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        "event_title":"Student Number Theory: An Introduction to Perfectoid Theory",
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        "combined_title":"Student Number Theory: An Introduction to Perfectoid Theory: Zijie Tao",
        "event_subtitle":"Zijie Tao",
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        "description":"Nearly fifteen years ago, Peter Scholze introduced perfectoid fields, establishing a bridge between deeply ramified fields of characteristic zero and characteristic (p). Perfectoid theory has since found important applications in arithmetic geometry. In this talk, I will follow Scholze\u2019s original paper to give an introduction to perfectoid fields and, if time permits, perfectoid spaces.",
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        "datetime_modified":"20261001T231756",
        "datetime_start":"20261014T150000",
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        "has_end_time":1,
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        "time_start":"15:00:00",
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        "time_zone":"America\/Detroit",
        "event_title":"Entropy-Regularized Finite-State Mean Field Control Problems",
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        "combined_title":"Entropy-Regularized Finite-State Mean Field Control Problems: Aditya Priya",
        "event_subtitle":"Aditya Priya",
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        "description":"We study an infinite-horizon discounted mean field control problem for finite-state nonlinear continuous-time Markov chains. Controls are locally integrable transition-rate matrices and are penalized by relative entropy with respect to a state-dependent reference generator. We prove that the value function is the unique bounded viscosity solution on the closed simplex of the Hamilton\u2013Jacobi\u2013Bellman equation by a direct dynamic-programming argument and a finite-state comparison principle. Under a uniform semiconvexity condition and C\u00b9,\u00b9 regularity, the value function is globally semiconvex and intrinsically C\u00b9,\u00b9 on the closed simplex and yields a Gibbs feedback that is the unique Hamiltonian minimizer at interior states. For positive, symmetric, possibly state-dependent reference rates, an entropy-plus-value free energy decays globally exponentially when the temperature parameter is compatible with the semiconvexity modulus. We also establish a discounted planner-to-game correspondence for potential mean field games under a state-independent reference generator.",
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