Alejandro Galvan
Introduction to Ramsey Theory
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Ramsey theory deals with questions of the type: "how big must some structure be to guarantee the existence of a certain substructure?" In particular, Ramsey's theorem states that, given two integers k, l > 1, there exists a number R(k,l) (called the Ramsey number) such that the following holds: any coloring of the edges of the complete graph of order R(k,l) with colors red and blue contains either a red clique of order k or a blue clique of order l.
In this talk, we will prove Ramsey's theorem and show bounds for the Ramsey numbers. Then, we will present an infinite version of Ramsey's theorem, which we will then use — together with some topology — to prove a generalization of the finite case.
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Rohan Kapoor
Rigged Hilbert Spaces
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This is an adapted version of my advancement talk, where I will discuss the rigged Hilbert space formalism: a method originally introduced by Gelfand for making the mathematics of quantum mechanics mathematically precise. I will do an example of its application to the quantum harmonic oscillator, and then, if time permits, describe two new techniques for rigged Hilbert spaces as applied to Koopman operator theory, one of which is joint work by Giannakis, Camper and me.
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Anna Vasenina
Bias, Structure, and Inference in Applied Network Analysis
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This dissertation develops mathematical and statistical methods for extracting reliable information from network data across biological applications, with an emphasis on understanding what observed network structure can and cannot resolve. The first study leverages protein-protein interaction network topology in the c-di-GMP signaling system of Pseudomonas fluorescens, showing that node centrality measures accurately classify protein domain types and that physical interaction structure contributes statistically significant predictive power for biofilm formation phenotypes across nearly 200 environments, while gene expression does not. The second study examines sampling bias in lemur-plant trophic interaction networks in Madagascar, demonstrating that differential detection of diurnal versus nocturnal species confounds trait-based predictors of dietary richness and that sampling effort mediates key functional trait effects. The third study formalizes this bias problem in a Bayesian framework, analyzing the identifiability of latent Poisson rates under unequal detection probabilities and characterizing the conditions under which informative priors on detection can and cannot disentangle true rate differences from observational artifacts.
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Graduating Students
Graduating Students' Advice Seminar!
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Each year, typically during the Spring Quarter, we host our Graduating Student Advice Seminar—an event that celebrates our graduating PhD cohort and honors their achievement of this significant career milestone. The seminar also gives our students an opportunity to share reflections on their time at Dartmouth, offering valuable insights and advice to the younger PhD cohorts.
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Michaela Polley and Alejandro Galvan
Departmental Checkin
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For GSS this week we will be having a department check-in. Come with your concerns, advice, and ideas! We will share updates regarding concerns raised in previous check-ins and also open the floor to any new concerns that people have.
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Alex Moon
What can you tell about a Kohnert poset or polynomial from its initial diagram?
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Kohnert posets and polynomials are a relatively simple construction with deep connections to representation theory, geometry, symmetric functions, and more. While it has been well-established that Kohnert polynomials capture Demazure characters and Schubert polynomials, the study of Kohnert posets and polynomials in their own right is fairly young. Given a fixed initial diagram, even questions of whether a Kohnert polynomial is monomial multiplicity free, whether the Kohnert poset is bounded or ranked, what its height is, and what the minimum length of a maximal chain have been poorly understood until the last few years. In this talk, I will discuss a joint work with Beth Anne Castellano and others in which we answer the first three questions in the case that the initial diagram is Northeast, and a work with Nicholas Meyers and Theo Koss in which we answer the last two questions in full generality.
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Jonathan Lindbloom
What can you tell about a Kohnert poset or polynomial from its initial diagram?
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Eigenvalues are one of the most important tools in linear algebra, but for some matrices they do not tell the whole story. In particular, nonnormal matrices can behave in surprising ways even when their eigenvalues at first look harmless. In this talk, I will discuss several ways to visualize the behavior of a matrix, including the spectrum, the numerical range, and the pseudospectra. While the numerical range gives a useful geometric “picture” of a matrix, pseudospectra provide a more refined view by studying how eigenvalues move under small perturbations, or equivalently where the resolvent becomes large. Through simple examples, I will explain how these ideas shed light on phenomena such as transient growth in the matrix exponential, bounds for norms of matrix functions (|f(A)|_2), numerical stability for PDEs, and convergence bounds for GMRES. Time permitting, I will also briefly discuss connections to open problems such as Crouzeix’s conjecture.
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