Weekly Bulletin

The Legendrian Contact Homology (LCH) differential graded algebra (DGA) was among the first, and is still among the most important, non-classical invariants of Legendrian knots. In this talk, I will tell a story that builds up ever more sophisticated analogues of Poincare Duality as we delve more deeply into the structure of the DGA. Despite the algebraic nature of the results, I promise pictures, examples, and analogies with Morse Theory. This is joint work with Jason Ma (Duke).

In 2003, A. Bressan proposed a conjecture concerning the mixing efficiency of incompressible flows, which remains open. I will survey progress toward its resolution and present a new result establishing the asymptotic validity of the conjecture for time-periodic velocity fields. This is achieved by adapting tools from smooth ergodic theory, such as Lyapunov exponents, to the non-smooth setting of DiPerna-Lions flows. This is joint work with Elia Brué and Maria Colombo.

<div dir="ltr" data-olk-copy-source="MessageBody">A link is a smoothly embedded closed 1-dimensional submanifold in the 3-dimensional sphere S^3. In this talk we study a particular class of links, known as RBG links, and present some results arising from this framework in low-dimensional topology,  the study of smooth manifolds of  dimension 4 or less.</div> <div dir="ltr"> </div> <div>We will see how RBG links can be used to construct pairs of knots sharing a specific topological feature, that is having orientation preserving diffeomorphic traces. As notable applications of this framework, we will see that the Conway knot is not smoothly slice and has a friend for every integer slope.</div>

<p><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">We show that the marked length spectrum determines the isometry class for surfaces with Anosov geodesic flows. This extends a result of Otal and Croke, known previously in negative curvature. This is joint work with Lefeuvre and Paternain.</span></p>

We will give an overview of the Sarkisov program in higher dimension and then describe it in the case of rational surfaces. We will define the Cremona group over the projective plane and some dynamical invariants for its elements. Finally, we will look at a very special dynamical invariant, namely the Sarkisov length.

<p>This talk introduces a novel IND-CCA2 Key EncapsulationMechanism (KEM) based on the weighted subset sum problem. The cryptosystem uses a decoding algorithm inspired by Goppa codes, adapted for the ring of integers. Decryption is efficiently handled through a truncated version of the extended Euclidean algorithm. By allowin non-binary alphabets, the design aims to mitigate traditional lattice-reduction attacks that have historically compromised knapsack schemes.</p> <p>Additionally, to protect the system against lattice-reduction attacks to the private key, two possible masking mechanisms of the public key are proposed. Since the system is based on basic integer arithmetic, it offers high computational performance and it is potentially secure against quantum attacks.</p>

A regular plane polynomial automorphism is a subclass of polynomial diffeomorphism f: C^2 - > C^2 of the complex affine plane with positive topological entropy. Every polynomial diffeomorphism of positive topological entropy is conjugated to a regular one. Dujardin and Favre showed that two such transformations cannot share infinitely many periodic points unless they share a common iterate. We show that this results holds uniformly in a family: Take two 1-parameter families of regular plane polynomial automorphisms (f_t) and (g_t), then there exists a constant D >0 such that for every parameter t except for a finite number of them f_t and g_t share at most D periodic points. This generalises a result of Mavraki and Schmidt who showed it for endomorphisms of the projective line. I will give a brief overview of the history of this problem and explain key tools of the proof which is based on the theory of adelic divisors and arithmetic equidistribution. This is joint work with Yugang Zhang.

<p>Data centres are one of the backbones of modern-day digital infrastructure, with the amount of processed and stored data growing year by year. In this context, coding techniques are employed to ensure the availability and reliability of the stored data.  One question that has recently emerged, is how to optimally store a certain number of data objects on a certain number of servers in a way that allows the system to serve certain amounts of requests for the objects. This problem is naturally phrased as a coding question, as it concerns the design of a redundancy strategy for the storage system. In this talk, we will introduce and formalize the service problem and the Service Rate Region as a measure of system flexibility. Connection to other well-established research topics will be outlined, such as PIR/batch codes and Majority Logic Decoding. The difference between systematic and non-systematic systems will be considered, and we will show how the latter can give a better performance when designed carefully.</p>

In this talk, I will present some recent results obtained with Nicolas Fournier. We are interested in a simple model describing the (random) motion of a particle in a gaz, namely the kinetic scattering equation. When the particle lives in $\mathbb{R}^d$, and when the equilibrium (for the velocities) is heavy-tailed, it is easy to see that the position process, correctly rescaled, converges weakly to an $\alpha$-stable process. We are interested in the case where the particle is confined into a convex smooth domain, by a certain reflection mechanism that I will describe. We show that, according to the boundary condition, two different types of reflected stable process may arise at the limit. After introducing the model and the results, I will mostly explain how we construct the limiting processes by piecing together their excursions inside the domain. If time permits, I will say a few words about the convergence.

Recently, sharp matrix concentration inequalities were developed using the theory of free probability. In this work, we design polynomial time deterministic algorithms to construct outcomes that satisfy the guarantees of these inequalities. As direct consequences, we obtain polynomial time deterministic algorithms for the matrix Spencer problem and for constructing near-Ramanujan graphs. Our proofs show that the concepts and techniques in free probability are useful not only for mathematical analyses but also for efficient computations.

General multivariate distributions are notoriously expensive to sample from, particularly the high-dimensional posterior distributions in PDE-constrained inverse problems. In this talk, I will present a measure transport based approach for Bayesian inverse problems based on low-rank surrogates in the tensor train format, a methodology that has been exploited for many years for scalable, high-dimensional density function approximation in quantum physics and chemistry. We build upon recent developments in the field of cross approximation algorithms in linear algebra to construct a tensor train approximation to the target probability density function using a small number of function evaluations. For sufficiently smooth distributions, the storage required for accurate tensor train approximations is moderate, scaling linearly with dimension. In turn, the structure of the tensor train (TT) surrogate allows sampling by an efficient conditional distribution method since marginal distributions are computable with linear complexity in dimension. Using this generic tool enables conditional sampling, construction of optimal biasing densities or even tractable Bayesian optimal experimental design. In order to keep the arising ranks in the TT approximations manageable, we furthermore propose a ‘deep’ version of the approach where the transport from reference to target distribution is approximated incrementally via intermediate bridging densities. The method is demonstrated in the context of complex PDE-constrained Bayesian inverse problems, computing expectations of functionals of the PDE solution with respect to high-dimensional posterior distributions, including rare event probabilities and optimal experimental designs.

<div>In this talk, we will consider the Vlasov equation on expanding isotropic homogeneous tori described by the Friedmann–Lemaître–Robertson–Walker cosmological spacetimes. For expansion rate $t^q$, with $0 < q \leq 1/2$ (excluding some exceptional values), we show that the difference of a time-normalised spatial density with the spatial average decays with a quantitative rate depending on the regularity of the initial datum.</div> <div> </div> <div class="elementToProof">This is joint work with Martin Taylor (Imperial College London).</div>

I will present a blockchain-based insurance scheme that integrates parametric and collaborative elements. A pool of investors, referred to as surplus providers, locks funds in a smart contract, enabling blockchain users to underwrite parametric insurance contracts. These contracts automatically trigger compensation when predefined conditions are met. The collaborative aspect is embodied in the generation of tokens, which are distributed to both surplus providers and policyholders. These tokens represent each participant's share of the surplus and grant voting rights for management decisions. The smart contract is developed in Solidity, a high-level programming language for the Ethereum blockchain, and deployed on the Sepolia testnet, with data processing and analysis conducted using Python. I will discuss both the technical details as well as the actuarial aspects.

The relative Langlands duality posed by Ben-Zvi-Sakellaridis-Venkatesh predicts that periods of automorphic form can sometimes equal to sum of several special values of automorphic L-functions rather than a single one. In this talk, I will present explicit evidence for this phenomenon. I will discuss families of automorphic periods on general linear groups and classical groups that are designed to detect specific types of Eisenstein series. These periods indeed evaluate to finite sums of L-values. This is joint work with Guodong Xi.

We study the codifferential of the Prym map at the double covers of smooth curves whose Prym varieties decompose nontrivially as a product of abelian varieties, to show that the fiber product of the Prym map R_g -> A_(g-1) and the product map A_k x A_(g-1-k) -> A_(g-1) for 1 ≤ k ≤ (g-1)/2 is smooth.