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.

We illustrate the construction of quantum field theories on supermanifolds and we provide a complete Cartan-calculus to deal with superdiffeomorphisms in curved space. We will briefly review the geometry of supermanifolds and we discuss the challenges related to quantum field theory applications.

In various areas of mathematics there exist "big fiber theorems", these are theorems of the following type: "For any map in a certain class, there exists a 'big' fiber", where the class of maps and the notion of size changes from case to case. We will discuss three examples of such theorems, coming from combinatorics, topology and symplectic topology from a unified viewpoint provided by Gromov's notion of ideal-valued measures. We adapt the latter notion to the realm of symplectic topology, using an enhancement of Varolgunes’ relative symplectic cohomology to include cohomology of pairs. This allows us to prove symplectic analogues for the first two theorems, yielding new symplectic rigidity results. Necessary preliminaries will be explained. The talk is based on a joint work with Yaniv Ganor, Leonid Polterovich and Frol Zapolsky.

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.

We study a broad class of nonlinear iterative algorithms applied to random matrices, including power iteration, belief propagation, approximate message passing, and various forms of gradient descent. We show that the behavior of these algorithms can be analyzed by expanding their iterates in an appropriate basis of polynomials, which we call the Fourier diagram basis. As the dimension of the input grows, this basis simplifies to the tree-shaped diagrams, that form a family of asymptotically independent Gaussian vectors. Moreover, the dynamics of the iteration restricted to the tree diagrams exhibit properties reminiscent of the assumptions of the cavity method from statistical physics. This enables us to "implement" heuristic cavity-based reasoning into rigorous arguments, including a new simple proof of the state evolution formula. Based on joint work with Chris Jones (https://arxiv.org/abs/2404.07881)

The talk will be about removable singularities for solutions of the Heat Equation and the Fractional Heat Equation in time varying domains. In order to talk about removability, some associated capacities will be introduced to study its metric and geometric properties. I will discuss onsome results obtained in joint work with X. Tolsa and J. Mateu and also mention some recent achievements with J. Hernández.

Hamiltonian systems form a fascinating class of dynamical systems exhibiting deterministic chaos. In my talk I will introduce some tools from modern symplectic geometry that can be used to approach interesting dynamical questions. In particular, I will speak about certain sequences of min-max spectral invariants of the Hamiltonian action functional which satisfy a symplectic analogue of the classical Weyl law for the spectrum of the Laplace operator. I will highlight some recent applications, for example smooth closing lemmas.

More information: https://eth-its.ethz.ch/activities/its-fellows--seminar/Oliver-Edtmair.htmlcall_made

Suppose you want to send a message to your friend. Can errors occur during the transmission? Unfortunately yes. It is then crucial to find ways to detect errors in the received message and possibly correct it. The goal of algebraic coding theory is to design ways of encoding messages (vectors over a finite field) with an algebraic structure that guarantees that, with a limited number of errors, the meaning of the original message is not compromised. In multi-cast communication, as in the streaming of data over the Internet, one deals with sending information to multiple receivers across a network with several intermediate nodes. To improve the network throughput, a coding technique called random linear network coding was developed. In this scenario, the intermediate nodes transmit a random linear combination of the vectors received. With this technique, it is possible to asymptotically achieve the maximum capacity of the network, without relying on its topology. In this talk, we will start studying the basic notions of coding theory with only one sender and one receiver, and then switch to the case of data transmission over a network and explain how in this case giving the messages the structure of a linear subspace helps correcting errors.

We'll recall how to attach to a periodic geodesic of H^2/PSL(2,Z) an arithmetical quantity, its discriminant. After discussing how to pick a geodesic at random, we will show that 'most' geodesics have large discriminant when ordered by length, and that 42% of them have a fundamental discriminant.

The Torelli morphism and its extension to the moduli space of stable curves

In nature one finds superconductors of varying critical temperatures and energy gaps. For weak superconductors, where the critical temperature is small, a universal phenomenon occurs: The ratio of the energy gap and critical temperature is a universal value, independent of the specific superconductor. I will present recent work on such universal phenomena in the BCS theory of superconductivity. Based on joint works with A. B. Lauritsen and B. Roos.

I want to give a survey about the rational cohomology of SL_n Z. This includes recent developments of finding Hopf algebras in the direct sum of all cohomology groups of SL_n Z for all n. I will give a quick overview about Hopf algebras and what this structure implies for the cohomology of SL_n Z. If time permits, I will explain a connection to graph homology.