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.

Geometric (pre)quantization can be performed only for integral symplectic manifolds. In 1989 Alan Weinstein conjectured that using the notion of Diffeology, as well as Noncommutative Geometry methods, one might obtain the representations required to “quantise the unquantisable” from a torus bundle rather than a line bundle. In joint work with P. Antonini, we showed that the obstruction to integrality can be lifted by adding extra dimensions and passing to the diffeological category. In fact, the added dimensions force the C*-algebra associated with this construction to be nothing else than the crossed product algebra associated with a torus action.

In the 1990s, two physisists, Gopakumar and Vafa, proposed an ideal way to count curves in a Calabi-Yau threefold, that is conjecturally equivalent to other curve counting theories such as Gromov-Witten theory. It is very recent that Maulik and Toda gave a mathematically rigorous definition of GV invariants. In this talk, I will review a recent progress on the GV theory, including chi-independence for GV invariants on local curves and degree two GW/GV equivalence for some smooth curves with generic normal bundles. This is based on a joint work with T. Kinjo and another work with B. Davison.

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.

A classical result in differential topology says that there are no nowhere vanishing vector fields on a 2-sphere. One may ask a similar question in algebraic geometry: does the tangent bundle to a sphere given by the equation x^2+y^2+z^2=1 over some field k have a nowhere vanishing section? Or more generally, when does the tangent bundle on an affine quadratic q=1 with q being a homogeneous degree 2 polynomial have a nowhere vanishing section? We give an essentially full answer to this question assuming that the quadric q=1 has a rational point. In particular, the 2-sphere x^2+y^2+z^2=1 over a field k has a nowhere vanishing vector field if and only if -1 is a sum of 4 squares in k. The proof uses a mixture of results from motivic homotopy theory, Chow-Witt rings and some constructions from the theory of quadratic forms. This is a joint work with Marc Levine.

In this talk, I will discuss vortex dynamics in the planar Euler equations, focusing on two key aspects. First, I will present a rigorous derivation of leapfrogging quartets of concentrated vortex patches near singular time-periodic relative equilibria of the point vortex system, using KAM theory. In the second part, I will show how to extend these techniques to desingularize time-periodic vortex orbits when the Euler equation is set in a generic bounded simply-connected domain. Specifically, we can prove that for a single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches.

We will discuss the smooth closing lemma for Hamiltonian diffeomorphisms with invariant Lagrangians.  Based on joint work with Erman Cineli & Shira Tanny.

The toroidal compactifications of A_g

We provide a concise introduction to CSS and CSS-T codes from the perspective of classical coding theory. We demonstrate that pairs of linear codes that yield a CSS code with good correction capability can be easily produced using a randomized construction when the cardinality of the base field is sufficiently large. Next, we prove that CSS-T codes exhibit the opposite behavior, showing that, under very natural assumptions, their rate and relative distance cannot be simultaneously large. We conclude with a simple construction of CSS-T codes derived from Hermitian curves. This is joint work with Elena Berardini and Alberto Ravagnani.

Bounded cohomology is a functional analytic analogue of group cohomology, with many applications in rigidity theory, geometric group theory, and geometric topology. A major drawback is the lack of excision, and because of this some basic computations are currently out of reach; in particular the bounded cohomology of some “small” groups, such as the free group, is still mysterious. On the other hand, in the past few years full computations have been carried out for some “big” groups, most notably transformation groups of R^n, where the ordinary cohomology is not yet completely understood. I will report on this recent progress, which will include joint work with Caterina Campagnolo, Yash Lodha and Marco Moraschini, and joint work with Nicolas Monod, Sam Nariman and Sander Kupers.

We present our recent developments on the asymptotic expansions of high-frequency multiple scattering iterations in the exterior of sound-hard scatterers. As in the sound-soft case, these expansions lead into wavenumber dependent estimates on the derivatives (of all orders) of the multiple scattering iterations which, in turn, allow for the design and analysis of Galerkin boundary element methods (BEM) for their frequency independent approximation. We also present preliminary theoretical developments related to the accurate approximation of the remaining infinite tail in the Neumann series formulation of multiple scattering problems. Time permitting, in the second part of the talk, we present our preliminary results on the frequency independent approximation of the sound-soft scattering amplitude based on Bayliss-Turkel type local approximations to the Dirichlet-to-Neumann operator. Joint with: Y. Boubendir (NJIT) and S. Lazergui (NJIT)

In this talk we study the BIKE cryptosystem. We will introduce a lattice over a polynomial ring which we construct from the public key. The private key is a vector in this lattice with special properties (being sparse). We study the properties of this lattice. Furthermore, using this framework we will reformulate a previous work about weak keys for BIKE in terms of this lattice and show that their weak keys correspond to shortest vectors in this rank 2 lattice. Instead of only finding the shortest vector, we are able to construct a reduced basis and generalize their attack. We also propose new problems which arise from the reformulation of their attack in terms of lattices.

In the past years, deep learning algorithms have been applied to numerous classical problems from mathematical finance and financial services. For example, deep learning has been employed to numerically solve high-dimensional derivatives pricing and hedging tasks, to provide efficient volatility smoothing or to detect financial asset price bubbles. Theoretical foundations of deep learning for finance, however, are far less developed. In this talk, we start by revisiting some recently developed deep learning methods. We then present our recent results on theoretical foundations for approximating option prices, solutions to jump-diffusion PDEs and optimal stopping problems using (random) neural networks. We address neural network expressivity, highlight challenges in analysing optimization errors and show the potential of random neural networks for mitigating these difficulties. Our results allow to obtain more explicit convergence guarantees, thereby making employed neural network methods more trustable. Based on joint works with F. Biagini, A. Jacquier, A. Mazzon, T. Meyer-Brandis, C. Schwab, R. Wiedemann

I will report on joint work with Andrea Brini in which we establish expected properties of the refined Gromov-Witten generating series of local P2 coming from B-model heuristics: modularity, the extended holomorphic anomaly equation, orbifold regularity and the leading order conifold asymptotics. The first three statements generalise results of Lho-Pandharipande and Coates-Iritani while the proof of the last property is new even in the unrefined limit. I will start off explaining why these four features are expected in the first place and will mention some of their consequences. Afterwards I will discuss aspects of their proof mostly focusing on the arguments proving the leading order conifold asymptotics.