one. Then I will explain a simple Coxeter-type classification of these examples, and why this “one step from toric” condition forces strong spherical behavior (in particular, it produces a large family of doubly-spherical Schubert varieties). Time permitting, I will also discuss toric Richardson varieties and a type-free combinatorial criterion: a Richardson variety is toric exactly when its Bruhat interval is a lattice (equivalently, it contains no subinterval of type $S_3$, under a mild dimension hypothesis).

January 28

William Newman, Ohio State University.
Chow rings of moduli spaces of genus 0 curves with collisions. abstract.

Simplicially stable spaces are alternative compactifications of $M_{g,n}$ generalizing Hassett’s moduli spaces of weighted stable curves. We give presentations of the Chow rings of these spaces in genus 0. When considering the special case of $\bar M_{0,n}$, this gives a new proof of Keel’s presentation of $CH(\bar M_{0,n})$.

February 4

Tai Ha, Tulane University.
An algebraic theory of Lojasiewicz exponents. abstract.

We develop a unified algebraic and valuative theory of Lojasiewicz exponents for pairs of graded families of ideals. Within this framework, analytic local Lojasiewicz exponents, gradient exponents, and exponents at infinity are all realized as asymptotic containment thresholds between appropriate filtrations.
The main theme is a finite-max principle: under verifiable algebraic hypothesis, the a priori infinite valuative supremum describing the Lojasiewicz exponent reduces to a finite maximum and attained by divisorial valuations. We identify two complementary mechanisms leading to this phenomenon: finite testing arising from normalized blowups and Rees algebra constructions, and attainment via compactness of normalized valuation spaces under linear boundedness assumptions. This finite-max principle yields strong structural consequences, including rigidity, stratification, and stability results. We also explain classical results/problems in toric and Newton polyhedral settings.

February 11

Louiza Fouli, New Mexico State.
Asymptotic Resurgence of Matroid Ideals. abstract.

We study two monomial ideals naturally associated to the independence complex of a matroid: the Stanley–Reisner ideal and the facet ideal. Focusing on their asymptotic resurgence, we establish general bounds and, for specific families of matroids, derive exact formulas. This is joint work with Michael DiPasquale and Arvind Kumar.

February 25

Nicholas Barvinok, Smith College.
A necessary and sufficient condition for detecting overlap in edge unfoldings of nearly flat convex caps. abstract.

By cutting a 3D convex polyhedron by a plane, we obtain a convex cap. By cutting on a boundary rooted spanning forest of the edge graph, we can unfold the cap into the plane. Nearly flat caps have unfoldings which are very close to their orthogonal projections. We take advantage of this to construct a necessary and sufficient condition for detecting overlap in the unfolding based on the orthogonal projection of the cap's edge graph. This is a recent result which is a joint work with Tyson Trauger. We also discuss two possible applications of this condition: a positive resolution to a special case of Durer's problem, and a necessary and sufficient condition for detecting overlap in infinitesimal edge unfoldings of arbitrary convex caps.

March 4

Stephen Landsittel, Hebrew University of Jerusalem and Harvard University.
Some results about saturation. abstract.

Given a local ring $R$ we can ask when saturation of ideals in $R$ commutes with other operations on ideals (such as extension to a ring containing $R$). We show that the condition that extension of ideals along a ring map $R \to S$ commutes with saturation controls inherent properties of the rings $R$ and $S$, such as Cohen-Macaulayness and unramifiedness.

March 11

March 16 - Note the unusual day

March 18

April 1

April 8

April 15

April 22

April 29