Algecom-XXI

Algebra, Geometry and Combinatorics Day (AlGeCom) is a one day, informal meeting of mathematicians from the University of Illinois, Purdue University, IUPUI, Washington University at St. Louis, Loyola University Chicago, DePaul University, University of Notre Dame, the University of Michigan and nearby universities, with interests in algebra, geometry and combinatorics (widely interpreted).

Algecom Committee:

Hal Schenck Alexander Yong David Speyer Uli Walther Saugata Basu Evgeny Mukhin Peter Tingley Chris Drupieski Laura Escobar Kyle Petersen

Date: April 10

Location: Virtual, Hosted by University of Notre Dame.

Local Organizers (questions related to Algecom XXI):

Zhao Gao (zgao1@nd.edu)

Lizda Moncada (lmoncada@nd.edu)

Claudiu Raicu (craicu@nd.edu)

Registration please use the registration link

Participants will have to register in order to receive the Zoom link for the talks!

Tentative Schedule:

(All times are in Eastern Time Zone units.)

9:30-10:30am Miruna-Ştefana Sorea (SISSA)

10:45-11:45am Nir Gadish (MIT)

12:45-1:45pm Nathan Pflueger (Amherst College)

2:00-3:00pm Patricia Klein (Minnesota)

Abstracts:

Nir Gadish (MIT): Möbius inversion in hömotopy theory Notes

Abstract:

Möbius inversion is classically a procedure in number theory that inverts summation of functions over the divisors of an integer. A similar construction is possible for every locally finite poset, and is governed by a so called Möbius function encoding the combinatorics. In 1936 Hall observed that the values of the Möbius function are Euler characteristics of intervals in the poset, suggesting a homotopy theoretic context for the inversion.

In this talk we will discuss a functorial 'space-level' realization of Möbius inversion for diagrams taking values in a category with homotopy equivalences. The role of the Möbius function will be played by hömotopy types whose reduced Euler characteristics are the classical values, and inversion will hold up to extensions (think inclusion-exclusion but with the alternating signs replaced by even/odd spheres). This provides a uniform perspective to many constructions in topology and algebra. Notable examples that I hope to mention include handle decompositions, Koszul resolutions, and filtrations of hyperplane arrangements.

Patricia Klein (Minnesota): Geometric vertex decomposition and liaison Notes

Abstract:

Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this talk, we will describe an explicit connection between these approaches. In particular, we describe how each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, how every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of graded lower bound cluster algebras.

This connection also gives us a framework for implementing with relative ease Gorla, Migliore, and Nagel’s strategy of using liaison to establish Gr\"obner bases. We describe briefly, as an application of this work, a proof of a recent conjecture of Hamaker, Pechenik, and Weigandt on diagonal Gr\"obner bases of Schubert determinantal ideals.

This talk is based on joint work with Jenna Rajchgot.

Nathan Pflueger (Amherst College): Enumerative Brill-Noether theory of twice-marked curves Notes

Abstract:

Classical Brill-Noether theory concerns the following question: given a smooth curve C and two positive integers d and r, what is the geometry of the space of degree-d line bundles on C with at least r+1 linearly independent sections? These spaces, called Brill-Noether varieties, have interesting geometry that is closely linked to the combinatorics of Young tableaux. For example, for a general curve C, when a Brill-Noether variety is 0-dimensional, its degree is equal to the number of standard Young tableaux on a rectangular partition. For a twice-marked curve (C,p,q), one considers the space of degree-d line bundles L with a prescribed "rank function" r(a,b) = h^0(C, L(-ap-bq)). For (C,p,q) general, when this locus is 0-dimensional, its degree is equal to the number of reduced words for a permutation associated to the rank function. I will discuss these results and related generalizations and conjectures, emphasizing the interplay between the underlying geometry and combinatorics.

Miruna-Ştefana Sorea (SISSA): Poincaré-Reeb trees of real Milnor fibres Notes

Abstract:

We study the real Milnor fibre of real bivariate polynomial functions vanishing at the origin, with an isolated local minimum at this point. We work in a neighbourhood of the origin in which its non-zero level sets are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, they may fail to be convex, as was shown by Coste.

In order to measure the non-convexity of the level curves, we introduce a new combinatorial object, called the Poincaré-Reeb tree, and show that locally the shape stabilises and that no spiralling phenomena occur near the origin. Our main objective is to characterise all topological types of asymptotic Poincaré-Reeb trees. To this end, we construct a family of polynomials with non-Morse strict local minimum at the origin, realising a large class of such trees.

As a preliminary step, we reduce the problem to the univariate case, via the interplay between the polar curve and its discriminant. Here we give a new and constructive proof of the existence of Morse polynomials whose associated permutation (the so-called “Arnold snake”) is separable, using tools inspired from Ghys’s work.