Weekly Bulletin

A couple of years ago, with M.~Goldman we devised a new approach to the regularity theory for Optimal Transportation that mimics De Giorgi's approach to the regularity theory of minimal surfaces in the sense that a harmonic approximation result is at its center: Under a non-​dimensional smallness condition, the displacement is close to the gradient of a harmonic function. Probably the main advantage of this variational regularity theory over the one based on maximum principle -- and attached to the name of Caffarelli -- is that it does not require any regularity of the involved measures. Hence it can be applied to the matching problem, where it allows for a regularity theory on microscopically large scales, and thus makes Parisi's linearization quantitative down to these scales. It thus provides a mesoscopic counterpart to macroscopic theory pioneered by Ambrosio and coworkers.

Mean-​field or McKean-​Vlasov type optimal control is closely related to the exciting program of mean-​field games as initiated by Larry and Lions. Dynamic programming approach to these control problems result in nonlinear partial differential equations on the space of probability measures. These equations not only require the solution to be differentiable but impose further regularity on the derivatives which are being on the dual of the set of measures are also functions themselves. Despite these difficulties, several approaches to characterize the value function of the control problems as the unique appropriate weak solutions have been developed. In this talk, I will first introduce the mean field games through an interesting example of Kuramoto type synchronization. Then, I will extend this example to a general setting and prove uniqueness of for a class of equations that are analogous to classical Eikonal equations. This talk is based on joint works with Rene Carmona and Qinxin Yan of Princeton, and Quentin Cormier of INRIA.

We initially consider a controlled reaction-​diffusion equation, modeling the spreading of an invasive population. By formally taking a sharp interface limit, a simpler model is derived, describing the controlled evolution of the contaminated region. This leads to a family of optimization problems for a moving set. Results on controllability, existence of optimal strategies, and necessary conditions for optimality will be given. Roughly speaking, optimal strategies are those that shrink the initial set to the empty set, keeping the perimeter as short as possible over time. Some open questions will be discussed.

Enumerating curves in algebraic varieties traditionally involves choosing a compactification of the space of smooth embedded curves in the variety. There are many such compactifications, hence many different enumerative invariants. I will propose a "universal" (very tautological) enumerative invariant which takes values in a certain Grothendieck group of 1-cycles. It is often the case with such "universal" constructions that the resulting Grothendieck group is essentially uncomputable. But in this case, the cluster formalism of Ionel and Parker shows that, in the case of threefolds with nef anticanonical bundle, this Grothendieck group is freely generated by local curves. This reduces the MNOP conjecture (in the case of nef anticanonical bundle and primary insertions) to the case of local curves, where it is already known due to work of Bryan--Pandharipande and Okounkov--Pandharipande.

The slicing problem by Bourgain is an innocent-​looking question in convex geometry. It asks whether any convex body of volume one in an n-​dimensional Euclidean space admits a hyperplane section whose (n-1)-​dimensional volume is at least some universal constant. There are several equivalent formulations and implications of this conjecture, which occupies a rather central role in the field. The slicing conjecture would follow from the isoperimetric conjecture of Kannan, Lovasz and Simonovits, which suggests that the most efficient way to partition a convex body into two parts of equal volume so as to minimize their interface, is a hyperplane bisection, up to a universal constant. In this lecture we will discuss progress from the last two years, showing that these two conjectures hold true up to factors that increase logarithmically with the dimension.

Whereas general existence results for minimizers of (vectorial) variational problems are clearly related to (coercivity) and Morreys quasiconvexity, the situation becomes much more constrained if also uniqueness of the minimizers is required for all linear pertubation of the energy. In this case a rather natural notion of functional convexity arises in a general Banach space context. We will discuss what are the specific implications for energy densities of integral cost functions. This is joint work with J. Campos Cordero (Mexico), J. Kollar (Prag) and J.Kristensen (Oxford).

The quantitative uniqueness and the geometric measure estimates for the nodal and critical point sets of solutions of second order elliptic equations depend crucially on the bound of the associated Almgren's frequency function. The latter is possible (only) when the leading coefficients of equations are uniformly Lipschitz. One does not have this uniform Lipschitz continuity for coefficients of equations in elliptic homogenization. Instead, by using quantitative homogenization, successive harmonic approximation and suitable L^2-​renormalization, we shall see how one can get a uniform estimate (independent of a small parameter characterizing the nature of homogenization) of co-​dimension two Hausdorff measure as well as the Minkowski content of the critical point sets. A key element is an estimate of "turning" for the projection of a non-​constant solution onto the subspace of spherical harmonics of order N, when the doubling index of solution on annular regions is trapped near N.

The works of Preiss and Preiss & Mattila provide criteria for the rectifiability of measures in terms of the existence of densities and principal values for the Riesz transform. Whether these results depend on the underlying metric in Euclidean space in the case of the density has been a puzzling problem for years. In this talk we will discuss some recent results in this direction as well as what happens under the assumption of the existence of principal values for the gradient of the fundamental solution to general divergence form operators. This is joint work with M. Goering and B. Wilson.

We discuss the role of variable and distributional (synthetic lower) bounds for the Ricci curvature in studying spectral gap and gradient estimates for the heat semigroup. In particular, we present a novel sharp lower bound for the spectral gap on a nonnegatively curved Riemannian manifold or, more generally, on an RCD(𝑘,𝑁) space with a variable Ricci bound 𝑘:𝑋𝑡𝑜𝑅+. Our estimate in terms of the 𝐿−𝑝 norm of 𝑘 for 𝑝=1−1/𝑁 improves upon the celebrated Lichnerowicz estimate (1958, case 𝑝=∞) and the estimate of Veysseire (2010, case 𝑝=1). Also, spectral gap estimates with negative Ricci curvature in the Kato class will be briefly discussed. Moreover, we present gradient estimates for the Neumann heat semigroup on non-​convex domains which leads to a negative, distribution-​valued Ricci curvature in the Kato class.

In this talk I will present recent results and open problems concerning the lowest eigenvalue of a Dirac operator with a general multi-​pole external electrostatic potential. They describe a relativistic quantum electron moving in the field of some (point-​wise or extended) nuclei, possibly in a molecule. One of the main questions we ask is whether the eigenvalue is minimal when the nuclear charge is concentrated at one single point. This well-​known property in non-​relativistic quantum mechanics (involving the Schrödinger operator) has escaped all attempts of proof in the relativistic case. This is work in collaboration with M. Lewin and E. Séré.

Area-​minimizing integral currents are a natural generalization of area-​minimizing oriented surfaces, a concept pioneered by De Giorgi and Federer-​Fleming. Famous examples of singular 7-​dimensional minimizers in ℝ8 and of singular 2-​dimensional minimizers in ℝ4 are known since long. Moreover a theorem which summarizes the work of several mathematicians in the 60es and 70es (De Giorgi, Fleming, Almgren, Simons, and Federer) and a celebrated work by Almgren in the 80es give dimension bounds for the singular set which match the one of the examples, in codimension 1 and in general codimension respectively. In codimension higher than 1 a recent result of Liu shows that the singular set can in fact be a fractal of any Hausdorff dimension 𝛼≤𝑚−2. In joint works with Anna Skorobogatova and Paul Minter we prove that the singular set is 𝑚−2. A proof of the same result has been discovered at the same time and independently by Brian Krummel and Neshan Wickramasekera. The theorem is the counterpart of a celebrated work of Leon Simon in the nineties for the hypersurface case, and indeed a byproduct of our proof is the uniqueness of the tangent cone at 𝑚−2-a.e. point.

I will review old results on the regularity of minimizers of Total-​Variation regularized inverse (denoising) problems. Addressing then the issue of the localization of the jump set of minimizers, for data with bounded variation, I will introduce a new and much simpler technique which provides some information for a much broader class of problems and regularizers (joint work with M. Lasica, mathematical institute of the polish academy of science).

While Einstein's theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-​definite signature, there is a highly successful theory of metric and metric-​measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under curvature and dimension bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while Ricci curvature is reformulated using entropic convexity along geodesics of probability measures. This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. We highlight how the null energy condition of Penrose admits a nonsmooth formulation as a variable lower bound on timelike Ricci curvature.

We study a tumor growth model driven by nutrition and the pressure variable generated by the density height constraint. Our focus is on the free boundary regularity of the tumor patch that holds beyond topological changes, starting from general initial data. We will discuss the problem and our analysis, which centers around an elliptic obstacle problem and a Hopf-​Lax formula for the pressure variable. Building on these ingredients, we show that the patch boundary is space-​time regular in \R𝑑×(0,∞) except on a set of Hausdorff dimension less or equal to 𝑑−𝛼.

We consider the Cauchy problem for the Navier-​Stokes equation with viscosity 𝜈 in 𝑅3, and the initial vorticity given by an idealized vortex ring of a given radius and zero thickness. It can be shown the problem has a unique solution in its natural symmetry class. The goal is to study the behavior of the solutions for 𝜈→0. Joint work with Thierry Gallay.

In the first part of the talk we will adress the problem of studying area variations of surfaces under pointwise Lagrangian constraint in ℂ2 (or any arbitrary Kähler Surface). We will explain the challenges of performing analysis (well posedness, existence, regularity...) with the associated Euler-​Lagrange Equation. Then in trying to find conserved quantities and monotonicity formula for this problem we will naturally be invited to "lift" our problem to 5 dimensions by introducing a fifth Legendrian coordinate and to work in the Heisenberg group (or any Sasakian 5-​manifold). The Lagrangian constraint is then converted into a Legendrian one. The area variation under pointwise Legendrian constraint consists in looking for critical points of the area among surfaces which are horizontal. This is a model of "extreme anisotropic" variational problem where one direction is forbidden while total isotropy holds in the remaining 4 directions (which are not integrable). We will derive a new monotonicity formula for this problem. Ultimately the main result we would like to explain is the following : In any 5 dimensional closed Sasakian manifold 𝑁5 (e.g. 𝑆5, 𝑆3×𝑆2, Heisenberg group ℍ2...etc) we prove that any minmax operation on the area among Legendrian surfaces is achieved by a continuous conformal Legendrian map from a closed riemann surface 𝑆 into 𝑁5 equipped with an integer multiplicity bounded in 𝐿∞. Moreover this map, equipped with this multiplicity, satisfies a weak version of the Hamiltonian Minimal Equation. We conjecture that any solution to this equation is a smooth branched Legendrian immersion away from isolated Schoen-​Wolfson conical singularities with non zero Maslov class. If time permits we will explain our motivation for studying such question in relation with the Willmore conjecture in arbitrary co-​dimensions.

The two dimensional infinity Laplacian PDE, the Euler-​Lagrange equation for a sup-​norm variational problem, can be formally converted into the heat equation by the Legendre transform. I will discuss how to reverse this process, to help understand the geometry of "soft'' shocks for solutions, along which first derivatives are continuous but second derivatives are unbounded.

First, I will explain the nuts and bolts of decentralized finance (DeFi). I will put DeFi in perspective by comparing it to some adjacent concepts such as ledgers, cryptocurrencies, smart contracts, web3, etc. Then we will then delve into the good aspects of DeFi, identify areas that require further improvement, and which parts of DeFi are downright ugly. In particular, I will present some spectacular attacks on the DeFi ecosystem, and what can be done to prevent such attacks in the future. Lastly, time permitting, I will explore how the underlying technology of DeFi can contribute to governance and even democracy.

Isotonic distributional regression (IDR) is a nonparametric distributional regression approach under a monotonicity constraint. It has found application as a generic method for uncertainty quantification, in statistical postprocessing of weather forecasts, and it is an integral part of distributional single index models. IDR has favorable calibration and optimality properties in finite samples. Furthermore, it has an interesting population counterpart called isotonic conditional laws that generalize conditional distributions with respect to σ-​algebras to conditional distributions with respect to σ-​lattices. In this talk, an overview of the theory and some applications of IDR are presented.

Data and data-​driven decisions have always been the lifeblood of the insurance industry. In recent years, data availability and technology have grown dramatically, creating new opportunities. To realise the full value, insurers can face challenges such as data silos, low data literacy, and new risks from data. Nora will talk about how we at Swiss Re got a "data flywheel" spinning, a virtuous cycle of faster, more efficient data use and smarter decisions. She will talk about tools and processes to "grease" the wheel, and share use cases and some recommendations to increase the maturity of data and analytics.

Investment performance measurement and attribution are key functions of portfolio management, bringing transparency in terms of the true drivers of portfolio risk and return. Performance analysts provide real added value and are often the only independent source equipped to understand the sources of risk and return of all portfolios and benchmarks operating within an asset management firm. Performance attribution aims to split the total outperformance of a portfolio versus its benchmark into fair contributions from the active individual decisions as well as from markets’ movements. Whereas on one hand this goal can be successfully achieved with attribution schemes that match as closely as possible the management structure of the portfolio, running different attributions on the other hand can help managers to identify unexpected, sometimes possibly unwanted sources of risk and return and adjust the management process accordingly. Performance attribution models vary from the simplest ones 1) total return or sector based attribution to the most sophisticated ones 2) factor based attribution based on a set of common risk factors and 3) hybrid performance attribution, which bridges 1) and 2). In all cases these frameworks need to accurately deal with the complexities of portfolio management: portfolio assets spread over different markets, asset classes and currencies, multiple investment horizons, multi-​period compounding, derivatives handling, dynamic management, bad or missing security prices and analytics, etc. In the first part of the presentation the focus will be on factor based attribution. Risk factor models are not only indispensable tools for risk forecasting or portfolio optimization, but also largely used for performance attribution purposes. With their choice of portfolio analytics providers investment management firms need to be very selective, since there is no such a thing like a perfect model and a proper model governance and understanding of the model strengths and weaknesses are key. We use the latest generation of risk factor model suites from Bloomberg and MSCI, both leading providers of critical decision support tools. In the second part of the presentation we show examples on how hybrid performance attribution can be used to analyse the performance of fixed income portfolios by splitting total outperformance into two parts: that explained by bottom-​up common factor aggregation and that in excess of common factors, which is further top-​down (recursively) split into allocation/sector management on a user-​defined hierarchical partition.

We extract contextualized representations of news text to predict returns using the state-​of-the-art large language models in natural language processing. Unlike the traditional word-​based methods, e.g., bag-​of-words or word vectors, the contextualized representation captures both the syntax and semantics of text, thus providing a more comprehensive understanding of its meaning. Notably, word-​based approaches are more susceptible to errors when negation words are present in news articles. Our study includes data from 16 international equity markets and news articles in 13 different languages, providing polyglot evidence of news-​induced return predictability. We observe that information in newswires is incorporated into prices with an inefficient delay that aligns with the limits-​to-arbitrage, yet can still be exploited in real-​time trading strategies. Additionally, we find that a trading strategy that capitalizes on fresh news alerts results in even higher Sharpe ratios.