Date/Time : 12 Aug 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Bjoern Bringmann, Princeton University

There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimension, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton.

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Date/Time : 18 Jun 2025 16:00
Venue : S17 #06-11 (SR6)
Speaker : Jaydeep Singh, Princeton University

In this talk we will discuss self-similar solutions to the Einstein-scalar field system, first studied by Christodoulou (’94). These are a family of C^{1,\alpha} solutions modeling spacetime singularities without an event horizon. In BV spaces their instability is well understood via a geometric blue-shift effect, and leads to the formation of black holes. We will share progress on extending these results to Hölder spaces, and discuss some of the analytical and conceptual obstacles to doing so.

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Date/Time : 30 Apr 2025 16:00
Venue : S17 #04-05 (SR2)
Speaker : Silu Yin, Hangzhou Normal University

In this talk, we investigate the ill-posedness of the non-strictly hyperbolic compressible ideal MHD (IMHD) system in two dimensions，which exhibits multiple propagation speeds. For scalar quasilinear wave equation, Smith-Tataru showed its local well-posedness in $H^{11/4+}(R^2)\times H^{7/4+}(R^2)$. Here, we construct a counterexample to the local existence of low-regularity solutions to the IMHD system in the borderline space $H^{7/4}(R^2)$. Our proof is based on a coalition of a carefully designed algebraic approach and Christodoulou’s geometric approach. We give a complete description of solutions’ dynamics up to the earliest singular event, when a shock forms. This talk is based on a joint work with Xinliang An and Haoyang Chen.

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Date/Time : 29 Apr 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Federico Pasqualotto, University of California San Diego

In this talk, I will first review the singularity formation problem in incompressible fluid dynamics, describing how particle transport poses the main challenge in constructing blow-up solutions for incompressible fluids. I will then outline a new mechanism that allows us to overcome the effects of particle transport, leveraging the instability seen in the classical Taylor–Couette experiment. Using this mechanism, we construct the first swirl-driven singularity for the incompressible Euler equations in R^3. This is joint work with Tarek Elgindi (Duke University).

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Date/Time : 23 Apr 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Luca Spolaor, University of California San Diego

I will discuss the boundary regularity for minimizers of the so-called Alt-Caffarelli problem and how it is related to a boundary unique continuation problem. This is joint work with L. Ferreri (SNS) and Bozhidar Velichkov (Pisa).

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Date/Time : 16 Apr 2025 16:00
Venue : S17 #04-04 (SR3)
Speaker : Weijun Xu, Peking University

We consider two types of stochastic perturbations to the one dimensional Allen-Cahn equation, one with spacetime white noise, and the other with more singular noise. In both models, the interface location, when suitably rescaled, converges to a diffusion process on the real line. Cancellations play important roles in both models, but they happen in very different manners.

In the first model, we construct “correctors” that are designed to cancel potential divergences. We thus recover classical results by Funaki (1995) and Brassesco-Butta-De Masi-Presutti (1998), but also shed some new thoughts. In the second model, the cancellation happens somewhat miraculously, and the result seems to be new. Based on joint works with Wenhao Zhao (EPFL) and Shuhan Zhou (PKU).

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Date/Time : 16 Apr 2025 15:00
Venue : S17 #05-12 (SR4)
Speaker : Mu-Tao Wang, Columbia University

Two black holes rotate about each other and eventually merge into a single black hole. How does one measure the angular momentum carried away by gravitational radiation during this merger? This has been a subtle issue since the 1960’s due to the existence of “supertranslation ambiguity”: the angular momentums recorded by two observers of the same system may not be the same.

In this talk, I shall describe how the mathematical theory of quasilocal mass and optimal isometric embedding identifies a new definition of angular momentum that is free of any supertranslation ambiguity. In addition, some recent development of the cross-section continuity of the angular momentum definition will also be discussed. This is based on joint work with Po-Ning Chen, Jordan Keller, Daniel Paraizo, Robert Wald, Ye-Kai Wang, and Shing-Tung Yau.

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Date/Time : 26 Mar 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Kim Sunghan, Uppsala University

Constraint maps are critical points to the Dirichlet energy subject to an image constraint. Namely, the constraint restricts these maps to lie outside a given (smooth, open-connected) set in the target space, which can then be thought of as an obstacle. The target space is multi-dimensional, making this problem a natural extension of the obstacle problem into the vectorial setting. The presence of the obstacle naturally induces free boundary, which is the interface between the set where the image of the maps lies away from the obstacle and the set where their image lies on the obstacle. Yet, constraint maps also develop singularities, such as discontinuous points and branch points which have been the main subjects in the theory of harmonic maps and minimal surfaces, respectively. In this talk, I will discuss the recent breakthroughs on the tantalizing interplay between the free boundary and the mapping singularities for energy-minimizing constraint maps, based on a series of my collaboration with Alessio Figalli, André Guerra (ETH), and Henrik Shahgholian (KTH).

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Date/Time : 17 Mar 2025 16:00
Venue : S16 #04-35
Speaker : Jiajie Chen , New York University

While it is known that a pre-shock or implosion singularity can form in finite time for the compressible Euler equations from smooth initial data, whether the vorticity blows up in finite time has remained an open problem. In this talk, we will explore the Euler equations with axisymmetry. Using (nearly) axisymmetric flow, we construct vorticity blowup in the compressible Euler equations in R^d, d >=2 with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity and is accompanied by a (nearly) axisymmetric implosion, in which the swirl velocity exhibits full stability as opposed to finite co-dimension stability.

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