Abstract: Topological entropy is a fundamental invariant of a dynamical system, measuring its complexity. In this talk, we discuss connections between the topological entropy of a Hamiltonian system, e.g., a geodesic flow, and the underlying filtered Morse or Floer homology viewed as a persistence module in the spirit of Topological Data Analysis. We introduce barcode entropy — a Morse/Floer theoretic counterpart of topological entropy — and show that barcode entropy is closely related to topological entropy and that, in low dimensions, these invariants agree. For instance, for a geodesic flow on any closed surface, the barcode entropy is equal to the topological entropy. The talk is based on joint work with Erman Cineli, Viktor Ginzburg, and Marco Mazzucchelli.
Abstract: We derive an asymptotic expansion of Laplace-type integrals in which dimension grows together with the large parameter in the exponent. This fills a gap in the theory between the classical fixed dimensional regime dating back to Laplace, and more recent work on the asymptotic expansion of infinite dimensional Laplace-type integrals due to Ben Arous. We also present related work on approximations of high-dimensional Laplace-type probability densities. These results resolve several long-standing open questions in the theory of both general asymptotic analysis and high-dimensional Bayesian statistics. Beyond their theoretical significance, our results are useful for statistical computation such as model selection and uncertainty quantification in Bayesian inverse problems and other data science settings.
Abstract: For every odd prime p, the number 2 + 2cos(2 pi/p) is an algebraic integer whose conjugates are all positive numbers; such a number is known as a totally positive algebraic integer. For large p, the average of the conjugates of this number is close to 2, which is small for a totally positive algebraic integer. The Schur-Siegel-Smyth trace problem, as posed by Borwein in 2002, is to show that no sequence of totally positive algebraic integers could best this bound.
In this talk, we will resolve this problem in an unexpected way by constructing infinitely many totally positive algebraic integers whose conjugates have an average of at most 1.899. To do this, we will apply a new method for constructing algebraic integers to an example first considered by Serre. We also will explain how our method can be used to find simple abelian varieties with extreme point counts.
Abstract: The study of matrix algebras is of fundamental importance in non-commutative analysis, especially in disciplines such as operator algebras. In recent decades, there has been a significant effort to uncover the analytic and probabilistic behavior of large N-limits of matrices. Voiculescu initiated his free probability theory with a crucial insight on the limiting joint spectral distribution of pairs of independent random Gaussian ensembles. This limiting distribution actually is concretely seen inside the von Neumann algebra associated to the free group on two generators. Since this result, there have been several deep contributions in this line of research, including theories of entropy, strong convergence, etc, with powerful applications to the study of operator algebras. In this talk I will describe my recent contributions to this area studying the so-called ultraproduct of matrix algebras, with applications to the internal structure of the free group von Neumann algebras, continuous model theory, and new considerations related to the famous Connes embedding problem which has been recently resolved using quantum complexity theory.
Abstract: Addressing contemporary problems of collective action—from pandemic management to climate change—requires that we understand the dynamic interplay between information and behavior. In this talk, I will discuss two models of cooperative behavior coupled with dynamics of information spread. In the first model, we will consider cooperation driven by the spread of social reputations. Using methods from evolutionary game theory and dynamical systems, we develop a mathematical model of cooperation that integrates a mechanistic description of how reputations spread through peer-to-peer gossip. We show that sufficiently long periods of gossip can stabilize cooperation by facilitating consensus about reputations. In the second model, we will examine the dynamics of prosociality under political polarization. We develop a stochastic model of game-theoretic opinion dynamics in a multi-dimensional space of political interests. We show that while increasing the diversity of interests can improve both cooperation and social cohesion, strong partisan bias reduces the effective dimensionality of the opinion space via self-sorting along party lines, yielding greater in-group cooperation at the cost of increasing polarization. Taken together, these studies contribute to our understanding of when and how communication and opinion contagion facilitate cooperation.
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