Seminars and Colloquia by Series

Global Optimization of Analytic Functions over Compact Domains

Series
Algebra Seminar
Time
Monday, September 18, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Georgy Scholten – Sorbonne Université

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am-11:30 am in Skiles 006.

In this talk, we introduce a new method for minimizing analytic Morse functions over compact domains through the use of polynomial approximations. This is, in essence, an effective application of the Stone-Weierstrass Theorem, as we seek to extend a local method to a global setting, through the construction of polynomial approximants satisfying an arbitrary set precision in L-infty norm. The critical points of the polynomial approximant are computed exactly, using methods from computer algebra. Our Main Theorem states probabilistic conditions for capturing all local minima of the objective function $f$ over the compact domain. We present a probabilistic method, iterative on the degree, to construct the lowest degree possible least-squares polynomial approximants of $f$ which attains a desired precision over the domain. We then compute the critical points of the approximant and initialize local minimization methods on the objective function $f$ at these points, in order to recover the totality of the local minima of $f$ over the domain.

The Principal Minor Map

Series
Algebra Seminar
Time
Monday, September 11, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Abeer Al Ahmadieh – Georgia Tech

The principal minor map takes an n x n square matrix and maps it to the 2^n-length vector of its principal minors. In this talk, I will describe both the fiber and the image of this map. In 1986, Loewy proposed a sufficient condition for the fiber to be a single point up to diagonal equivalence. I will provide a necessary and sufficient condition for the fiber to be a single point. Additionally, I will describe the image of the space of complex matrices using a characterization of determinantal representations of multiaffine polynomials, based on the factorization of their Rayleigh differences. Using these techniques I will give equations and inequalities characterizing the images of the spaces of real and complex symmetric and Hermitian matrices. This is based on joint research with Cynthia Vinzant.

Computing isotopy type of real zero sets faster for n-variate (n+k)-nomials

Series
Algebra Seminar
Time
Monday, August 28, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Weixun Deng – Texas A&M –
Suppose f is a Laurent polynomial in n variables with degree d, exactly (n+2) monomial terms, and all its coefficients in {-H,...,H} for some positive integer H. Suppose further that the exponent vectors of f do not all lie in an affine hyperplane: Such a set of exponent vectors is referred to as a circuit. We prove that the positive zero set of f is isotopic to the real zero set of an explicit n-variate quadric q, and give a fast algorithm to explicitly compute q: The bit complexity is (log(dH))^O(n). The best previous bit-complexity bounds were of the form (dlog(H))^{\Omega(n)} (to compute a data structure called a roadmap). Our results also extend to real zero sets of n-variate exponential sums over circuits. Finally, we discuss how to approach the next case up: n-variate polynomials with exactly (n+3) terms.

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 Leake – University 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.

Invariants of Matrices

Series
Algebra Seminar
Time
Monday, April 17, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Harm Derksen – Northeastern University –

The group SL(n) x SL(n) acts on m-tuples of n x n matrices by simultaneous left-right multiplication.  Visu Makam and the presenter showed the ring of invariants is generated by invariants of degree at most mn^4. We will also discuss geometric aspects of this action and connections to algebraic complexity and the notion of noncommutative rank.

CANCELLED

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.

Root-freeness and generalized log-concavity

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

In this talk, we explore the fractional log-concavity property of generating polynomials of discrete distributions. This property is an analog to the Lorentzian [Branden-Huh’19]/log-concavity [Anari-Liu-OveisGharan-Vinzant’19] property of the generating polynomials of matroids. We show that multivariate generating polynomials without roots in a sector of the complex plane are fractionally log-concave. Furthermore, we prove that the generating polynomials of linear delta matroids and of the intersection between a linear matroid and a partition matroid have no roots in a sector, and thus are fractionally log-concave. Beyond root-freeness, we conjecture that for any subset F of {0,1}^n such that conv(F) has constantly bounded edge length, the generating polynomial for the uniform distribution over F is fractionally log-concave.

Based on joint works with Yeganeh Alimohammadi , Nima Anari and Kirankumar Shiragur.

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 Polstra – University 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 Mandelshtam – University 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 DiPasquale – University 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).

Pages