Date/Time : 11 Dec 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jiaqi Liu, University of Southern California

The Guderley problem describes the behavior of a strong shock wave propagating radially in an ideal gas. A spherical shock converges radially inwards to the spatial origin, strengthening as it collapses. At the collapse point, the shock’s strength becomes infinite, leading to the formation of a new outgoing shock wave of finite strength, which then propagates outwards to infinity. In this talk, I will present recent work on the rigorous construction of the Guderley solution for $\gamma \in (1,3]$. These solutions are analytic away from the shock interfaces and the blow-up point. The proof relies on continuity arguments, nonlinear invariances, and barrier functions.

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Date/Time : 10 Dec 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Gianluca Faraco, University of Milano Bicocca

A translation surface is the datum of an abelian differential on a Riemann surface. Every such pair determines a representation called absolute period representation or period character. In this talk we discuss the realization of a given representation as the period character of some translation surface.

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Date/Time : 09 Dec 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Daniel Mathews, Monash University

Work of Penrose and Rindler in the 1980s developed a formalism for spinors in relativity theory. In their work they gave geometric interpretations of 2-component spinors as flags in Minkowski space. We present some extensions of this work involving hyperbolic geometry. In particular, we give a correspondence between between spinors and horospheres with certain decorations, and show how spinors can be used to calculate a complex-valued, 3-dimensional generalisation of Penner’s lambda lengths. Time permitting, we may discuss some further applications, including inter-cusp distances in knot complements, higher-dimensional generalisations, and circle packings.

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Date/Time : 05 Dec 2024 15:30
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Igor Balla, Masaryk University

Despite being elementary objects in geometry, large sets of equiangular lines appear naturally in a wide variety of fields including elliptic geometry, frame theory, Banach space theory, spectral graph theory, and quantum information theory with connections to algebraic number theory and Hilbert’s twelfth problem. In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in an r-dimensional Euclidean space with a given angle. In this talk, we will present new upper bounds on this problem that unify and significantly extend or improve essentially all previously known results, as well as being best possible either exactly or up to a factor of 2. These recent results were obtained independently or together with Matija Bucic.

Roughly speaking, our approach relies on new lower bounds on the second eigenvalue of a graph as well as new upper bounds on its multiplicity. In particular, we obtain the first extension of the Alon–Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to families of r+1 choose 2 equiangular lines in r dimensions. Time permitting, we will also mention some results in the complex setting.

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Date/Time : 05 Dec 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Hong Liu, Institute for Basic Science, South Korea

A well-known theorem of Nikiforov asserts that any graph with a positive $K_r$-density contains a logarithmic blowup of $K_r$. We explore variants of Nikiforov’s result and investigate when positive clique density condition implies the existence of a significantly larger blowup of a clique. Our results study such problems for families of ordered graphs with forbidden induced monotone path, obtaining optimal bounds. As corollaries, we strengthen a result of Pach and Tomon, and resolve a conjecture of Tomon in a strong form. To find a large blowup, we reduce the embedding problem to a certain Ramsey problem. For optimal lower bound constructions, we make use of concentration of measure and the isodiametric inequality on high dimensional spheres.

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Date/Time : 04 Dec 2024 14:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Federico Rodriguez Hertz, The Pennsylvania State University

In the 80’s two type of related rigidity results emerged, one of geometric flavor and the other dynamical, but both have a dynamical background. Otal and Croke showed that for negatively curved surface the marked length spectrum determines the isometry type of the surface. On the other hand, Shub ans Sullivan for expanding maps and later de la LLave, Marco and Moriyon for Anosov diffeomorphisms in dimension 2 showed that the marked Lyapunov spectrum determines the smooth isomorphism type of the system. In this talk I will discuss new developments along this line of problems, discussing a more general framework where these theory can be developed. This project is joint with A. Gogolev and M. Leguil.

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Date/Time : 04 Dec 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Thomas Karam, University of Oxford

This talk shall be essentially composed of four parts, each consisting of a short introduction to a research area followed by some contributions. We shall begin by discussing a central conjecture in Ramsey theory unifying several generalisations of the van der Waerden theorem. Thereafter, our focus shall move to the extension of basic properties of the rank of matrices to the different notions of rank of tensors, followed by the extension of basic properties of polynomials over finite fields to restrictions of these polynomials with Boolean variables. Finally, we shall study the Brunn-Minkowski inequality in more arbitrary ambient spaces than R^n, in particular in the torus (R/Z)^n for subsets satisfying various levels of constraints.

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Date/Time : 03 Dec 2024 15:30
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Federico Rodriguez Hertz, The Pennsylvania State University

In a joint work in progress with Pablo Carrasco we show that to any quasi-morphism for a hyperbolic group can be associated a function with the Bowen property for some hyperbolic dynamics. In this talk I plan to describe the construction and how to prove the CLT for a quasi-morphism w.r.t. some length function.

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Date/Time : 03 Dec 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : An Xinliang, National University of Singapore

In this talk, I will give an overview of my recent research. We prove a series of mathematical theorems that rigorously describe the entire process of black hole formation. With scale-critical short-pulse initial data, we solve Einstein’s equations and show that: “in the course of gravitational collapse, a whole trapped region and its boundary (apparent horizon) emerge from a singular spacetime point. The apparent horizon then expands as the spacetime evolves. Passing through a transition period, the spacetime later enters a regime, close to a slowly-rotating Kerr black hole solution. Nonlinear Kerr stability then comes into play and the spacetime metric eventually settles into the Kerr solution with small angular momentum.”

Meanwhile, I also introduce a new framework for proving the high co-dimensional nonlinear instability of naked-singularities. Specifically, I study the censorship of Christodoulou’s naked singularity and show that even a tiny anisotropic perturbation can trigger the formation of an anisotropic apparent horizon, thereby covering and censoring the naked singularity.

If time permits, sharp low-regularity illposedness and wellposedness results for nonlinear wave systems will also be presented.

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