Events of the Week

In this talk, we consider hexagonal domains on a triangular grid and use a method known as Kasteleyn theory (KTF method) to count the number of lozenge tilings of these domains. We will provide a proof of this enumeration result. Exact counts of tilings leads us to study the typical behavior of a uniformly random tiling in large domains, including the emergence of limit shapes such as frozen and liquid regions separated by an Arctic curve.

I became aware of the extent of the attacks on the publication and citation processes though Taylor & Francis, the publisher of Quality Engineering for which I am the editor.

All areas of science are being affected, from mathematics to medicine. The following are some of the primary concerns: authorships being sold, fake papers produced by paper mills, sham reviews, unethical behavior of guest editors for special issues, bribes offered to editors, the rise of predatory journals and conferences, citation cartels, plagiarism, misuse of AI, and the fabrication of data and images.

These and other issues, illustrated with numerous examples, will be discussed in this presentation. A related issue is the proliferation of junk science.

Efforts to protect the scientific literature will be discussed, such as the contributions by individual sleuths and the use of the STM Integrity Hub that was established by the major academic publishers. There will also be some discussion of the article retraction process. Over 10,000 scientific papers were retracted in 2023, a record number.

Let $(R, \mathfrak{m})$ be a Noetherian local ring of dimension $d$, and let $I\subseteq R$ be an ideal. Ulrich and Validashti defined the \emph{$\varepsilon$-multiplicity} of $I$ as
\[
\varepsilon(I) = \limsup_{n \to \infty}
\frac{\lambda_R\!\left(H^0_{\mathfrak{m}}(R/I^n)\right)}{n^d / d!}.
\]
This invariant may be viewed as a generalization of the classical Hilbert-Samuel multiplicity. Cutkosky showed that the $\limsup$ in this definition can be replaced by an actual limit when $R$ is analytically unramified. A surprising example due to Cutkosky, Hà, Srinivasan, and Theodorescu demonstrates
that this limit can be an irrational number even when $R$ is a regular local ring.

In this talk, we shall focus on the case of homogeneous ideals in a standard graded domain over a field. Motivated by Trivedi's approach to the Hilbert-Kunz multiplicity via density functions, we introduce a compactly supported continuous real-valued function, called the \emph{$\varepsilon$-density function}, whose integral recovers the $\varepsilon$-multiplicity. If time permits, we will present explicit examples and discuss applications in the context of integral closures.

This talk is based on joint work with Roy and Trivedi.

Inspired by a question posed by Lax in 2002, the study of metric entropy for nonlinear partial differential equations has received increasing attention in recent years. This talk demonstrates methods to obtain sharp upper and lower bounds on the metric entropy for a class of real-valued bounded total variation functions and then for a class of bounded total generalized variation functions taking values in a general totally bounded metric space. Thereafter we use each of these results to establish metric entropy estimates for the set of viscosity solutions to Hamilton-Jacobi equations with a uniformly directionally convex Hamiltonian and the set of entropy admissible weak solutions to scalar conservation laws with a weakly genuinely nonlinear flux, respectively. Estimates of this type could provide a measure of the order of resolution and complexity of a numerical method required to solve the corresponding equation.

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In recent years, algebraic geometers began exploring a fascinating phenomenon exhibited by configurations of points in projective space wherein their general projection into a hyperplane is a complete intersection. This phenomenon was called the "geproci" property. In an effort to investigate this geproci property, Brian Harbourne and Allison Ganger began to look at special groups induced by configurations of lines known as groups of groupoids. In this talk, we will give some background on the geproci property and how it relates to configurations of lines known as spreads and maximal partial spreads. We will then define the groupoid induced by a configuration of lines in P^3 (with lots of Desmos demonstrations!), and look at specific examples of this groupoid.

Given initial positions and velocities of $n$ celestial bodies, with only gravitation force in play, can you describe their trajectories?

This analytic question---the so-called $n$-body problem---prompts another problem: describe configurations of the celestial bodies that are in relative equilibrium, that is, the mutual distances don't change over time. The latter question can be set up purely algebraically, as a system of multivariate polynomial equations depending on masses of the bodies as parameters.

In a joint work with Anders Jensen, we investigate the following conjecture: for fixed $n$, up to natural symmetries, the set of planar relative equilibria for $n$ bodies with positive masses is finite. Although this statement may appear deceptively simple, it is the sixth problem on Smale’s list of problems for the 21st century and is fully resolved only for up to four bodies. We provide a computer-assisted proof for $n=5$ in the case of generic masses. The human component of the argument draws on several elementary ideas from tropical geometry, and our approach places the case $n=6$ within reach.

This talk is aimed at a general mathematical audience, including students. I will give an overview of basic celestial mechanics, basic tropical geometry, and basic rigidity theory. Examples for all of these will be in the plane.

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Species-level phylogenetic inference under the multispecies coalescent model remains challenging in the typical inference frameworks (e.g., the likelihood and Bayesian frameworks) due to the dimensionality of the space of both gene trees and species trees. Algebraic approaches intended to establish identifiability of species tree parameters have suggested computationally efficient inference procedures that have been widely used by empiricists and that have good theoretical properties, such as statistical consistency. However, such approaches are less powerful than approaches based on the full likelihood. Methods based on composite likelihood are a compromise between these two approaches that enable computationally efficient inference while maximizing use of the available sequence data. In this talk, I’ll describe the relationship between these two approaches, highlighting the strengths and weaknesses of each and providing directions for future work.

Fisher's linear discriminant analysis (LDA) is a foundational method of dimension reduction for classification that has been useful in a wide range of applications. The goal is to identify an optimal subspace to project the observations onto that simultaneously maximizes between-group variation while minimizing within-group differences. The solution is straightforward when the number of observations is greater than the number of features but difficulties arise in the high dimensional setting, where there are more features than there are observations. Many works have proposed solutions for the high dimensional setting and frequently involve additional assumptions or tuning parameters. We propose a fast and simple iterative algorithm for high dimensional multiclass LDA on large data that is free from these additional requirements and that comes with some guarantees. We demonstrate our algorithm on real data and highlight some results.