Termine

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.

Further information

Generative AI is rapidly transforming the landscape of mathematical research by solving many open mathematical problems. I will talk about these advances from the perspective of formalized mathematics. Many of these AI solutions come with an AI-generated formalization in Lean. These might not help to understand the mathematical argument, but do provide some guarantees about the correctness of the proof. I will discuss to what extent we can trust Lean formalizations and what there is left to check and my perspectives on autoformalization.