Date/Time : 25 Nov 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
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.

This is joint work with S. Cao.

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Date/Time : 15 Oct 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Kai Dong, National University of Singapore

In this talk, we give a new proof of the past nonlinear stability of Kasner solutions for $3+1$-dimensional Einstein vacuum equations with polarized $U(1)$-symmetry. Our method, following Beyer, Oliynyk, and Arturo Olvera-Santamaria \cite{beyer2021fuchsian}, relies on energy estimates for a class of first order hyperbolic PDEs, namely Fuchsian systems.

We show that the perturbed solutions are asymptotically Kasner, geodesically incomplete and crushing at the big bang singularity. This is achieved by reducing the $3+1$ Einstein vacuum equations to a Fuchsian system coupled with several constraint equations, with the symmetry assumption playing an important role in the reduction. Using Fuchsian theory together with finite speed of propagation, we obtain global existence and precise asymptotic behavior of the solutions up to the singularities.

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Date/Time : 15 Oct 2025 15:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Liang Zhao, National University of Singapore

In this talk, we will consider the simplification from the multi-dimensional (M-D) bipolar Euler–Poisson (2EP) equations to the unipolar Euler–Poisson (1EP) equations of electrons, which are both important models in plasma physics and semiconductors. Based on some rigorous convergence analysis of certain physical parameters, we are able to justify these simplifications for two classes of global-in-time solutions of the M-D 2EP system. The first is the smooth solution of the damped M-D 2EP system near the constant equilibrium state, and the second is the regular solution to the Cauchy problem for the undamped M-D 2EP system with far-field vacuum. The global-in-time error estimates between solutions to 2EP and 1EP are also established.

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Date/Time : 07 Oct 2025 13:00
Venue : S16 #05-18
Speaker : Jiakun Liu, The University of Sydney

In this talk, we will introduce the optimal transportation with some interesting applications in various fields including a reconstruction problem in cosmology; a brief proof of isoperimetric and Brunn-Minkowski inequalities in geometry; and an application in image recognition relating to a transport between hypercubes. We will also discuss our recent research on the regularity of free boundaries in optimal transportation.

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Date/Time : 30 Sep 2025 15:00
Venue : S16 #05-17
Speaker : Zhifei Zhu, Tsinghua University

In this talk, we show that the length of a shortest closed geodesic on a Riemannian manifold of dimension 4 with diameter D, volume v, and |Ric|<3 can be bounded by a function of v and D. In particular, this function can be explicitly computed if the manifold is Einstein. The proof of this result depends on a structural theorem proven by J. Cheeger and A. Naber. This is joint work with N. Wu.

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Date/Time : 30 Sep 2025 16:00
Venue : S16 #05-17
Speaker : Wenze Su, Tsinghua University

In this talk, I will present a recent result on the finite-time shock formation for the compressible Euler equations on the two-dimensional sphere $\mathbb{S}^2$. Unlike the flat Euclidean case $\mathbb{R}^2$, the geometry imposes new difficulties and the fluid dynamics is affected by the curved background.

To overcome these challenges, we modified the existing modulation method and employ a set of carefully constructed, time-dependent coordinates that precisely track the shock formation on $\mathbb{S}^2$. In particular, we apply time-dependent rotations on the sphere, stereographic projections for the rotated sphere, and straighten the steepening shock front to construct the desired coordinate system. Under these coordinates, the Euler’s equations on $\mathbb{S}^2$ can be recast into a new form that is suitable for self-similar analysis.

Within this framework, we then implement detailed bootstrap arguments and establish global well-posedness for the self-similar system. After translating these results back to the original physical system, we thereby demonstrate the finite-time shock formation on $\mathbb{S}^2$.

This talk is based on joint work with Xinliang An, Haoyang Chen, and Fulin Qi.

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Date/Time : 02 Sep 2025 14:00
Venue : S16 #05-17
Speaker : Wang Yamin, National University of Singapore

The classical Alt-Phillips problem, appearing in the study of permeable catalysts and population dynamics, was first studied by Phillips [1983] and Alt-Phillips [1986]. It embeds two of the most well studied elliptic free boundary problems as special cases, namely, the Alt-Caffarelli problem when the parameter $gamma=0$ and the obstacle problem when $gamma=1$.

In this talk, we study the boundary behavior of a solution to the fully nonlinear Alt-Phillips problem with $\gamma\in (1,2)$. We show that the free boundary intersects the fixed boundary tangentially where the Dirichlet data vanish. For this range of $\gamma$, our result is new even when the operator is the Laplacian. This is joint work with Yu Hui.

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Date/Time : 12 Aug 2025 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Matias Delgadino, University of Texas at Austin

We prove sharp estimates for the mean-field limit of weakly interacting diffusions with repulsive logarithmic interaction in arbitrary dimension. More precisely, we show that the associated partition function is uniformly bounded in the number of particles N for an arbitrary bounded base measure. Combined with the modulated free energy method, this amounts to a logarithmic improvement in N of the current best available closeness estimates in the literature. Our arguments are inspired by and borrow ideas from Nelson’s classical construction of the \Phi^4_2 Euclidean quantum field theory.

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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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