Events
Department of Mathematics and Statistics
Texas Tech University
The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. In this talk, I will discuss the motivation and methodology behind joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from the predictions of Kolmogorov's classical K41 phenomenological theory.
This Analysis seminar may be attended Monday the 28th at 4:00 PM CST (UT-6) via this Zoom link.
Meeting ID: 972 0263 8160
Passcode: 549916
We will discuss perturbative path integral quantization of scalar field theory with a polynomial interaction potential on two-dimensional compact Riemannian manifolds with boundary. The perturbative partition function is defined in terms of configuration space integrals on the surface. Moreover, partition functions can be organized into a functor, in the sense of Atiyah-Segal axiomatics, from the Riemannian cobordism category to the category of Hilbert spaces. A crucial role in the result is played by non-trivial behavior of tadpoles (short loops) under gluing. This is based on joint work with Pavel Mnev and Konstantin Wernli.In this talk, I will describe some progress towards the construction of the 3D Yang-Mills (YM) measure. In particular, I will introduce a state space of "distributional gauge orbits" which may possibly support the 3D YM measure. Then, I will describe a result which says that assuming that 3D YM theories exhibit short distance behavior similar to the 3D Gaussian free field (which is the expected behavior), then the 3D YM measure may be constructed as a probability measure on the state space. This is based on joint work with Sourav Chatterjee.
Watch online via this Zoom link.
The Ramanujan congruences provide beautiful and
explicit results on the arithmetic of the unrestricted partition
function, while Euler's pentagonal number theorem furnishes
a complete description of the parity of the number of partitions
into distinct parts. Motivated by the latter result, in this talk
we will consider the arithmetic of k-regular partitions, which
are partitions in which no part is divisible by k. We give
particular attention to cases where the theory of modular
forms can be used to give a complete (or nearly so)
characterization of this arithmetic.
Follow the talk this Zoom link
Meeting ID: 990 5902 7169
Passcode: 648002
Recent experimental studies in atomically thin materials such as graphene have offered insights into the collective motion of electrons in 2D. A wealth of intriguing optical phenomena can arise in these systems because of the coupling of the electron motion with incident electromagnetic fields.
In many applications of photonics at the nanoscale, 2D materials such as graphene may behave as conductors, and allow for the excitation and propagation of electromagnetic waves with surprisingly small length scales. These surface waves are tightly confined to the material. They can possibly beat the optical diffraction limit, in the sense that the wavelength of the excited surface waves can be much smaller than that of the incident wave in a frequency range of practical interest. A broad goal in mathematical modeling is to understand how distinct kinetic regimes of 2D electron transport can be probed, and even controlled, by electromagnetic signals.
In this talk, I will discuss recent work in describing the dispersion of electromagnetic modes that may propagate along edges of flat, anisotropic conducting sheets. Some emphasis will be placed on the role of the fractional Laplacian in this context. An emergent concept of topological character pertaining to the existence of such modes will be presented. The starting point is a boundary value problem for Maxwell’s equations coupled with the physics of the moving electrons in 2D structures. This problem will be converted to singular integral equations of the Wiener-Hopf type, and be solved explicitly. Some extensions of the analysis will be discussed, if time permits.
Please attend this week's Applied Math seminar at 4 PM Wednesday via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied
We develop an agent-based simulation of the catastrophe insurance and reinsurance industry
and use it to study the problem of risk model homogeneity.
The model simulates the balance sheets of insurance firms, who collect premiums from clients
in return for ensuring them against intermittent, heavy-tailed risks.
Firms manage their capital and pay dividends to their investors, and use either
reinsurance contracts or cat bonds to hedge their tail risk.
The model generates plausible time series of profits and losses and recovers stylized facts,
such as the insurance cycle and the emergence of asymmetric, long tailed firm size distributions.
We use the model to investigate the problem of risk model homogeneity.
Under Solvency II, insurance companies are required to use only certified risk models.
This has led to a situation in which only a few firms provide risk models, creating a
systemic fragility to the errors in these models.
We demonstrate that using too few models increases the risk of nonpayment and default
while lowering profits for the industry as a whole.
The presence of the reinsurance industry ameliorates the problem but does not remove it.
Our results suggest that it would be valuable for regulators to incentivize model diversity.
The framework we develop here provides a first step toward a simulation model of the insurance
industry for testing policies and strategies for better capital management.