Weekly Bulletin

Any open domain in the standard contact ${\mathbb R}^3$ which is diffeomorphic to ${\mathbb R}^3$ is contactomorphic to the standard contact ${\mathbb R}^3$. Until now it was unknown whether there are open subsets of the standard contact ${\mathbb R}^{2n+1}$ which are diffeomorphic but not contactomorphic to the standard contact ${\mathbb R}^{2n-1}$. In the talk I will show that there are continuous families of such pairwise non-contactomorphic domains for iany $n>2$. The main tool is a version of contact homology which can be defined for any open contact manifolds. This is a joint work with Kiran Ajij, Mahan Mj, Dishant Pancholi and Leonid Polterovich.

<p>Markov Chain Monte Carlo (MCMC) methods provide a general framework for numerical approximation and quantitative information analysis in settings where direct computation is infeasible. Relying on the long-time behavior of Markov chains, MCMC can be used to solve problems via random sampling of complex probability distributions. In this talk, we develop the intuition/theory behind this idea and show how it leads to practical computational methods. The Metropolis–Hastings algorithm is presented as a central example and, as an application, we revisit a real-life cipher-decoding problem. We also give a brief overview of further applications of MCMC across probability, statistics, and related fields.</p>

Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that meet the criteria. This lecture offers an invitation to the field of matrix concentration and its multifarious applications.

<div class="elementToProof" style="font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid; font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: #000000;">We prove ergodicity of conservative random dynamics satisfying a certain hyperbolicity condition. The new feature of our result is that we do not require the non-existence of zero Lyapunov exponents. As a particular application, we show that if \(R_1, R_2\) in \(\mathrm{SO}(d + 1), d ≥ 2\), generate a dense subgroup, then any pair \((f_1, f_2)\) of infinitely smooth volume preserving diffeomorphisms of the \(d\)-dimensional sphere that is sufficiently close to \((R_1, R_2)\) is ergodic with respect to the volume. Previously this was only known to hold when \(d\) is even by a result of Dolgopyat and Krikorian. Joint work in progress with Jonathan DeWitt and Dmitry Dolgopyat.</div>

<p>For an AG code \(C=C(D,G)\) on a curve \(\mathcal X/\mathbb{F}_q\), the structure of the hull \(C\cap C^\perp\) plays an important role in code equivalence, entanglement-assisted quantum codes, and efficient decoding. In general the hull need not be an AG code, but it becomes one when a certain divisor \(A\) associated to \(G\) and its dual divisor \(H\) is non-special.</p> <p>This reduces the problem to determining when effective divisors of the form<br>\[<br>A=\sum_{P\in \mathbb{P}(\mathcal X)}<br>   \max\{v_P(G),\,v_P(H)\}\,P<br>\]<br>are non-special. In this talk I give explicit criteria for non-speciality of several classes of effective divisors of small degree on \emph{Kummer extensions}. The characterization is obtained using the description of the Weierstrass semigroup at multiple points of these curves.</p> <p>These results yield new families where the hull of an AG code can be written again as an AG code, and illustrate how the geometry of Kummer extensions controls the Riemann--Roch behaviour of low-degree divisors.</p>

Multiscale, parameter-dependent partial differential equations (PDEs) pose severe computational challenges due to strong coefficient heterogeneity and high-dimensional parameter spaces. We develop a geometric interpolation approach within the multiscale generalized finite element method (MS-GFEM) that targets the most expensive component: computing parameter-dependent optimal local approximation spaces. Leveraging the spatial localization of MS-GFEM and assuming local parameter dependence, we decompose the global problem into parametrically low-dimensional local subproblems. The optimal subspaces for each parameter are identified as points on a Grassmann manifold and approximated via Grassmann interpolation on sparse grids, which preserves the geometric structure of these spaces while efficiently handling high-dimensional parameter spaces. The resulting localized model reduction method inherits the nearly exponential spatial convergence of MS-GFEM and the parametric convergence rates of sparse grids. Numerical experiments for elliptic problems confirm the theoretical convergence results.

<div>The problem of determining when two F<span style="font-size: xx-small;">q^m</span> -linear rank-metric codes are equivalent is important for reasons of both mathematical interest and for practical applications. Even though this problem can be efficiently solved computationally [2], finding structural properties that remain invariant under equivalence remains an active line of research. One of the main reasons behind that is the computational hardness of existing invariants, which act as distinguishers between equivalence classes. Beyond initial considerations such as minimum distance and weight enumerators, one of the most well-studied invariants is the generalised weights of a code [1, 3].</div> <p>In this talk we introduce a new invariant for vector rank metric codes based on the support dimensions of complete flags of subcodes. We will demonstrate that these invariants are related to the generalised weights, but that they are both different and comparatively easier to compute than the generalised weights. We also relate the new invariants to the corresponding q-system of the code.</p> <div>References:</div> <div>[1] T. H. Randrianarisoa. A geometric approach to rank metric codes and a classification of constant weight codes. Designs, Codes and Cryptography, 88:1331–1348, 2020</div> <div>[2] A. Couvreur, T. Debris-Alazard, and P. Gaborit. On the hardness of code equivalence problems in rank metric. arXiv:<span id="OBJ_PREFIX_DWT115_com_zimbra_phone" class="Object" role="link">2011.04611, 2020</span>.</div> <div>[3] G. Marino, A. Neri, and R. Trombetti. Evasive subspaces, generalized rank weights and near mrd codes. Discrete Mathematics, 346(12):<span id="OBJ_PREFIX_DWT116_com_zimbra_phone" class="Object" role="link">113605, 2023</span>.</div>

A cylinder is a the graph obtained by taking the product of a finite graph with the half-line N. We consider two aggregation models on cylinders, internal and external DLA, analyzing asymptotics as the size of the base graph grows to infinity. There is a stark contrast between the behavior of these two models. External DLA is notoriously very difficult to analyze, whereas for internal DLA we have very good understanding of the limiting shape and even a type of CLT for the fluctuations. This is based on joint works with Itai Benjamini, Ahmed Bou-Rabi and Vittoria Silvestri.

Understanding whether a treatment works differently across patient subgroups is a central question in drug development. However, answering it reliably is notoriously difficult. Clinical trials are typically powered for overall treatment effects, leaving subgroup analyses underpowered and vulnerable to both false positives and false negatives, as illustrated by multiple real cases where subgroup findings couldn't be confirmed in follow-up trials. Despite these severe limitations however questions around treatment effect heterogeneity are of great interest to sponsors, regulators, and clinicians alike and need to be addressed in drug development. In this talk, I will present WATCH (Workflow to Assess Treatment EffeCt Heterogeneity) for exploratory assessment of TEH in randomized clinical trials. The challenges resulting from insufficient data-based information on the question of interest (TEH) are approached via (i) providing a structured and pre-planned analysis approach for this exploratory question, providing more standardization and reducing analyst degrees of freedom, (ii) documenting a-priori, external evidence on potential effect modifiers upfront and as part of the workflow and (iii) utilizing flexible, multivariate analysis methods based on statistical learning methods. I will present the workflow steps, and discuss current analysis methods used to implement the workflow, as well as open methodological questions.

<div class="elementToProof"> <p class="elementToProof">In his celebrated proof of the weak cosmic censorship conjecture for the spherically symmetric Einstein-scalar field system, Christodoulou exploited the following property of that specific matter model: Naked singularities, when they arise, exhibit infinite blue-shift along the null geodesics terminating at the singularity. This behaviour is consistent with self-similarity: Even for more general spherically symmetric matter models, it can be shown that self-similar naked singularities must exhibit infinite blue-shift. Whether, for these more general models, all naked singularities have the infinite blue shift property (and hence are potentially subject to an instability mechanism analogous to that introduced by Christodoulou) still remains an open question.</p> </div> <div class="elementToProof"> <p class="elementToProof"> </p> </div> <div class="elementToProof"> <p class="elementToProof">In this talk, we will present the construction of a spherically symmetric solution to the Einstein-massless Vlasov system which contains a locally naked singularity with finite total blue-shift along its past null cone. The initial data giving rise to this solution have limited differentiability, but belong to a regularity class above the scale invariant threshold.</p> </div>

We develop a comprehensive mathematical finance framework for propagator models with transient linear price impact. These models lead to infinite-dimensional, non-Markovian control problems and fall outside the scope of classical arbitrage and duality theory. We establish a fundamental theorem of asset pricing, a superreplication theorem with liquidity-adjusted risk measures, and a full convex-duality approach to utility maximisation. Despite the non-linearity of preferences and the path-dependent impact structure, we show that optimal strategies can be obtained from an equivalent frictionless optimisation problem under a suitably constructed shadow price.

Let R be the ring of integers of a p-adic field with residue field k. Given an algebraic variety X over R one can consider the function that associates to any k-point x of X the p-adic volume of the ball of R-points specializing to x. Unless X has quotient singularities it is unknown whether this function is the trace of Frobenius of some interesting l-adic sheaf on X. In joint work with M. Groechenig and P. Ziegler we study this question for certain moduli spaces X appearing in enumerative geometry, where the sheaf in question turns out to be the intersection complex.