Weekly Bulletin

The geometric formulation of hydrodynamics by Arnold motivated the study of infinite-dimensional manifolds, and more precisely, half Lie groups — topological groups in which right multiplication is smooth while left multiplication is continuous. The main examples are groups of ( H^s ) or ( C^k ) diffeomorphisms of compact manifolds. In this talk, we will prove several Hopf–Rinow type theorems for right-invariant magnetic systems and certain Lagrangian systems on half Lie groups, extending the recent work of Bauer–Harms–Michor from the case of geodesic flows to this more general setting. Towards the end, we will show that on a half Lie group which is non-aspherical and equipped with a strong Riemannian metric, there always exists a contractible periodic geodesic. This is based on joint work with M. Bauer and F. Ruscelli.

Suppose we have a measure preserving flow - i.e. a space, equipped with a probability measure, evolving under continuous time in a way that preserves the probability measure. This induces a discrete time evolution on each cross section given by the return time map, that also can be shown to preserve an induced probability measure. What kind of different discrete time dynamical systems can be induced by same flow? Which non isomorphic flows can have cross section for which the return time maps provide isomorphic discrete time systems? It is not hard to see this latter question amounts to which non isomorphic flows can be made isomorphic if one makes a time change, i.e. speeds up or slow down the dynamics according to location. This is a classical subject, and there are very different systems that are equivalent under the change. In my talk I will present joint work with Daren Wei, showing that for a certain natural class of flows - unipotent flows - there is a sharp dichotomy: there is one explicit subclass of unipotent flows that are all equivalent to each other (this class contains uncountably many non isomorphic systems) and another explicit class in which equivalence under time change can only come from an isomorphism. This work builds upon and is motivated by seminal work of Marina Ratner from the late 1970s.

In a striking contrast with the classical case of real-valued Sobolev functions, a Sobolev map with values into a given compact manifold N need not be approximable with smooth N-valued maps. This observation, initially due to Schoen and Uhlenbeck (1983), gave rise to a whole area of research concerned with questions related to approximability properties of Sobolev mappings with values into a compact manifold. In this talk, I will give a broad overview of this research direction, its history, the main problems it is concerned with, important known results, as well as some recent contributions.

<p><span style="color: black; font-size: 12pt; font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif;" data-olk-copy-source="MessageBody">Temporal networks offer a unique lens on complex systems that constantly change, where connections form, dissolve, and reappear, much like friendships in our own lives. While static networks capture a single moment, like a photograph, temporal networks reveal how structures evolve over time, shaping the dynamics that unfold on them. This perspective is crucial for questions ranging from disease spread to predicting where a PhD student may end up after graduation. In my talk, I begin with homophily, how similarities influence our social interactions, and show how random walks provide a richer, more dynamic understanding of these patterns. I then extend this to temporal networks and introduce a method for quantifying homophily across time scales, addressing a long-standing gap in the field. Throughout, I highlight the iterative nature of complex systems research: refining questions, adapting mathematical tools, scaling analyses with distributed computing, and grounding insights in empirical data, because every dataset tells its own story.</span></p>

<p>Thurston’s characterisation theorem gives a necessary and sufficient condition for when a branched covering map of the sphere (for which the orbits of the branch points have finite cardinality) can be realised by a rational map. In spite of progress, an analogous result for entire maps of the complex plane seem to be not yet known. In this talk, I will discuss a somewhat different approach in the setting of real entire maps whose post-singular set is real and has finite cardinality.</p>

<p>Multivariate signature schemes are interesting for use cases where post-quantum security and short signatures are required. Currently, the most relevant multivariate signature schemes are UOV - which is already 25 years old -, and variants of the UOV scheme. In this talk, I will present recent insights on the physical security of UOV-based signature schemes, i.e., their security with respect to side channel and fault attacks.</p>

A hyperbolic structure on a surface S can be deformed by cutting along a curve c without self-intersections, twisting, and re-gluing. Such a deformation can also be seen as the Hamiltonian flow of the length function of c on the Teichmüller space of S, equipped with the Weil—Petersson symplectic form. In this talk, I will discuss Hamiltonian flows associated to length functions of self-intersecting curves, and their generalizations to higher rank Teichmüller spaces. This is partially joint work with J. Farre and A. Wienhard.

In this talk, we consider space-time Galerkin discretizations of the acoustic wave equation. Unlike time-stepping methods, the time variable is treated as an additional dimension. Our goal is to design numerical schemes that are unconditionally stable, achieve optimal convergence rates, and are suitable for efficient implementation. We seek formulations that guarantee these properties under minimal assumptions on the discrete spaces, ensuring broad applicability. In particular, we aim at methods that go beyond piecewise continuous polynomials in time and include spline functions of arbitrary regularity. We compare properties, advantages, and limitations of three continuous variational formulations for the acoustic wave equation, which differ in the treatment of the temporal component: - a second-order-in-time scheme without integration by parts in time, - a first-order-in-time formulation, - a second-order-in-time scheme with integration by parts in time. We present recent results and highlight some open problems. This talk is based on joint work with I. Perugia and E. Zampa.

<p class="p2">The Schwartz-Zippel Lemma is a fundamental tool, that provides a bound on the size of the zero set of multivariate polynomials.</p> <p class="p2">In this talk we present a classical proof of the Schwartz-Zippel Lemma based on Gröbner bases and elementary algebraic geometry, interpreting the lemma in terms of varieties, ideals and standard monomials.</p> <p class="p2">One can use the Schwartz-Zippel Lemma to lower bounds the minimum distance of Reed-Muller codes. </p>

Martingale Optimal Transport (MOT) offers a powerful, robust framework to price and hedge illiquid derivatives. The primal formulation requires admissible models to exactly match the market’s implied volatility (IV) surface, imposing hard constraints on the marginal distributions. However, in practice, the model’s IVs only need to fall within the observed bid-ask range. Mathematically, this translates into the model's marginal distributions being between the bid and ask marginals (when they exist) with respect to convex order, leading to a relaxation of MOT. By enlarging the set of admissible martingale couplings, Relaxed MOT generates wider, more realistic price bounds for illiquid instruments, including vanilla options whose payoffs exhibit mixed convexity. We finally introduce a weaker notion of mimicking martingales, coined mocking martingales, and extend Kellerer’s theorem to characterize their existence. Joint work with Shunan Sheng, Marcel Nutz (Columbia), and Bryan Liang (Bloomberg).

The Vertex Reinforced Jump Process (VRJP) is a continuous-time process closely related to the linearly edge reinforced random walk. The recurrence/transience of the VRJP can be caracterized by the asymptotic behavior of a positive martingale : the VRJP is recurrent when the limit is null and transient when it is positive. Besides, the L^p integrability of that martingale is related to the diffusive behavior of the VRJP. A large part of the talk will be devoted to recall some key properties of the VRJP and to explain how that martingale appears and how it can be interpreted as the partition function of a non-directed polymer in a very specific 1-dependent potential. At the end, we will show new results about the L^p integrability of the martingale, using the polymer interpretation and taking inspiration from some works of Junk on directed polymers, Based on a joint work with Q. Berger, A. Legrand and R. Poudevigne-Auboiron.

After revisiting the approach by Witten and Floer to Morse theory, that is, the construction of a cochain complex, using inspiration from classical and quantum mechanics, whose cohomological invariants provide relevant information about the supporting spaces, we will address the problem of incorporating a symmetry group to the picture and the study of the corresponding equivariant theory. This is part of a research program started in collaboration with E. Miranda.

<div class="elementToProof">We prove quantitative decay rates for the linearised gravitational potential around compactly supported equilibria of the Vlasov-Poisson system featuring an attractive point mass potential at the origin.  Such equilibria feature stably trapped particles with a stationary elliptic point of the characteristic flow. This presents a severe obstacle to any kind of dispersion. The problem is further complicated by the presence of an infinite-dimensional kernel. To handle these issues we combine tools from dynamical systems, Hamiltonian geometry, and scattering theory. </div> <div class="elementToProof"> </div> <div class="elementToProof">Our theorem can be viewed as a first quantitative proof of (linear) gravitational Landau damping. It gives in particular the first complete treatment of linear stability of the vacuum solution around the point mass - an essential ingredient for nonlinear stability.  Joint work with Matthew Schrecker.</div>

Cost-of-capital valuation is a well-established approach to the valuation of liabilities and is one of the cornerstones of current regulatory frameworks for the insurance industry. Standard cost-of-capital considerations typically rely on the assumption that the required buffer capital is held in risk-less one-year bonds. The aim of this work is to analyze the effects of allowing investments of the buffer capital in risky assets, e.g. in a combination of stocks and bonds. In particular, we make precise how the decomposition of the buffer capital into contributions from policyholders and investors varies as the degree of riskiness of the investment increases, and highlight the role of limited liability in the case of heavy-tailed insurance risks. We present a combination of general theoretical results, explicit results for certain stochastic models and numerical results that emphasize the key findings. The talk is based on joint work with Hansjörg Albrecher and Hervé Zumbach, available on arXiv: arxiv:2511.00895

Let $p$ be a prime. Bounding short Dirichlet character sums is a classical problem in analytic number theory, and the celebrated work of Burgess provides nontrivial bounds for sums as short as $p^{1/4+\varepsilon}$ for all $\varepsilon>0$. In this talk, we will first survey known bounds in the original and generalized settings. Then we discuss the so-called ``Burgess method'' and present new results that rely on bounds on the multiplicative energy of certain sets in products of finite fields.

The global mean surface temperature record—combining sea surface and near-surface air temperatures—is essential for understanding climate variability and change. Understanding of the past record is also key to constrain the uncertainty around future climate projections. In my talk, I will present a recent study (Sippel et al., 2024, doi:10.1038/s41586-024-08230-1) that aims to improve our understanding of the past record; and I will discuss implications and avenues to constrain near-future climate risk. Past temperature record: The early temperature record (before around 1950) is still uncertain due to evolving measurement methods, incomplete documentation, and sparse spatial coverage. Using independent reconstructions from land and ocean data, we find that historical ocean temperatures were likely measured too cold by about 0.26°C compared with land-based estimates, despite strong agreement in other periods. This cold anomaly is not explained by internal climate variability, and multiple lines of evidence—including climate attribution, timescale analysis, coastal observations, and palaeoclimate data—support a substantial cold bias in early ocean records. While global warming estimates since the mid-19th century remain unchanged, correcting this bias would yield a smaller early-20th-century warming trend, lower global decadal temperature variability, and closer alignment between models and observations. Constraining near-future climate risk: I will further discuss the implications of these findings for constraining near-future temperature projections. Moreover, I will present ongoing work on simulating physically plausible worst-case extreme events at the regional scale, using climate model simulations and so-called rare event simulation algorithms.

The combinatorial description of the moduli space of curves in terms of ribbon graphs was one of the crucial ingredients in Kontsevich’s proof of Witten’s conjecture. The same model was used by Harer–Zagier and by Norbury to compute the Euler characteristic and the number of lattice points, respectively. In this talk, we will introduce the analogous combinatorial model for the moduli space of real curves: the moduli of metric Möbius graphs. Each such graph is equipped with a measure of non-orientability, a quantity interpolating between the complex and the real worlds. We then provide recursive formulas for the volumes and the lattice point counts of this moduli space, refining the results of Kontsevich and Norbury. Finally, we compute the refined Euler characteristic, thereby answering a question posed by Goulden, Harer, and Jackson 25 years ago.