Weekly Bulletin

The parametrised Whitehead torsion is an invariant of families of manifolds, and can be viewed as a map to an algebraic K-theory space. A strong version of the nearby Lagrangian conjecture says that when applied to families of closed exact Lagrangians in a cotangent bundle, this invariant vanishes. Abouzaid and Kragh showed that in this case, this map lands in the trivial path component of the target, i.e. is trivial on \pi_0. Using generating functions, we find strong constraints on what this map does to higher homotopy groups. I'll illustrate this with some concrete consequences for the symplectic mapping class group of T^*T^n relative to the 0-section. This is based on joint work-in-progress with Sylvain Courte.

I will present a Lagrangian action on fields, the critical points of which lead to solutions of the Einstein-Yang-Mills equations, in the spirit of Kaluza-Klein theories. The novelty is that the a priori fiber bundle structure hypothesis is not required: fields are defined on a "space-time" $Y$ of dimension $4+r$ without any a priori principal bundle structure, where $r$ is the dimension of the structure group. If the latter group is compact and simply connected, to each solution of the Euler-Lagrange equations it corresponds a 4-dimensional pseudo-Riemannian manifold $X$ (which can be interpreted as our usual space-time) in such a way that $Y$ acquires a principal bundle structure over $X$ equipped with a connection. Moreover the metric on $X$ and the connection on $Y$ are solutions of the Einstein-Yang-Mills system. If the structure group is $U(1)$ (the case which corresponds to the Einstein-Maxwell system) the situation is slightly degenerated and supplementary hypotheses are necessary.

We will start this talk by describing how to model heat diffusion on graphs and the relation between this model and the eigenvalues of random matrices, in particular Wigner's semicircle law. These considerations will lead us to discuss free probability theory at an introductory level, highlighting its similarities with standard probability theory.

We consider generic translation surfaces of genus g>0 with marked points and take covers branched over the marked points such that the monodromy of every element in the fundamental group lies in a cyclic group of order d. Given a translation surface, the number of cylinders with waist curve of length at most L grows like L^2. By work of Veech and Eskin-Masur, when normalizing the number of cylinders by L^2, the limit as L goes to infinity exists and the resulting number is called a Siegel-Veech constant. The same holds true if we weight the cylinders by their area. Remarkably, the Siegel-Veech constant resulting from counting cylinders weighted by area is independent of the number of branch points n. All necessary background will be given.  This is joint work with Aaron Calderon, Carlos Matheus, Nick Salter, and Martin Schmoll.

Given a text, can we determine whether it was generated by a large language model (LLM) or by a human? Watermarking is a prominent method used to address this question. In this talk, we will analyze two different watermarking settings from a statistical perspective, focusing in particular on how mixtures and statistical-to-computational gaps can be useful for watermarking. This is a joint work with Roman Vershynin.

The crossing lemma for simple graphs gives a lower bound on the necessary number of crossings of any drawing of a graph in the plane in terms of its number of edges and vertices. Viewed through the lens of topology, this leads to other questions about arcs and curves on surfaces. In joint work with Alfredo Hubard, we provide estimates on the necessary number of intersections of any realization of <i>m</i> distinct homotopy classes of curves on a (fixed) surface. These estimates allow us to answer questions raised by Pach, Tardos, and Toth concerning a version of the crossing lemma for graph drawings with non-homotopic edges. Our approach uses the geometry of hyperbolic surfaces in an essential way.

We provide a finite element discretization of $\ell$-form-valued $k$ form in $n$ dimensions for general $k$, $\ell$ and $n$ and polynomial degree. The construction generalizes finite element Whitney forms for the de~Rham complex and their higher-order and distributional versions, the Regge finite elements and the Christiansen--Regge elasticity complex, the TDNNS element for symmetric stress tensors, the MCS element for traceless matrix fields, the Hellan--Herrmann--Johnson (HHJ) elements for biharmonic equations, and discrete divdiv and Hessian complexes in [Hu, Lin, and Zhang, 2025]. The construction discretizes the Bernstein--Gelfand--Gelfand (BGG) diagrams. Applications of the construction include discretization of strain and stress tensors in continuum mechanics and metric and curvature tensors in differential geometry in any dimension. This talk is based on a joint work with Kaibo Hu (Edinburgh).

We study the long time behavior for the distribution of a critical birth and death diffusion process, motivated by population dynamics in changing environment (cf. a recent paper by Calvez, Henry, Méléard, Tran). The birth rates are bounded but death rates are unbounded. Our analysis is based on the spectral properties of the associated Feynman Kac semigroup. We require a standard spectral gap property for this semigroup with a dominant eigenfunction vanishing at infinity. Some examples of diffusions, diffusions with jump, pure jump dynamics are given for which it is true. We consider situations where the underlying diffusion process doesn't come down rapidly from infinity but the compactness properties follow from the divergence of the death rate at infinity. We prove the convergence in law of the branching diffusion process suitably normalized and conditioned to non-extinction. We also prove the existence of the $Q$-process. The main tool is the convergence of suitably normalized moments of the process, which follows from recursive relations for these moments. This is a joint work with Pierre Collet and Jaime San Martin.

data.table is an R package with C code that is one of the most efficient open-source in-memory database packages available today. First released to CRAN by Matt Dowle in 2006, it continues to grow in popularity, and now over 1500 other CRAN packages depend on data.table. This talk will discuss basic and advanced data manipulation topics, and end with a discussion about how you can contribute to data.table.

In this talk I will discuss basic predictions for the replica symmetric regime of the Sherrington-Kirkpatrick spin glass model. For the model without external field, Talagrand conjectured that at high temperature the two point correlation matrix has operator norm that is bounded uniformly in the system size $N$. Based on suitable Schatten norm estimates, he was able to show that the operator norm grows at most logarithmically in $N$. I will present a different strategy that is based on ideas related to the TAP approach and that enables an exact computation of the operator norm as $N\to\infty$, valid in the full high temperature regime. I will conclude the talk with some related open questions for the SK model with non-zero external field. The talk is based on joint work with A. Schertzer, C. Xu and H.-T. Yau.

Recently, providers of variable annuity (VA) contracts have launched products which offer potentially higher guaranteed benefits through a ratcheting mechanism in conjunction with an array of investment options, including a cash fund. In some contract designs, the cash fund serves as an intermediate repository of earnings. For example, in a VA with a guaranteed minimum withdrawal benefit (GMWB), the policyholder has the option to withdraw less than the guaranteed withdrawal amount, with the difference being deposited into the cash fund, which appreciates at a benchmarked rate until the contract matures. We consider the valuation of a VA contract with a GMWB rider in which the policyholder has access to a cash fund. Assuming a ratcheting mechanism for the guarantee, we determine the optimal withdrawal strategy and provide numerical examples of cash flows emanating from the contract. We also investigate the implications of taxation on the value of the VA contract.

Aiming to unify the construction of the dual abelian variety with the Cartier duality of algebraic tori, one is naturally led to a duality theory for commutative group stacks. This construction was first considered by Laumon, and it led to the proof of geometric Langlands for tori. In this talk, I will present a conceptual construction of this duality and highlight intriguing phenomena that emerge—in particular, the dual of the additive group turns out to be a rather mysterious object. This is joint work with Zev Rosengarten.