Weekly Bulletin

Some 20 years ago, Chen, Gibney and Krashen introduced beautiful algebraic varieties parametrizing "pointed trees of projective spaces"; these varieties generalize the celebrated Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points. In the latter case, the homology operad encodes the tree level part of a cohomological field theory. It follows from the work of Givental and many others that the space of all such structures on a given graded vector space has a very rich symmetry group; this fact has been used in many different research areas. In this talk, I shall prove that the homology of the operad made of Chen-Gibney-Krashen spaces possesses many interesting properties, and in particular, there are higher-dimensional Givental symmetries that emerge in this story. Curiously enough, this leads to new results on classical Givental symmetries as well. This is joint work with Eduardo Hoefel, Sergey Shadrin and Grigory Solomadin.

This talk aims at providing gauge field theory, in particular colour–kinematics duality and the double copy construction, with the required higher algebraic structures. From the algebro-homotopical perspective, a classical field theory is encoded by a cyclic homotopy Lie algebra whose Maurer–Cartan functional defines the action. For many gauge theories of interest, including Chern–Simons and Yang–Mills, this algebra splits as a tensor product of a cyclic Lie algebra, the colour Lie algebra, and a cyclic homotopy-commutative algebra, the kinematic algebra. The duality between colour and kinematics, first observed by Bern–Carrasco–Johansson in the study of Yang–Mills amplitudes, suggests that the kinematic algebra carries a Lie-type structure. For theories with at most cubic interactions, this structure is captured by coexact Batalin-Vilkovisky (BV) algebras, algebraic objects dual to the exact BV algebras arising in Poisson geometry. To incorporate higher-order interactions, following ideas of M. Reiterer, a homotopy refinement of this notion becomes necessary. The purpose of this talk is to provide a conceptual definition of homotopy coexact BV algebras, expressed in relation to homotopy commutative and BV algebras, together with a concrete operadic model. Our framework gives explicit presentations in terms of generating operations and relations and enables the systematic application of homotopical methods—including homotopy transfer, rectification, infinity-morphisms, and deformation theory—to the resulting algebras. The quartic-level structures recently identified in Yang–Mills theory fits naturally into this framework. This is a joint work with Anibal Medina-Mardones.

Lagrangian tangles are cobordisms between smooth links that generalize the classical Arnol'd theory of Lagrangian cobordism and the theory of Lagrangian cobordisms between Legendrian links. In this talk, we will explore the symplectic geometry of Lagrangian tangle links in the product of a surface with the complex numbers. The main tool is a novel Floer theory for Lagrangian tangles inspired by Morse theory for manifolds with a gradient field tangent to the boundary. We show the existence of an LES of persistence modules giving quantitative obstructions to the existence of Lagrangian tangle links. This is joint work with Josh Sabloff (Haverford College).

After a brief historical review I will concentrate on recent and ongoing work on constructions of hypersurface doublings (with J. Zou), results on the index and nullity of minimal surface doublings in the round three-sphere (with Zou), results on some Yau questions for minimal surfaces in the round three-sphere (with Mcgrath), on general doubling constructions in the case the base surface has nontrivial kernel for the Jacobi operator (with Zou), and finally various open questions.

<p>Skeleta of Berkovich spaces link together the worlds of combinatorics, arithmetic geometry, analytic geometry, tropical geometry and resolution of singularities. Rather than getting lost into tedious details, this talk aims to give an overview of historic and recent developments of non-archimedean geometry in the sense of Tate and Berkovich in a way that is understandable by any graduate student in mathematics.<br>We will introduce non-archimedean fields and how they historically led to a new type of analytic space. Then we discuss analogies between geometry of Berkovich spaces and complex spaces, which is captured by skeleta. Finally we discuss different different ways of obtaining skeleta and canonical forms can play a key role in this, and further reduce the dependence on tools from algebraic geometry.</p>

<div>The dynamics of billiards in non-rational polygons</div> <div>is rather poorly understood for lack of tools, in particular </div> <div>the lack of a renormalization theory.  In this talk, we focus</div> <div>on results which can be proved by harmonic analysis tools,</div> <div>in particular results on the cohomological equation for the flow</div> <div>and results on dynamical properties of the holonomy foliation. </div> <div>This is joint work with N. Moll.</div>

We consider a family of boundary integral operators (BIOs) arising from the boundary reduction of well-posed problems—either Laplace or Helmholtz—defined on a collection of parametrically described domains. Our goal is to establish the holomorphic (analytic) dependence of these BIOs, as well as of the solutions to associated boundary integral equations, on perturbations of the domain shape, a property commonly referred to as shape holomorphy. To date, various results have addressed shape holomorphy under differing assumptions, particularly concerning the physical dimension of the problem and, most notably, the smoothness of the domain deformations. In this talk, we present recent results on the shape holomorphy of BIOs under Lipschitz-regular deformations. We also discuss the implications of these findings for the development of computational methods in forward and inverse uncertainty quantification, as well as model order reduction.

A graph property is called increasing if it is closed under addition of edges. For an increasing property, the probability of its occurrence in the binomial random graph G(n,p) transitions rapidly (in asymptotics) from 0 to 1 as p crosses the so-called probability threshold. Since the original paper of Erdős and Rényi the task of determining the asymptotic behaviour of threshold probabilities for increasing properties has been a central topic in probabilistic combinatorics. While the asymptotic order of the probability threshold has been determined for many natural increasing graph properties, a general solution remains unknown, and determining the exact asymptotics is even more challenging. In the talk I will provide an answer to the latter question for a class of increasing properties generated by d-regular graphs. This family of d-regular graphs contains asymptotically almost all d-regular graphs and, in particular, it contains the square of a cycle. This resolves a conjecture of Kahn, Narayanan, and Park, regarding the asymptotics of the threshold for the appearance of the square of a Hamilton cycle.

In this talk, we will prove that for any non-trivial knot K, infinitely many r-surgeries K(r) along K have a unique surgery description along a knot. Conversely, we will discuss the optimality of this result by providing new families of manifolds admitting different surgery descriptions. This is based on joint work with Misha Schmalian.

The assessment of monotone dependence between random variables $X$ and $Y$ is a classical problem in statistics and a gamut of application domains. Consequently, researchers have sought measures of association that are invariant under strictly increasing transformations of the margins, with the extant literature being splintered. Rank correlation coefficients, such as Spearman's Rho and Kendall's Tau, have been studied at great length in the statistical literature, mostly under the assumption that $X$ and $Y$ are continuous. In the case of a dichotomous outcome $Y$, receiver operating characteristic (ROC) analysis and the asymmetric area under the ROC curve (AUC) measure are used to assess monotone dependence of $Y$ on a covariate $X$. In this talk I demonstrate that the two thus far disconnected strands of literature can be unified and bridged, by developing common population level theory, common estimators, and common tests that apply to all types of linearly ordered outcomes. In case studies, we assess progress in artificial intelligence (AI) based weather prediction and evaluate methods of uncertainty quantification for the output of large language models. The talk is based on joint work with Eva-Maria Walz and Andreas Eberl.

<p>The Hartree equation is a quantum version of the nonlinear Vlasov equation in the sense that it is a mean-field approximation of the many-body dynamics (classical for Vlasov, quantum for Hartree). In this talk I will present results concerning the asymptotic stability of homogeneous equilibria for the Hartree equation, which is a quantum version of the "Landau damping" results for the Vlasov equation. This is a joint work in collaboration with Antoine Borie (Rennes) and Sonae Hadama (Kyoto).</p>

We use neural operators to learn the coefficient-to-solution map for nonlinear elliptic equations of double phase type. Since the solution is the unique minimizer of the convex splitting problem involving the two double phase functionals over a uniformly convex and uniformly smooth Banach space, we approximate the coefficient-to-solution map by a generative equilibrium operator (GEO). For training, we concatenate the GEO with the reconstruction formula of the coefficient to learn its right-inverse on the coefficient space. Once trained, this approach allows us to efficiently predict solutions of the double phase equation for other coefficients. Further applications of these neural operators are presented in the context of portfolio optimization and quadratic hedging.

I will present a topological perspective on the order of the Tate-Shafarevich group up to squares as an application of a formula established in previous joint work with A. Venkatesh which gives a cohomological expression for the central value of symplectic L-functions on curves. I will sketch ideas of the proofs, and in particular the analogous picture in topology that is crucial in the arithmetic proofs.

The moduli space of pointed rational curves carries a natural action of the symmetric group permuting the marked points. In this talk, I will present combinatorial and recursive formulas for the induced representations on its cohomology. These formulas are obtained by relating the representation to that of the moduli space with one additional marked point fixed under the symmetric group action, which has a particularly nice and tractable structure. Taking the invariant subspace, in particular, leads to an effective inductive formula for the Betti numbers of the moduli space of rational curves with unordered markings. I will also discuss several interesting conjectural properties of these representations. This talk is based on joint work with Jinwon Choi and Young-Hoon Kiem.