Seminars and Colloquia by Series

Lorentzian polynomials on cones

Series
Algebra Seminar
Time
Monday, April 24, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Jonathan LeakeUniversity of Waterloo

We show how the theory of Lorentzian polynomials extends to cones other than the positive orthant, and how this may be used to prove Hodge-Riemann relations of degree one for Chow rings. If time permits, we will show explicitly how the theory applies to volume polynomials of matroids and/or polytopes. Joint work with Petter Brändén.

TBD

Series
Algebra Seminar
Time
Monday, April 17, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Harm DerksenUniversity of Michigan

TBD

Algebra and combinatorics of intersection bodies of polytopes.

Series
Algebra Seminar
Time
Monday, April 10, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Chiara Meroni Max Planck Institute for Mathematics in the Sciences

Intersection bodies are a popular construction in convex geometry. I will give an introduction on these objects, convex algebraic geometry, and starshaped sets in general. Then, we will analyze some features of intersection bodies and focus on the polyotopal case. Intersection bodies of polytopes are always semialgebraic sets and they are naturally related to hyperplane arrangements, which reveal their boundary structure. Finally, we will investigate their convexity, in the two-dimensional case. The exposition will be enriched by examples and computations. This is based on joint works with Katalin Berlow, Marie-Charlotte Brandenburg and Isabelle Shankar.

TBD

Series
Algebra Seminar
Time
Monday, April 3, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Thuy-Duong VuongStandford University

TBD

On the weak implies strong conjecture

Series
Algebra Seminar
Time
Monday, March 27, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Thomas PolstraUniversity of Alabama

A fundamental conjecture of tight closure theory is every weakly F-regular ring is strongly F -regular. There has been incremental progress on this conjecture since the inception of tight closure. Most notably, the conjecture has been resolved for rings graded over a field by Lyubeznik and Smith. Otherwise, known progress around the conjecture have required assumptions on the ring that are akin to being Gorenstein. We extend known cases by proving the equivalence of F -regularity classes for rings whose anti-canonical algebra is Noetherian on the punctured spectrum. The anti-canonical algebra being Noetherian for a strongly F -regular ring is conjectured to be a vacuous assumption. This talk is based on joint work with Ian Aberbach and Craig Huneke.

Macdonald polynomials and the multispecies zero range process

Series
Algebra Seminar
Time
Monday, March 13, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Olya MandelshtamUniversity of Waterloo

Macdonald polynomials are a family of symmetric functions that are known to have remarkable connections to a well-studied particle model called the asymmetric simple exclusion process (ASEP). The modified Macdonald polynomials are obtained from the classical Macdonald polynomials using an operation called plethysm. It is natural to ask whether the modified Macdonald polynomials specialize to the partition function of some other particle system.

We answer this question in the affirmative with a certain multispecies totally asymmetric zero-range process (TAZRP). This link motivated a new tableaux formula for modified Macdonald polynomials. We present a Markov process on those tableaux that projects to the TAZRP and derive formulas for stationary probabilities and certain correlations, proving a remarkable symmetry property. This talk is based on joint work with Arvind Ayyer and James Martin.

Saturating the Jacobian ideal of a line arrangement via rigidity theory

Series
Algebra Seminar
Time
Monday, March 6, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Michael DiPasqualeUniversity of South Alabama

A line arrangement is a collection of lines in the projective plane.  The intersection lattice of the line arrangement is the set of all lines and their intersections, ordered with respect to reverse inclusion.  A line arrangement is called free if the Jacobian ideal of the line arrangement is saturated.  The underlying motivation for this talk is a conjecture of Terao which says that whether a line arrangement is free can be detected from its intersection lattice.  This raises a question - in what ways does the saturation of the Jacobian ideal depend on the geometry of the lines and not just the intersection lattice?  A main objective of the talk is to introduce planar rigidity theory and show that 'infinitesimal rigidity' is a property of line arrangements which is not detected by the intersection lattice, but contributes in a very precise way to the saturation of the Jacobian ideal.  This connection builds a theory around a well-known example of Ziegler.  This is joint work with Jessica Sidman (Mt. Holyoke College) and Will Traves (Naval Academy).

Crossing the transcendental divide: from translation surfaces to algebraic curves

Series
Algebra Seminar
Time
Monday, February 27, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Yelena MandelshtamUC Berkeley

A translation surface is obtained by identifying edges of polygons in the plane to create a compact Riemann surface equipped with a nonzero holomorphic one-form. Every Riemann surface can be given as an algebraic curve via its Jacobian variety. We aim to construct explicitly the underlying algebraic curves from their translation surfaces, given as polygons in the plane. The key tools in our approach are discrete Riemann surfaces, which allow us to approximate the Riemann matrices, and then, via theta functions, the equations of the curves. In this talk, I will present our algorithm and numerical experiments. From the newly found Riemann matrices and equations of curves, we can then make several conjectures about the curves underlying the Jenkins-Strebel representatives, a family of examples that until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves.

Excluding a line from complex-representable matroids

Series
Algebra Seminar
Time
Monday, February 13, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Zach WalshGeorgia Institute of Technology

The extremal function of a class of matroids maps each positive integer n to the maximum number of elements of a simple matroid in the class with rank at most n. We will present a result concerning the role of finite groups in minor-closed classes of matroids, and then use it to determine the extremal function for several natural classes of representable matroids. We will assume no knowledge of matroid theory. This is joint work with Jim Geelen and Peter Nelson.

Central Curve in Semidefinite Programming

Series
Algebra Seminar
Time
Monday, February 6, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Isabelle ShankarPortland State University

The Zariski closure of the central path (which interior point algorithms track in convex optimization problems such as linear and semidefinite programs) is an algebraic curve, called the central curve. Its degree has been studied in relation to the complexity of these interior point algorithms.  We show that the degree of the central curve for generic semidefinite programs is equal to the maximum likelihood degree of linear concentration models.  This is joint work with Serkan Hoşten and Angélica Torres.

 

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