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.

Abstract: Motivated by heat kernel renormalization of perturbative quantum field theory, we study the cutting and gluing behaviour of the heat kernel on a Riemannian manifold $M$ which is cut along a compact hypersurface $\gamma$ into two Riemannian manifolds $M_1$, $M_2$. Under a certain assumption on $(M,\gamma)$ (which we conjecture to be true for all Riemannian manifolds), we prove a gluing formula for the heat kernel which involves only the heat kernels on $M_1, M_2$ and $\gamma$. This is a joint work with Pavel Mnev, arxiv:2404.00156.

Floer homology in its Monopole, Heegard and ECH version’s has proved a powerful tool for the study of the geometry and topology of three manifolds. Their older cousin instanton Floer homology also has its share of trophies but remains more mysterious and harder to compute. This talk will survey a few highlights of applications of the theory and problems that remain to be solved and prospects for the future.

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.

Moduli spaces of stable maps and admissible covers of a curve give rise to interesting cycles on moduli spaces of stable curves. The talk will be about a wall-crossing formula relating these two types of cycles. Based on a joint work with Max Schimpf. 

manifolds, both from a complex analytic and algebraic point of view. This kind of manifolds plays a central role in classification problems in complex algebraic geometry, being fundamental building blocks of Ricci-flat manifolds, and their theory has become a very popular research topic in the last years. Together with their definition(s) and their main properties, we will see the four known families of deformation classes of this kind of manifolds and we will approach the problem of their bimeromorphic classification.

I will show how the study of non-tautological classes on the moduli space of abelian varieties helps explain the structure of the tautological ring of the moduli space of curves of compact type. On the curves side, this is joint work with Hannah Larson and Johannes Schmitt, and on the abelian varieties side, it is joint with Dragos Oprea and Rahul Pandharipande.

As our technology advances, the need for \(b\)-symbol read channels that can handle messages with high-density data becomes crucial. The problem with conventional read channels is that they are more likely to overlap multiple information units, also called symbols, while reading messages with high-density data. The idea behind \(b\)-symbol read channels is that these channels consider all \(b\) consecutive symbols from the sent message as one symbol. This protects the message from being read with overlapping symbols. Considering a code word of some code \(\mathcal{C}\) as the sent message, the message read by some \(b\)-symbol read channel is called a \(b\)-symbol code word. In this thesis, we investigate \(b\)-symbol codes over semiprimitive irreducible cyclic codes and their Hamming weight distribution. The \(b\)-symbol Hamming weight distribution of semiprimitive irreducible cyclic codes is determined up to an invariant that we call \(\mu(b)\) and \(\mu_l (b)\). These invariants depend on \(b\) and on the choice of the primitive element that we use to describe the irreducible cyclic code\(\mathcal{C}\). This thesis aims to find lower and upper bounds of the average values of these invariants. To do so, we use algebraic function field theory and number theory. In this thesis, we obtain several lower and upper bounds, test their performance for smaller fields, and compare them. Furthermore, we are able to improve those bounds due to a number theoretic approach. Using these bounds, we are able to deduce a very good estimate for the average values of the invariants \(\mu(b)\) resp. \(\mu_l (b)\). These results provide a better understanding of the \(b\)-symbol Hamming weight distribution of semiprimitive irreducible cyclic codes.

The Square Peg Problem (SPP), formulated by Toeplitz in 1911 and still unsolved, asks whether every Jordan curve contains four points at the vertices of a square. We shall discuss how, when the Jordan curve is smooth, SPP and related peg problems can be interpreted as questions in symplectic geometry, and deduce some consequences both for smooth curves and for other classes. Joint work with Josh Greene.

In this talk, we investigate a particular family of graphs with large girth and look at their applications to low-density parity-check (LDPC) codes. Specifically, we study a family of Cayley graphs with large girth over finite fields with prime order, constructed by Margulis. We analyse Margulis' lower bound for the girth of these graphs and compute the exact girth for several prime numbers. Our results show that the actual girth exceeds the bound by an average of about 87% in all cases studied. We show an improvement to Margulis' bound which leads to an average error rate of about 56%. Additionally, we present further observations which suggest that an even better bound may still exist. Furthermore, we show how LDPC codes can be constructed based on these graphs and simulate their error correction performance over an additive white Gaussian noise channel.

We study a new notion of trace and extension operators for abstract Hilbert complexes.

It is often said that we live in an "information society". But what exactly is meant by information? A sequence of symbols 0 and 1? Currently, inside the most miniaturized computer, a binary digit, commonly called a bit, is a complex physical device with billions of interacting particles. What happens to information processing when each bit is carried by a single quantum particle, such as an atom, an electron or a photon? Conversely, can we see the movement of elementary particles as a calculation that the universe is performing? The physics of the last thirty years has been particularly rich in the development of ideas and experiments that have illustrated the crucial role of information in physical laws. A new type of computer, the quantum computer, still in the prototype phase, has been invented. This lecture, which is aimed at non-specialists, will explain the merits of such quantum machine and some of the questions it can tackle. In particular, one crucial aspect of its development, namely the progress in fault-tolerant operations, will be discussed.

In this talk we will look at some recent developments about integer partitions, including coloured partitions, semi-prime partitions and r-prime partitions. We will give asymptotic formulae for the number of these partitions. We will also look at some interesting elements of the proof of these formulae, without going into too much technical detail. In particular, we have a look at strange and pseudo-differentiable functions.

The accuracy of logical operations on quantum bits (qubits) must be improved for quantum computers to surpass classical ones in useful tasks. To that effect, quantum information needs to be made robust to noise that affects the underlying physical system. Rather than suppressing noise, quantum error correction aims at preventing it from causing logical errors. This approach derives from the reasonable assumption that noise is local: it does not act in a coordinated way on different parts of the physical system. Therefore, if a logical qubit is adequately encoded non-locally in the larger Hilbert space of a composite system, it is possible, during a limited time, to detect and correct the noise-induced evolution before it corrupts the encoded information. We will present an experiment implementing a logical qubit in a superconducting cavity coupled to a transmon synthetic atom – the latter employed here as an auxiliary non-linear element [1]. Error correction involves a novel primitive operation [2] and feedback control based on reinforcement learning [3]. Recently, we have stabilized in real-time a logical qubit manifold spanned by the Gottesman-Kitaev-Preskill grid states, reaching a correction efficiency such that the lifetime of the encoded information was prolonged by more than a factor of two beyond the lifetime of the best physical qubits composing our system.

In this talk, we study dependence uncertainty and the resulting effects on tail risk measures, which play a fundamental role in modern risk management. We introduce the notion of a regular dependence measure, defined on multi-marginal couplings, as a generalization of well-known correlation statistics such as the Pearson correlation. The first main result states that even an arbitrarily small positive dependence between losses can result in perfectly correlated tails beyond a certain threshold and seemingly complete independence before this threshold. In a second step, we focus on the aggregation of individual risks with known marginal distributions by means of arbitrary nondecreasing left-continuous aggregation functions. In this context, we show that under an arbitrarily small positive dependence, the tail risk of the aggregate loss might coincide with the one of perfectly correlated losses. A similar result is derived for expectiles under mild conditions. In a last step, we discuss our results in the context of credit risk, analyzing the potential effects on the value at risk for weighted sums of Bernoulli distributed losses. The talk is based on joint work with Corrado De Vecchi and Jan Streicher.

Readout and parametric gate operations in qubits implemented in quantum superconducting circuits are performed by applying microwave drive tones to the circuit. The simultaneous pursuit of fidelity and speed of these operations by increasing drive strength is limited by unwanted drive-induced state transitions arising from the circuit non-linearity. We experimentally address the origin of these adverse state transitions in a driven transmon by measuring transition probabilities as a function of drive frequency and power. We show that there are three distinct mechanisms for adverse transitions caused by the drive: 1) AC Stark shift of qubit frequency into resonance with lossy degrees of freedom in the qubit environment, 2) excitation of intrinsic resonances linking computational and non-computational states within the transmon spectrum, 2) non-linear, Raman-like processes involving extrinsic degrees of freedoms such as packaging or transmission line modes. Our findings provide insights into the improvement of readout and gate operations on superconducting qubits.