Termine

In the theory of G-expectation, uncertainty about the volatility of a process is modeled by taking a supremum over all admissible choices of the volatility. This can be formulated as a stochastic control problem, whose value function is described by a Nisio semigroup and a Hamilton?Jacobi?Bellman equation. Such semigroups can be approximated by a Chernoff-type time-stepping scheme, and the computed value functions converge to that of the control problem. Less clear is what happens to the associated optimal, piecewise constant controls in the limit: even in simple examples they oscillate ever faster (?chattering?) and do not converge to a control in the classical sense. In this talk, based on my master?s thesis, I show how this problem can be handled using Young measures and relaxed controls via the martingale problem. The main result states that the laws of the optimal controls of the approximating schemes, together with their state processes, are tight, and that every accumulation point induces a relaxed control that maximizes the relaxed problem.

The aim of this talk is to discuss a vanishing result for the top-dimensional rational cohomology of arithmetic Chevalley groups such as SL_n(Z). I will introduce this class of groups, which can be thought of as the algebraic counterpart of semisimple Lie groups, together with the Tits building, a geometric object on which they act. Via Borel?Serre duality, understanding the top-dimensional rational cohomology reduces to a combinatorial question about the homology of the building. We will see that this cohomology vanishes for G(O), where G is any semisimple Chevalley group with irreducible root system and O is a Euclidean ring of integers. This completes previous work of Tóth and of Brück, Santos Rego and Sroka.

Abstract: I will report on ongoing joint work with Devalapurkar and Schlank. We construct a telescopic analog of the Beilinson fiber square, providing a simple formula for the difference between the T(n)-local K-theory of a ring spectrum R and that of its T(n-1)-localization. In fact, there is a highly structured explanation for this square in terms of the T(n)-localization of the category of noncommutative motives, which we explain. There are many applications to our understanding of redshift, of which I will discuss some. In particular, we find explicit formulas for the K(n)-local K-theory of many K(n-1)-local ring spectra in terms of Galois cohomology.

The Kähler?Ricci flow is a powerful parabolic tool for studying the geometry of Kähler manifolds. Its interplay with complex geometry suggests that singularity formation and continuation through singularities can be described by canonical geometric models. In this talk, I will focus on non-collapsed singularities, particularly in complex dimension two, and present a quantitative description of their geometry. I will explain how shrinking solitons, Kähler cones, and expanding solitons fit together to model the flow before, at, and after a singular time. This talk is based on joint work with Max Hallgren and Lucas Lavoyer.

By a connection with a formal parameter, we mean a connection on a scheme $X$ that is relatively smooth over the ring $R = k[[t]]$, where $k$ is a field of characteristic $0$. The monodromy of such a connection is a flat affine group scheme over $R$. We introduce the notion of prudent group schemes in terms of their representations; this property turns out to be equivalent to the projectivity of the coordinate ring (a result that holds over any complete DVR). In this talk, we discuss the question: When is the monodromy group of a connection prudent? Our main tool is the concept of hyper-prudence, which involves the $t$-adic completion of the representation category. This is a joint work with J.P. dos Santos (Montpellier) and Dao Van Thinh, Pham Thanh Tam (Hanoi).

Extremal black holes have long been regarded as unphysical objects that cannot arise via gravitational collapse in physically plausible scenarios. In the asymptotically flat setting, this paradigm was overturned by Kehle and Unger, who used characteristic gluing to construct regular, spherically symmetric solutions to the Einstein?Maxwell-charged scalar field system that collapse to form an exactly extremal Reissner?Nordström black hole in finite time. In this talk, we explore how these techniques can be extended to include a positive cosmological constant, yielding solutions to the Einstein-scalar field system in which an exactly extremal Schwarzschild?de Sitter black hole is formed. We also discuss the dynamical formation of the related extremal Schwarzschild?de Sitter white hole and Nariai spacetimes.