Young Scholars Forum 2022, Institute of Mathematical Sciences at ShanghaiTech University

2022年上海科技大学数学科学研究所青年学者论坛议程

Agenda of  Young Scholars Forum 2022, Institute of Mathematical Sciences at ShanghaiTech University

Saturday, November 26th, 2022

9:00-10:00

Title:Langevin dynamics-based algorithms for sampling and optimization problems with applications in machine learning and finance

Speaker：Zhang, Ying

Abstract:In this talk, I will present three numerical algorithms, namely, SGLD, TUSLA, and e-Theo POULA, with applications in machine learning and finance. For the SGLD algorithm, we obtain convergence results under relaxed conditions compared to existing results in the literature. This allows us to extend the applicability of the SGLD algorithm to applications including index tracking optimization and CVaR minimization. However, SGLD cannot be applied to optimization problems with highly-nonlinear objective functions. We address this problem by proposing the TUSLA algorithm. We describe the conditions under which the theoretical guarantees can be obtained for TUSLA, and then provide the main convergence results. An optimization problem involving ReLU neural network is provided to illustrate the applicability of TUSLA. Finally, I will talk about the e-Theo POULA algorithm, which combines the advantages of the Langevin dynamics-based algorithms and the adaptive learning rate methods. An example from multi-period portfolio optimization is presented to show the powerful empirical performance of e-Theo POULA.

10:00-11:00

Title:Stationary solutions for 1D Burgers equations and KPZ scaling

Speaker：Li, Liying

Abstract: In the first part, we will talk about the stationary solutions for 1D stochastic Burgers equations and their ergodic properties.  We will classify all the ergodic components, establish the ``one force---one solution'' principle, and obtain the inviscid limit. The key objects to study are the infinite geodesics and infinite-volume polymer measures in random environments, and the ergodic results have their counterparts in the geodesic/polymer language. In the second part, we will present a random point field model that is motivated by the coalescing and monotone structure of the optimal paths in random environments that arise in many KPZ models. The 2/3 transversal exponent from the KPZ scaling becomes a natural parameter for the renormalization action in this model, and can be potentially extended to values other than 2/3. Some preliminary results are given.

11:00-12:00

Title:Efficient numerical method and reduced order model for transport dominant problems

Speaker：Peng, Zhichao

Abstract:The transport phenomena arise in many important areas of applications, such as electromagnetic, nuclear engineering and quantum physics. In this talk, we will focus on efficient numerical method and reduced order models for transport dominant problems with various computational challenges.

In the first part, we will focus on the high frequency Maxwell’s equation and present a flexible and efficient frequency-domain solver built from time-domain solvers. Two challenges of solving the frequency-domain Maxwell’s equation at high frequencies are its indefinite nature and high resolution requirement. The proposed method converts efficient time-domain solvers to efficient frequency-domain solvers, and it always leads to a better conditioned linear system which is proved to be symmetric positive definite for special cases.

In the second part, we will discuss a reduced order model (ROM) for the time-dependent radiative transfer equation (RTE) which is a high dimensional and multiscale kinetic transport equation. To mitigate the curse of dimensionality, we utilize the underlying low-rank structure of the RTE to construct a ROM with the reduced basis method (RBM). The proposed ROM reduces the degrees of freedom by projecting the RTE to low-dimensional reduced order subspaces. Key components of our ROM include an equilibrium-respecting strategy to construct low-dimensional subspaces and a reduced quadrature rule to handle the collisional operator of the RTE.

14:00-15:00

Title:Causal Effect of Functional Treatment

Speaker：Tan, Ruoxu

Abstract:Functional data often arise in the areas where the causal treatment effect is of interest. However, research concerning the effect of a functional variable on an outcome is typically restricted to exploring the association rather than the casual relationship. The lack of definition of probability density function for functional data poses a challenge for consistent estimation of causal effect. To overcome the difficulty, we propose a well-defined functional stabilized weight and develop a novel estimator for it. Based on the functional linear model for the average dose-response functional, we propose three estimators, namely, the functional stabilized weight estimator, the outcome regression estimator and the doubly robust estimator, each of which has its own merits. We study their theoretical properties, which are corroborated through extensive numerical experiments. A real data application on electroencephalography data and disease severity demonstrates the practical value of our methods.

Sunday, November 27th, 2022

9:00-10:00

Title:Computations in equivariant algebraic topology

Speaker：Zou, Foling

Abstract:Modern algebraic topology sees equivariance arising in unexpected context. Equivariant cohomology carries rich structures but is much harder to compute. In 2009, Hill, Hopkins, and Ravenel solved the 50-year-old Kervaire invariant problem about framed manifolds (for p = 2), which has nothing to do with group actions a prior, using equivariant computation. Their work was related to the computation of the dual Steenrod algebra for the group Z/2 by Hu and Kriz. We compute the dual Steenrod algebra for the group Z/p for odd p. It turns out that the case of odd primes has interesting new components. We hope to use it to tackle the odd primary Kervaire problem, which remains open for p = 3. I will also talk about equivariant factorization homology and its application in the computation of the Real topological Hochschild homology.

10:00-11:00

Title:The Fermi Acceleration Problem

Speaker：Zhou, Jing

Abstract:In this talk, I will briefly introduce the Fermi-Ulam acceleration problem and the existing results on the subject. In particular, I will present my work on several variants of the Fermi-Ulam models: the bouncing ball in gravity and the billiard with moving platforms. We use techniques from elliptic as well as hyperbolic dynamics with singularities to study the ergodic and statistical properties of these systems on infinite-volume phases.

11:00-12:00

Title:open dynamical systems

Speaker：Su, Yaofeng

Abstract:Open dynamical systems describe hitting processes of orbits through a target in phase space. One of the main questions of open systems is to study a statistical property (called a Poisson approximation) of the hitting process. I will present some new results for it, applications include some dissipative systems and hyperbolic billiards.

14:00-15:00

Title:Bounding irrationality of degenerations of Fano fibrations

Speaker：Qu, Santa

Abstract:In this talk, I will introduce a recent result about bounding degrees of irrationality of degenerations of klt Fano fibrations of arbitrary dimensions.  This proves the generically bounded case of a conjecture proposed by C. Birkar and K. Loginov for log Fano fibrations of dimensions greater than three.  Our approach depends on a method to modify the klt Fano fibration to a toroidal morphism of toroidal embeddings with bounded general fibres.  Moreover, we show that every fibre of the toroidal morphism is bounded and has mild singularities if we replace the birational modifications by alterations.  This is a joint work with Prof. C. Birkar.

15:00-16:00

Title:Enumerative geometry from 3d gauge theory and beyond

Speaker：Zhou, Zijun

Abstract:In this talk I will give an introduction of the recent development of  K-theoretic enumerative algebraic geometry, motivated from 3d supersymmetric gauge theories. We will mainly focus on the invariants called vertex function, and the geometric object called elliptic stable envelope, both introduced by A. Okounkov. These invariants are interesting from different aspects. On the on hand, they are K-theoretic versions of Gromov-Witten-type theories, which count quasi-maps, leading to deep mathematical realizations of the physics phenomenon called 3d mirror symmetry. On the other hand, they admit close connection to new constructions of geometric representation theory, e.g. BFN’s construction of Coulomb branch. I will introduce my work on both aspects, as well as future outlook for combining them.