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Hsuan-Yi Liao, Penn State University
A Lie pair $(L,A)$ consists of a Lie algebroid $L$ together with a Lie subalgebroid $A$. A wide range of geometric situations can be described in terms of Lie pairs including complex manifolds, foliations, and manifolds equipped with Lie algebra actions. We establish the formality theorem for Lie pairs. As an application, we obtain Kontsevich-Duflo type theorems for Lie pairs. In this talk, I'll start with the case of $\mathfrak{g}$-manifolds, i.e., smooth manifolds equipped with Lie algebra actions. After that I'll explain formality theorem and Kontsevich-Duflo theorem for Lie pairs and other geometrical situations.
Shif Berhanu, Temple University
TBA
Wai-Tong (Louis) Fan, University of Wisconsin-Madison
Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge, which is fundamental in any multi-scale modeling approach for complex systems, is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.
In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE, in particular, why naively adding diffusion terms to ordinary differential equations might fail to account for spatial dynamics in population models. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics.
Theresa Anderson, University of Wisconsin
Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior. The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example. In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to. We further quantify this with explicit bounds for naturally occurring maximal functions, which connects classical tools from harmonic analysis with analytic number theory. This is joint work with Cook, Hughes, and Kumchev.
Ed Letzter, Temple University
Tarik Aougab, Brown University
Title/abstract TBA
Kathryn Lund, Temple University
Abstract TBA
There are no conferences next week.