Weekly Bulletin

Floer homology and the related invariants of 4-manifolds has given us deep insight in smooth differential topology in dimensions 3 and particularly 4. The theory has yielded insights like existence of exotic differentiable structures on 4 dimensional euclidean space, complex curves minimize genus in complex projective space, killing the Hauptvermuntung, there even appear to be connection to the 4 color map theorem. This course will build up Floer homology of three manifolds from scratch. The focus will be on Instanton Floer homology but we will mention other versions and develop applications as the course goes on.

I will present recent work concerning a Floer theory persistence module associated to a Reeb flow and a contact isotopy \phi. The persistence module depends on a filling of the contact manifold. The spectrum of the persistence module is contained in the set of lengths of translated points of \phi relative the chosen Reeb flow. Under certain assumptions on the symplectic cohomology of the filling, the spectral invariants can be used to define a non-degenerate spectral norm on the contactomorphism group, and also spectral capacities for open sets. Various results, variations on the construction, and open questions concerning these structures will be discussed.

In this course we will investigate questions of weak turbulence theory by using as explicit example of wave interactions the solutions to periodic and nonlinear Schrödinger equations. We will start with Strichartz estimates on periodic setting, then we will move to well-posedness. We will then present two different ways of introducing the evolution of the energy spectrum. We will first work on a method proposed by Bourgain and involving the growth of high Sobolev norms. Then, we will give some ideas of how to derive rigorously the effective dynamics of the energy spectrum itself (wave kinetic equation), when one considers weakly nonlinear dispersive equations.

In this talk I will present some recent results concerning non-radial implosions for compressible Euler and Navier-Stokes equation and non radial blow up for certain defocusing supercritical nonlinear Schrodinger equations. This work is a non radial generalization of the breakthrough results of Merle-Raphael-Rodnianski-Szeftel. The work presented is in collaboration with Gonzalo Cao-Laboratories, Javi Gomez-Serrano and Jia Shi.

A mathematically rigorous description of fluid motion plays an important role, for example in understanding weather patterns or ocean dynamics. In this talk, we will derive the fundamentals equations of motions for an inviscid incompressible fluid -- the Euler equations. We will present known results as well as related open problems. Finally, we will discuss stability for 2D stratified flows relevant in geophysics.

Two non-isomorphic ergodic measure preserving flows can become isomorphic if one of the systems undergoes an appropriate time change. In this case we will say that these flows are Kakutani equivalent to each other. We say that an ergodic flow is loosely Kronecker if it is Kakutani equivalent to the straight line flow on (say) a two torus in an irrational direction (the exact direction is immaterial as these are all equivalent to each other). Landmark work of Ratner from the late 70s (that paved the way to her even more famous results on orbit closures and equidistribution of unipotent flows) establishes that 1) the horocycle flow on any finite area surface of constant negative curvature is loosely Kronecker. 2) the product of two such flows is not loosely Kronecker. It remained an open problem whether e.g. products of two horocycle flows are Kakutani equivalent to each other. We show unipotent flows are very rigid under time changes, and indeed unless the flows are loosely Kronecker, two unipotent flows are Kakutani equivalent if and only if they are isomorphic as measure preserving flows. This is a joint work with Elon Lindenstrauss

Code-based cryptography is an area of post-quantum cryptography. Several code-based candidates compete in the standardization process of the National Institute of Standards and Technology. A common drawback of code-based cryptosystems is their often large public key size. An approach to decrease this size is the usage of convolutional codes. The idea behind this is that the public key sizes scales linearly in the maximum degree of its polynomial entries. However, thus far the complexity of generic attacks has been exponential in the size of the sliding generator or parity-check matrix. We give a framework which allows us to iteratively decode convolutional codes with information-set decoding. This method relies on the choice of several parameters whose choice affects the success probability of the algorithm and the running time, so we give tools to determine whether a choice of parameters is suitable. We also discuss reasons why the algorithm sometimes doesn't terminate and how to circumvent said issues. Finally, we use an implementation of our algorithm to attack two cryptosystem based on convolutional codes. For the first cryptosystem, we managed to recover about 74% of messages in our experiment, each in less than 10 hours. For the second proposal, we used our algorithm to give estimates on the running time and success probability. In two cases where the estimate was low we also verified the result by running the full algorithm. 80% of the experiments were successful and they indicate that the algorithm has a complexity significantly below the security estimates provided by the authors.

There is a broad body of work devoted to proving theorems of the following form: spaces with infinitely many special sub-spaces are either nonexistent or rare. Such finiteness statements are important in algebraic geometry, number theory, and the theory of moduli space and locally symmetric spaces. I will talk about joint work with Simion Filip and David Fisher proving a finiteness statement of this kind in a differential geometry setting. Our main theorem is that a closed negatively curved analytic Riemannian manifold with infinitely many closed totally geodesic hypersurfaces must be isometric to an arithmetic hyperbolic manifold. The talk will be more focused on providing background and context than details of proofs and should be accessible to a general audience.

In this talk, we will focus on solving time-harmonic, acoustic, elastic and polarized electromagnetic waves scattered by multiple finite-length open arcs in unbounded two-dimensional domain. We will first recast the corresponding boundary value problems with Dirichlet or Neumann boundary conditions, as weakly- and hyper-singular boundary integral equations (BIEs), respectively. Then, we will introduce a family of fast spectral Galerkin methods for solving the associated BIEs. Discretization bases of the resulting BIEs employ weighted Chebyshev polynomials that capture the solutions' edge behavior. We will show that these bases guarantee exponential convergence in the polynomial degree when assuming analyticity of sources and arc geometries. Numerical examples will demonstrate the accuracy and robustness of the proposed methods with respect to number of arcs and wavenumber. Moreover, we will show that for general weakly- and hyper-singular boundary integral equations their solutions depend holomorphically upon perturbations of the arcs' parametrizations. These results are key to prove the shape holomorphy of domain-to-solution maps associated to BIEs appearing in uncertainty quantification, inverse problems and deep learning, to name a few applications. Also, they pose new questions you may have the answer to!

We study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity θ ∈ (0, 1/2]. We obtain an exact expression for the asymptotic probability of the existence of a cluster crossing a given annulus of radii r and rs as r → 0 (s > 1 fixed). Relying on this result, we then show that the probability for a macroscopic cluster to hit a given disc of radius r decays like | log r|−1+θ+o(1) as r → 0. Finally, we characterise the polar sets of clusters, i.e. sets that are not hit by the closure of any cluster, in terms of logα-capacity.