The Elusive Singularity

I will describe the open problems of singularity formation in incompressible fluids. I will discuss a list of related models, some results, and some more open problems.

Date: Monday, 11 November 2024 

Time: 5pm GMT

Location: Lecture Theatre 1, Mathematical Institute 

Speaker: Professor Peter Constantin            

Peter Constantin is the John von Neumann Professor of Mathematics and Applied and Computational Mathematics at Princeton University. Peter Constantin received his B.A and M.A. summa cum laude from the University of Bucharest, Faculty of Mathematics and Mechanics. He obtained his Ph.D. from The Hebrew University of Jerusalem under the direction of Shmuel Agmon.

Constantin’s work is focused on the analysis of PDE and nonlocal models arising in statistical and nonlinear physics. Constantin worked on scattering for Schr¨odinger operators, on finite dimensional aspects of the dynamics of Navier-Stokes equations, on blow up for models of Euler equations. He introduced active scalars, and, with Jean-Claude Saut, local smoothing for general dispersive PDE. Constantin worked on singularity formation in fluid interfaces, on turbulence shell models, on upper bounds for turbulent transport, on the inviscid limit, on stochastic representation of Navier-Stokes equations, on the Onsager conjecture. He worked on critical nonlocal dissipative equations, on complex fluids, and on ionic diffusion in fluids.

Constantin has advised thirteen graduate students in mathematics, and served in the committee of seven graduate students in physics. He mentored twenty-five postdoctoral associates. 

Constantin served as Chair of the Mathematics Department of the University of Chicago and as the Director of the Program in Applied and Computational Mathematics at Princeton University.

Constantin is a Fellow of the Institute of Physics, a SIAM Fellow, and Inaugural Fellow of the American Mathematical Society, a Fellow of the American Academy of Arts and Sciences and a member of the National Academy of Sciences

Since Hawking first discovered that black holes radiate, the evaporation of black holes has been a subject of great interest. In this talk, based on [2411.03447], we review some recent results about the evaporation of charged (Reissner-Nordström) black holes. We consider in particular the difference between neutral and charged particle emission, and explain how this drives the black hole near extremality, as well as how evaporation is then changed in that limit.

I will discuss the preprint arXiv:2409.18130 describing an interesting connection between 4d N=2 SCFTs and 2d chiral CFTs (alias: Vertex Operator Algebras) by way of 3d EFTs.

Abstract: I will introduce and explain a new symmetry structure for type IIA string theory, called string^h. Using string^h I will explain  how some objects of stable homotopy theory relating to elliptic cohomology enter into type IIA string theory.

The monadic second-order theory S1S of (ℕ,<) is decidable (it essentially describes ω-automata). Undecidability of the monadic theory of (ℝ,<) was proven by Shelah. Previously, Rabin proved decidability if the monadic quantifier is restricted to Fσ-sets.
We discuss decidability for Borel sets, or even σ-combinations of analytic sets. Moreover, the Boolean combinations of Fσ-sets form an elementary substructure. Under determinacy hypotheses, the proof extends to larger classes of sets.

Kirchhoff's celebrated matrix tree theorem expresses the number of spanning trees of a graph as the maximal minor of the Laplacian matrix of the graph. In modern language, this determinantal counting formula reflects the fact that spanning trees form a regular matroid. In this talk, I will give a short historical overview of the tree-counting problem and a related quantity from electrical circuit theory: the effective resistance. I will describe a characterization of effective resistances in terms of a certain polytope and discuss some recent applications to discrete notions of curvature on graphs. More details can be found in the recent preprint: https://arxiv.org/abs/2410.07756

Creating networks of statistical dependencies between brain regions is a powerful tool in neuroscience that has resulted in many new insights and clinical applications. However, recent interest in higher-order interactions has highlighted the need to address beyond-pairwise dependencies in brain activity. Multivariate information theory is one tool for identifying these interactions and is unique in its ability to distinguish between two qualitatively different modes of higher-order interactions: synergy and redundancy. I will present results from applying the O-information, the partial entropy decomposition, and the local O-information to resting state fMRI data. Each of these metrics indicate that higher-order interactions are widespread in the cortex, and further that they reveal different patterns of statistical dependencies than those accessible through pairwise methods alone. We find that highly synergistic subsystems typically sit between canonical functional networks and incorporate brain regions from several of these systems. Additionally, canonical networks as well as the interactions captured by pairwise functional connectivity analyses, are strongly redundancy-dominated. Finally, redundancy/synergy dominance varies in both space and time throughout an fMRI scan with notable recurrence of sets of brain regions engaging synergistically. As a whole, I will argue that higher-order interactions in the brain are an under-explored space that, made accessible with the tools of multivariate information theory, may offer novel insights.