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Upcoming Seminars

  1. Fellowship of the Ring, National Seminar: Reflection arrangements, syzygies, and the containment problem

    Location: MSRI: Online/Virtual
    Speakers: Alexandra Seceleanu (University of Nebraska)

    Invariant theory, that is the art of finding polynomials invariant under the action of a given group, has played a major role in the historical development of commutative algebra. In this theory reflection groups are singled out for having rings of invariants that are isomorphic to polynomial rings. From a geometric perspective, reflection groups give rise to beautiful and very symmetric arrangements of hyperplanes termed reflection arrangements.

    This talk will take a close look at the ideals defining the singular loci of reflection arrangements, which are in turn symmetric subspace arrangements. We describe their syzygies in terms of invariant polynomials for the relevant reflection groups. We leverage this information to settle many aspects of the containment problem asking for containments between the ordinary and the symbolic powers of the ideals in this family. This talk is based on joint work with Ben Drabkin.

    Updated on Jul 10, 2020 08:18 AM PDT
  2. Fellowship of the Ring, National Seminar: Lagrangian Geometry of Matroids

    Location: MSRI: Online/Virtual
    Speakers: Federico Ardila (San Francisco State University)

    Matroid theory had its origins in linear algebra and graph theory. In recent years, the geometric roots of the field have grown much deeper, bearing many new fruits. The interplay between matroid theory and algebraic geometry has opened up interesting research directions at the intersection of combinatorics, algebra, and geometry, and led to the solution of long-standing questions.

    This talk will discuss my recent joint work with Graham Denham and June Huh. We introduce the conormal fan of a matroid M. Inside its Chow ring, we find simple interpretations of the Chern-Schwartz-MacPherson cycle of M (a tropical geometric construction) and of the h-vector of M (a combinatorial invariant). We then use the Hodge-Riemann relations to prove Brylawski's and Dawson's conjectures that the h-vector of a matroid is log-concave.

    I will make the talk as self-contained as possible, and assume no previous knowledge of matroid theory.
     

    Updated on Jul 10, 2020 08:11 AM PDT
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Past Seminars

  1. Seminar Fellowship of the Ring, National Seminar: Commutative Algebra with S_n-invariant monomial ideals

    To attend this seminar, you must register in advance, by clicking HERE.

    Consider a polynomial ring in n variables, together with the action of the symmetric group by coordinate permutations. In my talk I will describe many familiar notions in Commutative Algebra in the context of monomial ideals that are preserved by the action of the symmetric group. These include Castelnuovo-Mumford regularity, projective dimension, saturation, symbolic powers, or the Cohen-Macaulay property. My goal is to explain how changing focus from minimal resolutions to Ext modules can lead to a simplified picture of the homological algebra, and to provide concrete combinatorial recipes to determine the relevant homological invariants.

    Updated on May 01, 2020 12:25 PM PDT
There are more then 30 past seminars. Please go to Past seminars to see all past seminars.