Events of the Week

We will return to the numerical experiments, to carefully develop intuition. And explore some more precise open problems. Then we will return to complete the proof of the fundamental relation between eigenvalues and Fredholm determinants.

The presentation provides an introduction to Chagas disease, a parasitic infection caused by Trypanosoma cruzi, primarily transmitted by Triatomine bugs. It outlines key transmission pathways, including vector-borne, congenital, and less common mechanisms. The discussion then shifts to the application of a stochastic SIS (Susceptible-Infected-Susceptible) model, illustrating how randomness can enhance epidemic modeling by accounting for variability and uncertainty in disease dynamics.

Symmetric three-term recurrences (STRs) naturally arise in the study of orthogonal polynomials, iterative methods for symmetric matrices and numerical complex analysis. While deceptively simple, STRs allow for many extremely effective numerical methods. This talk will review some classical methods and uses and connect to more recent developments related to the computation of Cauchy integrals, computing matrix functions and spectral density estimation for random matrices.

Ear decomposition is a classical notion in Graph Theory. It has been shown in 1, 2 that this notion can be used to solve difficult problems on homological properties of edge ideals in Combinatorial Commutative Algebra.
This talk presents the main combinatorial ideas behind these results.
1) H.M. Lam and N.V. Trung, Associated primes of powers of edge ideals and ear decompositions of graphs, Trans. AMS 372 (2019)
2) H.M. Lam, N.V. Trung, and T.N. Trung, A general formula for the index of depth stability of edge ideals, Trans. AMS, to appear.

In a striking series of papers, titled Geometry of KdV(1) - Geometry of KdV(5), Henry Mckean formulated a precise notion of what should be the function space foliation by invariant sets for the Korteweg-deVries evolution. This is meant to pertain to initial data that is smooth but otherwise only required to be bounded below. This foliation should generalize the picture of (typically infinite dimensional) Arnold-Liouville torii familiar from the particular case of periodic initial data. The proposed answer is phrased in terms Kodaira’s elegant extension of the classical Weyl-Titchmarsh theory for spectral weights of Schrodinger operators.

The goal of the talk will be to first present an overview of McKean’s conjecture and then to describe some recent work, joint with Dylan Murphy, on analogous investigations for the Toda lattice and Jacobi operators.

This talk will highlight techniques from discrete dynamical systems theory that have led to significant advances in map enumeration. We will start with a brief overview of generating functions for map counts and their relation to solutions of the discrete Painlevé I equation. We will then present computational and analytical results that lead to specific generating functions and, in the case of 4-valent maps, derive explicit expressions for map counts as functions of the number of vertices and the genus of the surface on which the map is embedded. We will conclude with open as well as recently solved questions associated with this research program. This is joint work with Nick Ercolani and Brandon Tippings.

In the study of commutative rings, several algebraic properties are captured by numerical invariants which are defined in terms of ideals and their powers. Among these, of particular relevance are the reduction number and analytic spread of an ideal, which control the growth of the powers of the given ideal for large exponents. Unfortunately, these invariants are usually difficult to calculate for arbitrary ideals, and different methods might be required depending on the specific features of the class of ideals under examination.

In this talk, I will discuss a combinatorial method to calculate the reduction number of an ideal, based on a homological characterization in terms of the regularity of a graded algebra. This is part of ongoing joint work with Louiza Fouli, Kriti Goel, Haydee Lindo, Kuei-Nuan Lin, Whitney Liske, Maral Mostafazadehfard and Gabriel Sosa.

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Homeostasis occurs in a biological system when a chosen output variable remains approximately constant despite changes in an input variable. In this work we specifically focus on biological systems which may be represented as chemical reaction networks and consider their infinitesimal homeostasis, where the derivative of the input-output function is zero. The specific challenge of chemical reaction networks is that they often obey various conservation laws complicating the standard input-output analysis. We derive several results that allow to verify the existence of infinitesimal homeostasis points both in the absence of conservation and under conservation laws where conserved quantities serve as input parameters. In particular, we introduce the notion of infinitesimal concentration robustness, where the output variable remains nearly constant despite fluctuations in the conserved quantities. We provide several examples of chemical networks which illustrate our results both in deterministic and stochastic settings.

Title and abstract to be announced

Title and abstract to be announced

Title and abstract to be announced