Carolina Araujo

Title: The geometry of tensors

Abstract: 

      Tensors are fundamental objects in multilinear algebra, with important applications to computability and complexity of algorithms, algebraic statistics, phylogenetics, signal processing, among many others. In applications, one is usually interested in decomposing a given tensor into a linear combination of indecomposable tensors. The smallest integer r needed to write a tensor T as a linear combination of r indecomposable tensors is called the rank of T. The problem of determining the rank of a given tensor has received great attention in recent years, and admits a beautiful geometric interpretation.

      In this talk, I will introduce tensors, explain some of their applications, and interpret the problem of tensor decomposition from the point of view of Algebraic Geometry. If time permits, I will also preset recent results on ranks of tensors, obtained using algebraic geometric methods in collaboration with Alex Massarenti and Rick Rischter.

Geovani Nunes Grapiglia

Title: First and zeroth-order implementations of the regularized Newton method with lazy approximated Hessians

Abstract: 

      In this work, we develop first-order (Hessian-free) and zeroth-order (derivative-free) implementations of the Cubically regularized Newton method for solving general non-convex optimization problems. For that, we employ finite difference approximations of the derivatives. We use a special adaptive search procedure in our algorithms, which simultaneously fits both the regularization constant and the parameters of the finite difference approximations.  It makes our schemes free from the need to know the actual Lipschitz constants. Additionally, we equip our algorithms with the lazy Hessian update that reuse a previously computed Hessian approximation matrix for several iterations. Specifically, we prove the global complexity bound of $mathcal{O}( n^{1/2} epsilon^{-3/2})$ function and gradient evaluations for our new Hessian-free method, and a bound of $mathcal{O}( n^{3/2} epsilon^{-3/2} )$ function evaluations for the derivative-free method, where $n$ is the dimension of the problem and $epsilon$ is the desired accuracy for the gradient norm.

      These complexity bounds significantly improve the previously known ones in terms of the joint dependence on $n$ and $epsilon$, for the first-order and zeroth-order non-convex optimization.

Liliane Basso Barichello

Title: Recent Studies on the Solution of the Linear Boltzmann Equation: Direct and Inverse Problems

Abstract: 

      Optical tomography modeling and nuclear safeguards are some of the relevant applications of the particle transport theory. In the first case, assuming the radiative transfer equation as the basic model for the direct problem, one is interested in estimating, in general, two parameters of the model: the absorption and scattering coefficients of the biological tissue. In nuclear safeguards analysis, a primary goal is reconstructing the source term in the neutron transport equation. Both models refer to variations of the linear Boltzmann equation, also known as the particle transport equation. The broad applicability of such an equation balances its high complexity and the challenge of deriving its solution. One of the very well-known techniques to deal with the transport equation is the discrete ordinates method, where the discretization of the particles’ directions, is assumed. Such an approach allows the transformation of the integrodifferential model into a system of differential equations. 

      In this talk, we present recent advances in analytical and spectral techniques for solving the discrete ordinates approximation of the linear Boltzmann equation. We introduce multidimensional quadrature schemes on the unit sphere and discuss their influence on the solution of large linear systems of algebraic equations arising in the derivation. Direct and iterative methods for solving such linear systems are presented. Numerical results for optical tomography and neutron transport problems are discussed. We also analyze the asymptotic convergence of the spatial and angular discretization of the solution in two-dimensional media. Furthermore, the analytical character of the solution is combined with either iterative Tikhonov regularization or Bayesian inference to solve the inverse problem of source reconstruction.

Kening Lv

Title: Turbulence, Lyapunov exponents, and SRB measures in infinite-dimensional dynamical systems

Abstract:

      In this talk, I will report several results concerning Lyapunov exponents, SRB measures, entropy, and horseshoes for infinite-dimensional dynamical systems. Additionally, I will present recent work focusing on the ergodicity and statistical dynamics of the 2-D Navier-Stokes equation, which is driven by both deterministic and stochastic forces. Furthermore, I'll explore the connection between SRB measures and turbulence.

Paolo Piccione

Title: Conformal Curvatures

Abstract:  

      In this talk, we will explore several notions of curvature, beginning with classical curves and surfaces in Euclidean space and advancing through the framework of Riemannian geometry. We will then shift our focus to conformal geometry, where the concept of curvature takes on new forms under angle-preserving transformations. The final part of the talk will highlight recent developments in Yamabe-type problems, discussing their geometric and analytical significance in the study of conformal curvatures.

Xiaoyun Wang

Title: Lattices and Post-Quantum Cryptography

Abstract: 

      Post-quantum cryptography (PQC) mainly refers to public key cryptosystems based on mathematical hard problems that are secure against both quantum and classical computers. Lattice-based cryptography, built on the mathematical hard problems in high-dimensional lattice theory, is a primary post-quantum cryptography family. Over the past three decades, significant advancements have been made in lattice-based cryptography research and lattice-based cryptography becomes a promising alternative to classical public key cryptography based on integer factorization and discrete logarithm problems. 

      In this talk, I will provide an overview of the mathematical foundations of lattice-based cryptography. Additionally, I will discuss recent developments in the practical design of lattice-based cryptosystems and offer insights into the exciting field of fully homomorphic encryption (FHE), which has promising applications in privacy computing.

Zaiwen Wen

Title: Exploring Learning-Based Algorithms and Theories in Mathematical Optimization

Abstract: 

      This talk will explore new paradigms for integrating data, models, algorithms, and theories in mathematical optimization. Firstly, we try to understand acceleration methods through ordinary differential equations (ODEs). Under convergence and stability conditions, we formulate a learning optimization problem that minimizes stopping time. This involves transforming the rapid convergence observed in continuous-time models into discrete-time iterative methods based on data. Next, we introduce a Monte Carlo strategy optimization algorithm for solving integer programming problems. This approach constructs probabilistic models to learn parameterized strategy distributions from data, enabling the sampling of integer solutions. Lastly, we discuss the vision of advancing automated theorem proving through formalization assisted by artificial intelligence.

Tong Yang

Title: Some recent study on kinetic equations in perturbative framework

Abstract: 

      There are two basic models in Kinetic theory, the Boltzmann equation and the Landau equation. Between these two models, the grazing limit of the Boltzmann equation to Landau equation is well-known and has been justified by using cutoff near the grazing angle with some suitable scaling. In the first part of the talk, we will present a new approach by applying a natural scaling on the Boltzmann equation and an improved well-posedness theory for the Boltzmann equation without angular cutoff in the regime with an optimal range of parameters to justify the grazing limit. In the second part of the talk, we will focus on the well-posedness of the Landau equation in some critical function spaces that capture its essential structure of scaling invariance. The talk is based on some recent joint works with Yu-Long Zhou on the first topic and Ke Chen and Quoc-Hung Nguyen on the second topic.