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.

An important quantity for the orbit complexity of a dynamical system is its topological entropy. Recently, several fruitful approaches were found to express topological entropy of Hamiltonian or Reeb dynamics in terms of Floer theory or contact homology. For a 3D Reeb flow and a link of periodic orbits, Alves and Pirnapasov introduced the notion of the homotopical growth rate in the link complement, defined by counting certain "essential" homotopy classes of periodic orbits in the complement of that link. In my talk, I will explain how topological entropy of a Reeb flow can be recovered through homotopical growth rates in complements of links of periodic orbits, given some mild assumptions on the flow. I will moreover explain some applications of this result and discuss stability features of the topological entropy of Reeb flows. Partly based on joint work with M. Alves, L. Dahinden, and A. Pirnapasov.

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.

I will introduce the problem of minimizing area among Lagrangian submanifolds in a Riemannian symplectic manifold, and the related problem for Legendrian submanifolds in a contact manifold. This constrained variational problem was introduced by Schoen and Wolfson with the aim of constructing Lagrangian minimal surfaces and Special Lagrangians. I will explain the connection between the Lagrangian and Legendrian problem, what is known in two dimensions, what goes wrong in higher dimension and some new progress in this direction.

I will discuss the existence and construction of smooth models for certain fibered partially hyperbolic systems. Fibered partially hyperbolic systems are partially hyperbolic diffeomorphisms that have an integrable center bundle, tangent to a continuous invariant fibration by invariant submanifolds. I will explain how under certain restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that descends to a hyperbolic nilmanifold automorphism on the base. I will then discuss how the restrictions on the fiber can be replaced by certain dynamical restrictions on the behavior of the fibered system in the center direction. This is part of a joint work in progress with Jon Dewitt and Oliver Wang.

There are many compactifications of the moduli space of abelian varieties. In a series of papers by Kajiwara, Kato and Nakayama, they construct a particularly nice modular compactification, consisting of log abelian varieties, certain proper group objects in the category of log spaces. In this talk, I will recall the definitions of a tropical abelian variety and a log abelian variety, and give an overview of the moduli space of log abelian varieties (with level structure). I will also explain how it compares to the toroidal compactifications, such as the second Voronoi compactification. I will also discuss how classical constructions like duality and the Poincare bundle generalise to the logarithmic setting.

Planar functions are functions defined over finite fields of odd characteristics such that their (nonzero) derivatives are permutations. They have connections with various areas of mathematics with applications in coding theory, cryptography, combinatorics, etc. In this seminar, we will introduce planar functions, their connection with commutative semifields, and some equivalence notions used to classify them. We will recall a conjecture of 1968 by Dembowski and Ostrom stating that all planar functions are quadratic and we will see some recent results about this conjecture.

In my talk I will explain how to use probabilistic methods to show that the number of generators of a higher rank lattice is sublinear in the covolume. We will look at the properties of Voronoi tessellations of symmetric spaces with seeds at a very sparse Poisson point process. In higher rank, one can use dynamics to prove that the Voronoi cells have very elongated walls. This phenomenon is missing in the rank one cases and is the key ingredient for the upper bound on the number of generators. Based on a joint work with Sam Mellick and Amanda Wilkens.

The development of efficient reduced order models (ROMs) from a deep learning perspective enables users to overcome the limitations of traditional approaches [1, 2]. One drawback of the techniques based on convolutional autoencoders is the lack of geometrical information when dealing with complex domains defined on unstructured meshes. The present work proposes a framework for nonlinear model order reduction based on Graph Convolutional Autoencoders (GCA) to exploit emergent patterns in different physical problems, including those showing bifurcating behavior, high-dimensional parameter space, slow Kolmogorov-decay, and varying domains [3]. Our methodology extracts the latent space’s evolution while introducing geometric priors, possibly alleviating the learning process through up- and down-sampling operations. Among the advantages, we highlight the high generalizability in the low-data regime and the great speedup. Moreover, we will present a novel graph feedforward network (GFN), extending the GCA approach to exploit multifidelity data, leveraging graph-adaptive weights, enabling large savings, and providing computable error bounds for the predictions [4]. This way, we overcome the limitations of the up- and down-sampling procedures by building a resolution-invariant GFN-ROM strategy capable of training and testing on different mesh sizes, resulting in a more lightweight and flexible architecture. References [1] Lee, K. and Carlberg, K.T. (2020) ‘Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders’, Journal of Computational Physics, 404, p. 108973. Available at: https://doi.org/10.1016/j.jcp.2019.108973. [2] Fresca, S., Dede’, L. and Manzoni, A. (2021) ‘A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs’, Journal of Scientific Computing, 87(2), p. 61. Available at: https://doi.org/10.1007/s10915-021-01462-7. [3] Pichi, F., Moya, B. and Hesthaven, J.S. (2024) ‘A graph convolutional autoencoder approach to model order reduction for parametrized PDEs’, Journal of Computational Physics, 501, p. 112762. Available at: https://doi.org/10.1016/j.jcp.2024.112762. [4] Morrison, O.M., Pichi, F. and Hesthaven, J.S. (2024) ‘GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applications’, Computer Methods in Applied Mechanics and Engineering, 432, p. 117458. Available at: https://doi.org/10.1016/j.cma.2024.117458.

In my previous research, during my PhD and several years as a postdoc, I studied endomorphisms of the projective line. More specifically, I focused on rational functions that can be represented as the quotient of two single-variable polynomials, working primarily over number fields or function fields (often defined over finite fields). The main objects of interest were the periodic and preperiodic points of these rational functions. Periodic points are those whose forward orbits form cycles, while preperiodic points have finite forward orbits but are not necessarily part of a cycle. As we will see, many concepts arising in this context can be naturally interpreted within the theory of elliptic curves. In addition, I investigated particular families of rational functions characterized by a natural notion of good reduction, which enables us to reduce the problem to a setting over finite fields. Throughout this work, moduli spaces play a central role in understanding the structure and classification of such dynamical systems.

Every closed, oriented 3-manifold is obtained by a Dehn surgery on a link in the three-sphere. It is natural to ask about the minimal number of components of a link that admits a Dehn surgery to a given 3-manifold. In this talk, we use Furuta's 10/8-theorem to provide new examples of 3-manifolds with the same integral homology as the lens space L(2k, 1), while not surgery on any knot in the three-sphere.

Let K be a global field and p be a prime number with p\neq char K. A classical theorem in algebraic number theory asserts that when L varies in Z/pZ-extensions of K, the p-rank of the class group of L is unbounded. It is expected that similar unboundenss results also hold for other arithmetic objects. For example, K. Česnavičius proved that for fixed abelian variety A over K, the p-Selmer group of A over L is unbounded when L varies in Z/pZ-extensions of K. And he raised a further problem: in the same setting, does the Tate-Shafarevich group of A also grow unboundedly? Using a machinery developed by B. Mazur and K. Rubin, we give a positive answer to this problem. This is a joint work with Yi Ouyang.

The gluing maps on the moduli space of curves are integral to much of the enumerative geometry of curves. For example, Gromov-Witten invariants satisfy recursive relations with respect to the gluing maps. For log GW invariants, counting maps with tangency conditions, this fails, as gluing maps do not respect the logarithmic structure hence the pullback is not well-defined. I will describe a certain logarithmic enhancement of Mgn that does admit gluing maps. For the logarithmic double ramification cycle, a certain curve count controlling the log Gromov-Witten invariants of toric varieties, a formula for the pullback along the separating gluing map was known. I will present a formula for pulling back the log DR cycle along the non-separating gluing map in terms of smaller log DR cycles and piecewise polynomials, and explain how this gives a formula for the classical pullback of the classical double ramification cycle. I will sketch how this story extends to logarithmic Gromov-Witten invariants.