Seminars and Colloquia by Series

TBD by Benjamin Lovitz

Series
Algebra Seminar
Time
Monday, April 22, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Benjamin LovitzNortheastern University

TBD by Ada Wang

Series
Algebra Seminar
Time
Monday, April 8, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ada WangHarvard University

TBD by Frank Sottile

Series
Algebra Seminar
Time
Monday, March 25, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Frank SottileTexas A&M University

TBD by Martin Helmer 

Series
Algebra Seminar
Time
Monday, March 4, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Martin HelmerNorth Carolina State University

TBD by Mahrud Sayrafi

Series
Algebra Seminar
Time
Monday, February 26, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mahrud SayrafiUniversity of Minnesota

Permutation action on Chow rings of matroids

Series
Algebra Seminar
Time
Monday, February 19, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Anastasia NathansonUniversity of Minnesota

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am to 11:30 am in Skiles 005.

Given a matroid and a group of its matroid automorphisms, we study the induced group action on the Chow ring of the matroid. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincar\'e duality andthe Hard Lefschetz theorem.  We lift these to statements about this permutation action, and suggest further conjectures in this vein.

Determinantal zeros and factorization of noncommutative polynomials

Series
Algebra Seminar
Time
Monday, February 12, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jurij VolčičDrexel University

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am to 11:30am in Skiles 005.

Hilbert's Nullstellensatz about zero sets of polynomials is one of the most fundamental correspondences between algebra and geometry. More recently, there has been an emerging interest in polynomial equations and inequalities in several matrix variables, prompted by developments in control systems, quantum information theory, operator algebras and optimization. The arising problems call for a suitable version of (real) algebraic geometry in noncommuting variables; with this in mind, the talk considers matricial sets where noncommutative polynomials attain singular values, and their algebraic counterparts.

Given a polynomial f in noncommuting variables, its free (singularity) locus is the set of all matrix tuples X such that f(X) is singular.  The talk focuses on the interplay between geometry of free loci (irreducible components, inclusions, eigenlevel sets, smooth points) and factorization in the free algebra. In particular, a Nullstellensatz for free loci is given, as well as a noncommutative variant of Bertini's irreducibility theorem and its consequences.

Two short talks

Series
Algebra Seminar
Time
Monday, February 5, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
May Cai and Matt BakerGeorgia Tech

This special algebra seminar will feature short talks by our very own May Cai and Matt Baker, who will speak on the following topics: 

May Cai: The completion problem asks one to take a partial observation of some underlying object, and try to recover the original observation. Concretely, we have some object of interest, and a point in the image of that object under a projection map, and want to understand the fiber of this point under this map. In particular, for log-linear models, which are the restrictions of toric varieties to the probability simplex, under certain mild conditions, when this fiber is finite it turns out to have exactly either one or two entries. This is joint work with Cecilie Olesen Recke and Thomas Yahl.

Matt Baker: The determinant of a skew-symmetric matrix has a canonical square root given by the Pfaffian. Similarly, the resultant of two reciprocal polynomials of even degree has a canonical square root given by their reciprocant. Computing the reciprocant of two cyclotomic polynomials yields a short and elegant proof of the Law of Quadratic Reciprocity.

Polynomials with Lorentzian Signature over Cones, and Perron-Frobenius Theorem

Series
Algebra Seminar
Time
Monday, January 29, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Papri DeyGeorgia Tech

Please Note: There is no pre-seminar this time.

 The classical theorems of Perron and Frobenius, which explore spectral properties of nonnegative matrices, have been extensively examined and generalized from various perspectives, including a cone-theoretic (geometric) viewpoint. Concurrently, in the past decade, there has been a notable effort to fuse the techniques of algebraic geometry and combinatorics in an exploration of Lorentzian polynomials by Brändén and Huh, also known as completely log-concave polynomials (CLC) by Anari et.al. or strongly log-concave polynomials by Gurvits.

 

In this talk, I will discuss my ongoing joint work with Greg Blekherman regarding the class of polynomials with Lorentzian signature (PLS) defined over closed convex cones. This class encompasses various special polynomials, including Lorentzian polynomials over the nonnegative orthant and hyperbolic polynomials over hyperbolicity cones. We establish a compelling connection between PLS over a self-dual cone K and the generalized Perron Frobenius theorem over K. This connection enables us to provide an alternative necessary and sufficient condition to characterize the Lorentzian polynomials.

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