Events of the Week

A generating function is a series whose coefficients are a sequence of interest. They are one of the most important tools in analytic number theory. I will give an introduction to generating functions and talk about an example from my research, the generating function for k-regular partition numbers.

We discuss existence and volume equals multiplicity for generalized Hilbert Kunz Multiplicities for p-families of ideals. We also exhibit Minkowski inequalities for p-families.

Recent improvements in technology have enabled neuroscientists to simultaneously record activity of many neurons at high spatial and temporal resolution. This has allowed them to discover that activity is organized into a variety of spatial patterns such as plane waves, bullseyes, and rotating waves. In this talk, I want to distinguish two different classes or wave-like activity: (1) evoked waves or "trigger waves", and (2) phase waves. In the former, the onset of activity in one area requires prior activity in a neighboring area, while in the latter, the apparent wave motion is a consequence of timing differences between areas. I will present some recent results on the role of inhibition in controlling the propagation and stability of trigger waves. Next, I will consider coupled phase equations that describe spatio-temporal activity in intrinsically oscillatory media. I will describe recent work where we are able to extract hidden waves from human cortical recordings. Finally, I will present some work showing how ongoing phase waves can promote the propagation of trigger waves in an anisotropic manner.

My talk compares two within-host models of Trypanosoma cruzi infection that differ in their modeling of immune regulation (linear decay vs. logistic growth), revealing how nonlinear immune dynamics shape long-term outcomes. Both models can exhibit bistability, characterized by either two chronic infection equilibria or the coexistence of a healthy state and a chronic infection, but the more complex logistic growth model also produces a critical backward bifurcation when the basic reproduction number is below one (R_0 < 1), allowing a stable chronic infection to persist even under conditions expected to lead to clearance.

In this talk, the estimand framework followed in the pharmaceutical industry will be discussed, using the language of causal inference. The definition of causal estimand and statistical estimand will be addressed, as well as the differences between randomized controlled trials (RCT) and non-interventional studies (NIS). Two basic identification strategies—the G-formula strategy and the weighting strategie—will be presented. With regard to estimation of the estimand, doubly robust methods will be discussed.

Additionally, a brief description will be provided of the difference between FDA submissions, which are focused on regulatory approval for marketing a product, and HTA submissions, which are focused on gaining reimbursement and demonstrating value to payers and healthcare systems. In particular, methods for direct and indirect comparisons will be discussed.

In summary, this talk is intended to serve as a high-level introduction to important causal inference topics for students interested in pursuing a career in the pharmaceutical industry.

Please join this seminar via: https://tulane.zoom.us/j/97158638464

Deligne connects the weight-zero compactly supported cohomology of a complex variety to the combinatorics of its compactifications. In this talk, we use this to study the moduli space of n-marked hyperelliptic curves. We use moduli spaces of G-admissible covers and tropical geometry to give a sum-over-graphs formula for its weight-0 compactly supported Euler characteristic, as a virtual representation of S_n. This is joint work with Melody Chan and Siddarth Kannan.

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I hope you can participate.

I hope you can participate.

I hope you can participate.

e explore connections between intersection theory and multiplicity theory over Noetherian local rings. We begin by constructing an intersection product for schemes that are proper and birational over a Noetherian local ring, using the theory of rational equivalence developed by Thorup and the Snapper-Mumford-Kleiman intersection theory. This yields a new proof of a classical theorem of Rees on degree functions and leads to a generalization of Ramanujam’s formula for Hilbert-Samuel multiplicities to arbitrary Noetherian local rings. We also examine multiplicities and degree functions associated to graded families of m-primary ideals, especially divisorial filtrations in dimension two. This is joint work with Steven Dale Cutkosky.

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Recent progress in multiplexed tissue imaging is advancing the study of tumor microenvironments to enhance our understanding of treatment response and disease progression. Despite its popularity, there are significant challenges in data analysis, including high computational demands that limit feasibility for large-scale applications and the lack of a principled strategy for integrative analysis across images. To overcome these challenges, we introduce a spatial topic model designed to decode high-level spatial architecture across multiplexed tissue images. Our method integrates both cell type and spatial information within a topic modelling framework, originally developed for natural language processing and adapted for computer vision. We benchmarked its performance through various case studies using different single-cell spatial transcriptomic and proteomic imaging platforms across different tissue types. We show that our method runs significant faster on large-scale image datasets, along with high precision and interpretability. It consistently identifies biologically and clinically significant spatial “topics”, such as tertiary lymphoid structures.

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