Until the 2010’s the only « comparison geometry » result for compact Riemannian manifolds (M^n,g) with scal≥n(n-1) was Greene’s upper bound on the injectivity radius. Moreover, it is known that classical metric invariants (volume, diameter) cannot be controlled by a lower bound on the scalar curvature alone. It has only recently been discovered that some more subtle invariants, such as 2-systoles, can be controlled under a lower bounds on scal provided M has enough topology. We will present some results of Bray-Brendle-Neves (in dim 3), Zhu (in dim≤7) for S^2xT^(n-2), some version for S^2xS^2 and some conjecture with more general topology which we show to hold true under the additional assumption of Kaehlerness.

In a classical work, Bowen and Margulis proved the equidistribution of
closed geodesics in any hyperbolic manifold. Together with Jeremy Kahn
and Vladimir Marković, we asked ourselves what happens in a
three-manifold if we replace curves by surfaces. The natural analog of a
closed geodesic is then a minimal surface, as totally geodesic surfaces
exist only very rarely. Nevertheless, it still makes sense (for various
reasons, in particular to ensure uniqueness of the minimal
representative) to restrict our attention to surfaces that are almost
totally geodesic.

The statistics of these surfaces then depend very strongly on how we
order them: by genus, or by area. If we focus on surfaces whose *area*
tends to infinity, we conjecture that they do indeed equidistribute; we
proved a partial result in this direction. If, however, we focus on
surfaces whose *genus* tends to infinity, the situation is completely
opposite: we proved that they then accumulate onto the totally geodesic
surfaces of the manifold (if there are any).

We propose a novel framework for exploring weak and $L_2$ generalization errors of algorithms through the lens of differential calculus on the space of probability measures. Specifically, we consider the KL-regularized empirical risk minimization problem and establish generic conditions under which the generalization error convergence rate, when training on a sample of size $n$ , is $\matcal{O}(1/n)$. In the context of supervised learning with a one-hidden layer neural network in the mean-field regime, these conditions are reflected in suitable integrability and regularity assumptions on the loss and activation functions.

In this talk, we will present a loop expansion for lattice gauge theories and its application to prove ultraviolet stability in the Abelian Higgs model. We will first describe this loop expansion and how it relates to earlier works of Brydges-Frohlich-Seiler. We will then show how the expansion leads to a quantitative diamagnetic inequality, which in turn implies moment estimates, uniform in the lattice spacing, on the Holder-Besov norm of the gauge field marginal of the Abelian Higgs lattice model. Based on Gauge field marginal of an Abelian Higgs model, which is joint work with Ajay Chandra.

The Steinberg scheme and the equivariant coherent sheaves on it play a very important role in Geometric Representation theory. In this talk we will discuss various t-structures on the equivariant derived category of the Steinberg of importance for Representation theory in positive characteristics. Based on arXiv:2302.05782.

This seminar is part of our Frontiers in Quantitative Finance. Attendance is free of charge but requires prior online registration.

Abstract
Empirical studies consistently find that the price impact of large trades approximately follows a nonlinear power law. Yet, tractable formulas for the portfolios that trade off predictive trading signals, risk, and trading costs in an optimal manner are only available for quadratic costs corresponding to linear price impact. In this paper, we show that the resulting linear strategies allow to achieve virtually optimal performance also for realistic nonlinear price impact, if the “effective” quadratic cost parameter is chosen appropriately. To wit, for a wide range of risk levels, this leads to performance losses below 2% compared to the numerical Viterbi algorithm of Kolm and Ritter (2014) run at very high accuracy. The effective quadratic cost depends on the portfolio risk, but can be computed without any sophisticated numerics by simply maximizing an explicit scalar function.
Read more on this work here.

Central clearing counterparty houses (CCPs) play a fundamental role in mitigating the counterparty risk for exchange traded options. CCPs cover for possible losses during the liquidation of a defaulting member's portfolio by collecting initial margins from their members. In this article we analyze the current state of the art in the industry for computing initial margins for options, whose core component is generally based on a VaR or Expected Shortfall risk measure. We derive an approximation formula for the VaR at short horizons in a model-free setting. This innovating formula has promising features and behaves in a much more satisfactory way than the classical Filtered Historical Simulation-based VaR in our numerical experiments. In addition, we consider the neural-SDE model for normalized call prices proposed by [Cohen et al., arXiv:2202.07148, 2022] and obtain a quasi-explicit formula for the VaR and a closed formula for the short term VaR in this model, due to its conditional affine structure.