Speaker: Prasit Bhattacharya (U. Virginia)
Title: Stable homotopy groups of spheres, finite CW-complexes and periodic self-maps
Abstract: Patterns in the stable homotopy groups of spheres are hard to detect. However chromatic homotopy theory gives a theoretical framework which justifies existence of a robust pattern. In theory, elements of stable homotopy groups are arranged in layers called the chromatic layers (one for each natural number). However, not much is known beyond chromatic layer 1. One way to detect elements in the stable homotopy groups is via finite CW-complexes which admit special self-maps, called v_n-self-maps. This talk will introduce a new class of CW-complexes which has the potential to detect elements in chromatic layer 2 of the stable homotopy group localized at the prime 2.
Speaker: Anna Puskas (Kavli Institute for the Physics and Mathematics of the Universe)
Title: Demazure-Lusztig operators and Metaplectic Whittaker functions
The study of objects from Number Theory such as metaplectic Whittaker
functions has led to surprising applications of Combinatorial
Representation Theory. Classical Whittaker functions can be expressed in
terms of symmetric polynomials, such as Schur polynomials via the
Casselman-Shalika formula. Tokuyama’s theorem is an identity that links
Schur polynomials to highest-weight crystals, a symmetric structure that
has interesting combinatorial parameterizations.
Approaches to generalizing the Casselman-Shalika formula resemble the
two sides of Tokuyama’s identity. Connecting these approaches with
purely combinatorial tools motivates the search for a generalization of
Tokuyama’s theorem. This talk will discuss how the introduction of
certain algebraic tools (Demazure and Demazure-Lusztig operators) yields
such a result. We shall see how these tools can further be used to
investigate questions in the infinite-dimensional setting.
Speaker: Vasu Tewari (U. Penn)
Title: Divided symmetrization and Schubert polynomials
Abstract: Divided symmetrization is an algebraic operation that takes a multivariate polynomial as input and outputs a scalar, which in many cases is a combinatorially interesting quantity. In this talk, I will describe how divided symmetrization arises in different areas of mathematics, ranging from discrete geometry, where it is intimately tied to computing volumes of permutahedra, to algebraic geometry, where it makes an appearance in the cohomology class of a certain variety.
I will then focus on the divided symmetrization of Schubert polynomials. The emphasis throughout is on the combinatorics involved.
The first meeting of the logic seminar will be today at 2:30–3:20 in Keller 314. Our speaker will be Jack Yoon, who will give an introductory lecture on reverse mathematics. An abstract for his talk is below.
I will introduce the basics of reverse mathematics and begin Hunter’s paper on higher order reverse topology, which can be found here: https://www.math.wisc.edu/
Reverse mathematics is a study of foundations of mathematics by assessing the “strength” of the theorems from ordinary mathematics. Rather than starting from given axioms to prove a theorem, it asks a reverse question “which axioms are necessary to prove the theorem?”. Traditionally, reverse mathematics has played out within the second order arithmetic, but further progress has been made on higher order systems as well. For example, Hunter’s paper above branches out to higher order systems to study the theorems of topology.
Speaker: Liu Liu (UT Austin)
Title: A Bi-fidelity method for multiscale kinetic equations with uncertainties
In this talk, we introduce a bi-fidelity numerical method for solving high-dimensional parametric kinetic equations. We first briefly discuss about the Boltzmann equation and its fluid dynamic limit, then introduce a bi-fidelity stochastic collocation method for its uncertainty quantification problem. By combining computational efficiency of the low-fidelity model–chosen as the compressible Euler system–with high accuracy of the high-fidelity (Boltzmann) model, our bi-fidelity approximation can successfully capture well the macroscopic quantities of solution to the Boltzmann equation in the random space. A uniform error estimate of the bi-fidelity method, based on a series of our theoretical work on hypocoercivity for the uncertain kinetic equations, will be shown. Lastly we present numerical results to validate the efficiency and accuracy of our proposed method.
Mathematical Analysis and Numerical Methods for an Underground Oil Recovery model
Ying Wang (University of Oklahoma)
In this talk, I will discuss a new class of entropy solutions of the modified Buckley-Leverett equation, which models underground oil recovery. This model includes a third-order mixed derivatives term resulting from the dynamical effects in the pressure difference between the two phases. Analytic study on the computational domain reduction will be provided. Strong stability preserving operator splitting method will be introduced. A variety of numerical examples will be given. They show that the solutions may have many different profiles depending on the initial conditions, diffusion parameter, and the third order mixed derivatives parameter. The results are consistent with the study of traveling wave solutions and their bifurcation diagrams.
Sui Tang (Johns Hopkins University)
Title: Machine learning on dynamic data
Abstract: High-dimensional dynamical data arise in many fields of modern science and introduce new challenges in statistical learning and data recovery. In this talk, I will present two sets of problems. One is related to the data-driven discovery of dynamics in systems of interacting agents. Such kind of systems is ubiquitous in science, from the modeling of particles in physics to prey-predator in Biology, to opinion dynamics in social sciences. Given only observed trajectories of the system, we are interested in estimating the interaction laws between the agents using tools from statistical/machine learning. We show that at least in particular circumstances, where the interactions are governed by (unknown) functions of distances, the high-dimensionality of the state space of the system does not affect the learning rates. We can achieve an optimal learning rate for the interaction kernel, equal to that of a one-dimensional regression problem. The other one is related to the dynamical sampling: a new area in sampling theory that deals with processing a linear time series of evolving signals and aims at recovering the initial state and the forward operator from its coarsely sampled evolving states. We provide mathematical theories to show how the dynamics can inform feasible space-time sampling locations and the fundamental limit of space-time trade-off.