Events

In 1984 Yoccoz proved that any two C^3 orientation-preserving circle homeomorphisms, with the same quantity of non-flat critical points and the same irrational rotation number, are topologically conjugated. For maps with only one critical point, it has been shown that the conjugacy is, in fact, a C^1-diffeomorphism. Moreover, in a total Lebesgue measure set of irrational rotation numbers the conjugacy can be improved to be C^{1+\alpha}. In this talk, we will discuss some recent results concerning the smoothness of the conjugacy for maps with more than one critical point, and how these results are obtained using renormalization.

An Enriques surface is the quotient of a K3 surface by a fixed point-free involution. Klemm and Marino conjectured a formula expressing the Gromov-Witten invariants of the local Enriques surface in terms of automorphic forms. In particular, the generating series of elliptic curve counts on the Enriques should be the Fourier expansion of (a certain power of) Borcherds automorphic form on the moduli space of Enriques surfaces. In this talk I will explain a proof of this conjecture. The proof uses the geometry of the Enriques Calabi-Yau threefold in fiber classes. If time permits, I will also discuss various conjectures about non-fiber classes.

我们考虑稀疏线性混合注册的问题ression with two components, where two sparse signals are observed through n unlabelled noisy linear measurements. Prior work has shown that the problem suffers from a significant statistical-to-computational gap, resembling other computationally challenging high-dimensional inference problems such as Sparse PCA and Robust Sparse Mean Estimation. We establish the existence of a more extensive computational barrier for this problem through the method of low-degree polynomials, but show that the problem is computationally hard only in a very narrow symmetric parameter regime. We identify smooth information-computation tradeoffs in this problem and prove that a simple linear-time algorithm succeeds outside of the narrow hard regime. To the best of our knowledge, this is the first thorough study of the interplay between mixture symmetry, signal sparsity, and their joint impact on the computational hardness of mixed sparse linear regression. This is joint work with Ramji Venkataramanan. https://proceedings.mlr.press/v195/arpino23a.html.

We consider elliptic equations with periodic high contrast coefficients and study the asymptotic analysis when the periodicity is sent to zero and/or the contrast parameters are sent to extreme values. Those coefficients model small inclusions that have very different physical properties compared to the surrounding environment. Homogenization captures the macroscopic effects of those inclusions. We report some quantitative results such as the convergence rates of the homogenization (with proper correctors), uniform regularity for the solutions of the heterogeneous equations, and so on. The talk is based on joint works with Mr. Xin Fu.

What is the probabilistic core of life insurance modelling? What kind of mathematical objects are premiums and reserves? Which modelling assumptions are really necessary and which are not? The presentation will take the audience on a journey through old concepts and modern interpretations. Traditions will be challenged and alternative views presented.

BPS invariants were introduced by Gopakumar-Vafa on Calabi-Yau 3-folds, Klemm-Pandharipande on CY 4-folds and Pandharipande-Zinger on CY 5-folds. They are conjectured to be integers (proven in many cases) and have correspondence with Gromov-Witten invariants. On holomorphic symplectic 4-folds, (ordinary) GW and hence BPS invariants vanish, one can consider reduced GW invariants which are usually nontrivial rational numbers. In this talk, we will introduce BPS invariants for such a reduced theory. Joint works with Georg Oberdieck and Yukinobu Toda.