Events
Department of Mathematics and Statistics
Texas Tech University
In the past twenty years, the method of convex integration has attracted considerable attention as a means to construct weak solutions with "wild" energy profiles to prominent equations in fluid dynamics. This led to the resolution of the Onsager conjecture (2018) and a proof of non-uniqueness for weak solutions of the 3D Navier-Stokes equations (2018).
I will give an introduction to the theory of convex integration, tracing the analytic and geometric developments back to its roots, namely the celebrated 1954 work of John F. Nash Jr. on $C^1$ embeddings. I will provide an overview of the embedding problem and Nash's solution as well as some comments on the modern convex integration techniques in the context of fluid mechanics.
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here.
Bordism is an equivalence relation on manifolds and has been a powerful tool in algebraic topology for the last 60 years. A genus is a $\mathbb{Q}$-valued function on equivalence classes of manifolds (actually it's a ring homomorphism); genera have played an important role in 4-manifold topology, index theory, and homotopy theory. In this talk, I will recall the cobordism ring and discuss generators in terms of explicit manifolds. Next, I will introduce some examples of genera and then recall a general construction due to Hirzebruch. Finally, I will present an interpretation of genera through the lens of quantum field theory and share some recent computations.This is joint work with Eyal Neuman.
Polymer models give rise to some of the most challenging problems in
probability and statistical physics. We typically model a polymer using a
random walk, where the time parameter n of the walk represents distance
along the polymer starting from one end. That is, we imagine that the
polymer is built up by adding molecules one by one at random angles. We
usually include a self-avoidance term, reflecting the idea that different parts
of the polymer cannot be in the same place at the same time. A difficult
problem, unsolved in the most important physical cases, is to predict the
end-to-end distance or radius of the polymer.
In this talk, I will discuss two extensions of the random polymer model.
1) Moving polymers can be modeled by stochastic partial differential equations. If the polymer takes values in one-dimensional Euclidean space,
we give fairly sharp upper and lower bounds for its radius. We find
that there is more stretching than in typical one-dimensional polymer
models that do not have time dependence.
2) Random surfaces can be modeled by elastic manifolds, also called discrete Gaussian free fields. These models originate in quantum field
theory. If the dimensions of the parameter space and the range are
the same, we can derive bounds on the radius of the polymer. These
bounds are fairly sharp in two dimensions.
We will explain the models mentioned above and give an outline of our
proof techniques.
Watch online via this Zoom link.
In the 1980's, Greene introduced $_nF_{n-1}$ $\text{hypergeometric
functions}$ over finite fields using normalized $\text{Jacobi
sums}$. The structure of his theory provides that these functions
possess many properties that are analogous to those of the classical
hypergeometric series studied by Gauss, Kummer and others. These
functions have played important roles in the study of Apery-style
supercongruences, the Eichler-Selberg trace formula, Galois
representations, and zeta-functions of arithmetic varieties. In this
talk we discuss the value distributions of simplest families of these
functions. For the $_2F_1$ functions, the limiting distribution is
semicircular, whereas the distribution for the $_3F_2$ functions is
$\text{Batman}$ distribution. This is a joint work with Ken Ono and Hasan
Saad.
Stochastic orders formalize preferences among random outcomes and are widely used in statistics and economics.
We analyze stochastic optimization problems involving stochastic-order relations as constraints,
which compare performance functionals, depending on our decisions, to benchmark random outcomes.
We discuss the relation of univariate and multivariate stochastic orderings to utility functions,
conditional value at risk, and to coherent measures of risk.
Necessary and sufficient conditions of optimality and duality theory for problems with stochastic order
constraints involve expected utility theory, dual (rank-dependent) utility theory, and coherent measures of risk.
The model provides a link between various approaches for risk-averse optimization.
Some attention will be paid to the numerical solution of the problems and their applications.