Department of Mathematics and Statistics
Texas Tech University
The
Pure Mathematics Colloquium: Current Advances in Mathematics is dedicated to different topics mostly in pure mathematics, and not limited to any specific areas. Therefore, it will be of interest of all faculty whose research may include algebra, number theory, topology, logic, geometry, and analysis. The goal of the Colloquium is to further promote research in pure mathematics in our department and to develop and maintain communications with outside experts about current advances in mathematics. We will invite mathematicians from our department and from around the world to deliver online lectures on recent progress of significance.
Website.
Commutative Noetherian local rings are the abstract form of the ring of germs of regular functions at a point on an algebraic variety. Accordingly, textbooks organize local rings in I hierarchy according to the character of their singularity. However, this “classification” misses “most” rings: A generic local ring does not resemble a local ring of a point on any reasonable variety. In the talk I will describe recent progress in the understanding of generic local rings of codimension 3, in a way the smallest open case.
In this talk, we discuss ideas, techniques, and results on regularity estimates in Sobolev spaces for solutions of elliptic and parabolic equations. We demonstrate the core ideas in the study by considering the Laplace equations. We then move on to study the class of equations in which the coefficients can be singular or degenerate. Several recent results in my joint work with Hongjie Dong (Brown University) will be discussed. Watch Monday the 28th at 4 PM via this zoom link.We discuss several problems on the total loss of heat by $n$
spherical particles in $\mathbf{R}^3$ as well as questions on the
losses of heat by individual particles. Main question here is
whether or not particles lose less heat if they move closer to
each other in a certain sense. These questions were inspired by
observations on the behavioral habits of sleeping armadillos and
some other warm-blooded animals.The celebrated Riemann hypothesis can be reformulated as a simply-stated criterion concerning least-squares approximation. While carrying out computations related to this criterion, we have observed a curious phenomenon: for no apparent reason, at least the first billion entries of a certain infinite triangular matrix associated to the Riemann zeta function are all positive. In this talk, I shall explain the background leading to this observation, and make a conjecture. (Joint work with Hugues Bellemare et Yves Langlois.)I will consider an oscillator with one degree of freedom and systems which are close to this simple object. For these examples, recent results on long-time influence of small deterministic and stochastic perturbations will be demonstrated. Such problems as stochasticity of long-time behavior of pure deterministic systems, metastability, and stochastic resonance will be considered. These results are based on the modified averaging principle, limit theorems for large deviations, diffusion approximation. Then I will describe a general approach to this type of problems and shortly consider (if I have time) various PDE problems like linear and nonlinear PDE's with a small parameter in higher derivatives, homogenization, pattern formation in reaction-diffusion equations.In 1984 Michael Berry discovered that an isolated eigenstate
of an adiabatically changing periodic Hamiltonian acquires a phase,
called the Berry phase. Barry Simon gave an interpretation of this
phase in terms of the holonomy of a certain Hermitian line bundle.
There are several situation described in physical literature when the
Berry phase is claimed to be equal to the phase of the determinant of
the corresponding imaginary-time Schroedinger operator. However not
only rigorous proofs but even the accurate formulations of these
results are missing in the literature. In this talk we establish and
prove the precise relationship between the phase of this determinant
and the Berry phase under the most general assumption on the
Hamiltonian. The previously known examples are the special cases of
this formula.