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.
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.
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
This talk, based on joint work with Alexander Zlokapa, concerns Bayesian inference with neural networks.
I will begin by presenting a result giving exact non-asymptotic formulas for Bayesian posteriors in deep linear networks. A key takeaway is the appearance of a novel scaling parameter, given by # data * depth / width, which controls the effective depth of the posterior in the limit of large model and dataset size.
Additionally, I will explain some quite recent results on the role of this effective depth parameter in Bayesian inference with deep non-linear neural networks that have shaped activations.
One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and infinite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We discuss some dynamic games of this type, from static to two player to the associated mean field game, with some numerical and existence-uniqueness results.
Based on works with Andrew Ledvina and with Emre Parmaksiz.
I will explain a formalism for anticyclotomic Euler systems for a large class of Galois representations and explain how to prove analogs of Kolyvagins' celebrated "rank one" results. A novelty of this approach lies in the use of primes that split in the CM field. This is joint work with Dimitar Jetchev and Jan Nekovar. I will also describe some higher-dimensional examples of such Euler systems.
I will explain how to construct an Euler system for imaginary quadratic fields using Eisenstein series and their cohomology classes. This illustrates a template for a construction that should yield many new Euler systems.
I will explain a new construction of an Euler system for the symmetric square of an eigenform and its connection with L-values. The construction makes use of some simple Eisenstein cohomology classes for Sp(4) or, equivalently, SO(3,2). This is an example of a larger class of similarly constructed Euler systems. This is a report on joint work with Marco Sangiovanni Vincentelli.
I’m excited to share with everyone some new, unpublished work that we are just in the process of wrapping up and could use everyone’s reactions. It is a reconciliation of evolutionary game theory and ecological dynamics that I have wrestled with since I moved from an evolution program into an ecology-heavy department. It always seemed like, depending on the problem I was thinking about, I had to change my perspective and approach it as either an evolutionary game theorist, or an ecologist; and yet I had this nagging feeling that, at its core, the problem was often one and the same, and therefore one theoretical framework should suffice. So when should one write down an n-type replicator equation and when should one write down an n-species Lotka-Volterra system; and what does it mean mathematically and biologically when one has made such a choice? In the process of reconciling, I also got a deeper appreciation of what is and is not a proper game, such as a Prisoner’s Dilemma. These findings can help shed light on previously puzzling empirical findings.
Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.
In 1983 Thomassen conjectured that every graph of sufficiently large average degree has a subgraph of large girth and large average degree. While this conjecture remains open, recent evidence suggests that something stronger might be true; perhaps the subgraph can be made induced when a clique and biclique are forbidden. We overview our proof for removing 4-cycles from $K_{t,t}$-free bipartite graphs. Moreover, we discuss consequences to tau-boundedness, which is an analog of chi-boundedness.