Spreading (diffusion) of new products is a classical problem. Traditionally, it has been analyzed using the compartmental Bass model, which implicitly assumes that all individuals are homogeneous and connected to each other. To relax these assumptions, research has gradually shifted to the more fundamental Bass model on networks, which is an agent-based model for the stochastic adoption decision of each individual. In this talk, I will present the emerging mathematical theory for the Bass model on networks. The main focus will be on the effect of network structure. For example, which networks yield the slowest and fastest adoption? I will also discuss the effect of heterogeneity among individuals: Does it always slow down the adoption? Can it be neglected?

]]>The Fontaine-Mazur conjecture predicts which two-dimensional p-adic representations of the absolute Galois group of Q arise from modular forms. In this talk, I will explain this conjecture by some concrete examples and report some recent progress which relies on developments of p-adic geometry.

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]]>Let C be an algebraic curve over Q, i.e., a 1-dimensional complex manifold defined by polynomial equations with rational coefficients. A celebrated result of Faltings implies that all algebraic points on C come in families of bounded degree, with finitely many exceptions. These exceptions are known as isolated points. We explore how these isolated points behave in families of curves and deduce consequences for the arithmetic of elliptic curves. This talk is based on joint work with A. Bourdon, Ö. Ejder, Y. Liu, and F. Odumodu.

]]>Abstract: Main goal of this talk is to introduce an exciting new area of mean field games modeling interactions between large number of identical particles. In this formalism, instead of positing the dynamics of the individual particles, one lets them endogenously determine their behaviors by minimizing a given cost functional and hopefully, settling in a Nash equilibrium. Initiated by Larry & Lions, and Huang, Malhame, & Caines in 2006, mean field games has found an amazing range of applications. This talk uses the specific example of classical Kuromato synchronization to introduce the novel approach and its potential. Originally motivated by systems of chemical and biological oscillators, the Kuramoto system is the key mathematical model to describe self organization in complex systems. These autonomous oscillators are coupled through a nonlinear interaction term which plays a central role in the long term behavior of the system. While the system is unsynchronized when this term is not sufficiently strong, fascinatingly, they exhibit an abrupt transition to a full synchronization above a critical value of the interaction parameter. Mean field approach also delivers same type of results including the phase transition from incoherence to synchronization.

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