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Yelena Mandelshtam, IAS Princeton
The Kadomtsev-Petviashvili (KP) equation is a partial differential equation whose study yields fascinating connections between integrable systems, algebraic geometry, and combinatorics. In this talk I will describe some of the various approaches to connecting KP solutions to algebraic objects such as algebraic curves (due to Krichever) and the positive Grassmannian (due to Sato and later Kodama-Williams). I will then discuss recent and ongoing work to build bridges between these approaches.
Katrina Morgan, Temple University
A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the so-called "string". This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that forward solutions to the wave equation (in an appropriate microlocal sense) do exist. Our techniques involve proving a statement on propagation of singularities and using the resulting estimates to show existence of solutions. This is joint work with Jared Wunsch.
Mert Gurbuzbalaban, Rutgers University
Langevin algorithms, integral to Markov Chain Monte Carlo methods, are crucial in machine learning, particularly for Bayesian inference in high-dimensional models and addressing challenges in stochastic non-convex optimization prevalent in deep learning. This talk delves into the practical aspects of stochastic Langevin algorithms through three illuminating examples. First, it explores their role in non-convex optimization, focusing on their efficacy in navigating complex landscapes. The discussion then extends to decentralized Langevin algorithms, emphasizing their relevance in distributed optimization scenarios, where data is dispersed across multiple sources. Lastly, the focus shifts to constrained sampling, aiming to sample from a target distribution subject to constraints. In each scenario, we introduce new algorithms with convergence guarantees and showcase their performance and scalability to large datasets through numerical examples.
Kristina Wicke, New Jersey Institute of Technology
Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of speciation and reticulate evolutionary events such as hybridization or horizontal gene transfer. The inference of phylogenetic networks from biological sequence data is a challenging problem, with many theoretical and practical questions still unresolved. In this talk, I will give an overview of the state of the art in phylogenetic network inference. I will then discuss a novel divide-and-conquer approach for inferring level-1 networks under the network multispecies coalescent model. I will end by discussing some open problems and avenues for future research.
Parts of this talk are based on joint work with Elizabeth Allman, Hector Banos, and John Rhodes.
There are no conferences this week.