Analysis Seminar, 2019--2020
Fall Semester
- September 16, 23, Dongning Song (University of Wyoming)
Title: An introduction to von Neumann subfactors, I, II.
Abstract: This is a continuation of my talk of last semester. I will briefly review what a von Neumann algebra is and what basic classification and examples are. Then, I will start to talk about the modules over a von Neumann algebra and their classification, the M-dimension, and the index of a subfactor.
- September 30, October 7, 14, Chunguang Li (Northeast Normal University, China)
Title: C* exponential length of commutators unitaries in AH algebras
Abstract: In this talk, I will start with an introduction to the C* exponential length of unitaries in C* algebras. After a brief review on some work about exponential length of unitaries in AH algebras, I will introduce our main result, which says that the supreme of the exponential lengths of the commutators unitaries is at least 2π in an AH algebra with slow dimension growth whose real rank is not zero. This work is joint with Liangqing Li and Ruiz, I. V.
(The first talk is an introduction to exponential length, the second talk is an overview of the work above, and the third talk is on the details.)
- October 18 (joint with Applied Mathematics Seminar), Annie Millet (University of Paris La Sorbonne)
Title: Behavior of Solutions in Stochastic Critical and Supercritical Focusing nonlinear Schrödinger Equation
Abstract:
We study nonlinear Schrödinger (NLS) equation with focusing nonlinearity, subject to additive or multiplicative stochastic perturbations driven by an infinite dimensional Brownian motion. Under the appropriate assumptions on the space covariance of the driving noise, previously A. de Bouard and A. Debussche established the H1 local well-posedness for energy sub-critical nonlinearity, and global well-posedness in the mass-subcritical case. In our work we study the L2-critical, intercritical and energy (H1)-critical cases of stochastic NLS, and obtain quantitative estimates on the blow-up time when the mass, energy and L2-norm of the gradient of the initial condition are controlled by similar quantities of the ground state. This completes blow-up results proved by A. de Bouard and A. Debussche for energy sub-critical nonlinearities.
- October 21, 28, Ping Zhong (University of Wyoming)
Title: Subordination functions in free probability (I), (II)
Abstract: The subordination relation was first proved by Voiculescu in one of his free entropy papers under some generic conditions. The result was then extended by Biane and Voiculescu to a very general setting. It turns out to be very useful in the study of the noncommutative distributions in free probability that include free convolution of two free random variables, the distribution of self-adjoint polynomials of free random variables, and also the Brown measure of non-normal operators. The subordination relation was also upgraded to study random matrix models.
I will discuss some basic techniques of subordination ideas. I will follow the approach developed by Belinschi and Bercovici, which involves some complex analytic techniques, in particular the Denjoy-Wolff theorem.
No knowledge of free probability is required. All new terms will be explained. All are welcome.
- November 8 (Fisk Lecture), Hari Bercovici (Indiana University)
Title: Outlying Eigenvalues of Random Matrices
Abstract:
Given two independent large random matrices A and B, the bulk of the eigenvalues of A+B or, more generally, of an arbitrary polynomial p(X,Y) can be predicted with great accuracy if, for instance, A has an invariant distribution relative to unitary equivalence. We will discuss work (joint with Belinschi, Fevrier, and Capitaine) concerning the eigenvalues outside the bulk, also called outliers. These can also be predicted under certain circumstances, and their positions are related with those of the outliers of A and B through the intermediary of certain analytic functions that appear in noncommutative probability theory. Most of the talk will be about sums. If time permits, we will discuss how the problem for arbitrary polynomials can be solved using sums of related random matrices.
- November 11, Jiun-Chau Wang (University of Saskatchewan)
Title: George Boole and his map
Abstract:
The name George Boole is often associated with mathematical logic, mainly because the man himself made a fundamental contribution to the field. Not many people know that in 1857 Boole actually discovered that the map Tx=x-1/x (called Boole's map) preserves the Lebesgue measure on the real line, providing a simple, explicit example of infinite measure-preserving dynamical systems. In this talk we will discuss the ergodic theory of Boole's map, based on a noncommutative probabilistic approach.
- December 9, Thomas Dean (University of Wyoming)
Title: Spectra of perturbations of self-adjoint operators
Abstract:
The Schrödinger operator with nonnegative potentials can be viewed as perturbations of (unbounded) self-adjoint operators. Rellich, Trotter, and Weyl-von Neumann theorems give structural results on such perturbations, but they are not specific on what the spectrum of the perturbed operator can be. In this seminar, we look at rank-one perturbations and the Borel transform of a positive measure in order to develop an understanding of the spectral theory of self-adjoint operators. This leads to a major results -- the Aronszajn-Donoghue theorem, which gives insight to the eigenvalues of rank-one perturbations of self-adjoint operators.
Spring Semester
- February 3, Dongning Song (University of Wyoming)
Title: An introduction to Groupoid
Abstract:
In this talk, I will introduce the notion of "Groupoid", which is the generalization of "Group". Also I will cite some basic examples of Groupoid, and then relate it to C*-algebras.
- February 17, Chunguang Li (Northeast Normal University, China)
Title: Complex symmetric operators and related topics
Abstract:
Complex symmetric operators are generalization of complex symmetric matrices on the finite dimensional Hilbert space. I will introduce the definition and properties of such operators. Further, I will give some examples. If possible, I will characterize the complex symmetry of partial isometries and weighted shifts.
- February 24, March 9, Hakima Bessaih (University of Wyoming)
Title: Invariant measures for stochastic differential equations. I, II.
Abstract:
Stochastic differential equations (SDESs) driven by a Brownian motion (BM) will be introduced. The definition of the stochastic integral will be given. The asymptotic behavior of the solution will be studied using the notion of invariant measures. These techniques will be used for the study of stochastic partial differential equations as well.
- April 6, Hongjun Guo (University of Wyoming)
Title: Transition fronts in unbounded domains
Abstract:
In this talk, I will present some results of transition fronts of bistable reaction-diffusion equations in unbounded domains. The notion of transition fronts generalizes the standard notion of traveling fronts. Under this general framework, we will show some properties of the propagation speed and the large time behavior for the transition fronts.
- April 13, 20, Thomas Dean (University of Wyoming)
Title: An Introduction to Scattering Theory. I, II
Abstract:
Scattering theory is at the heart of studying the scattering of waves and particles, while also being a mathematically rich topic. We will define the Wave Operator and discuss the motivation of why we study Wave Operators along with examples. Scattering theory is closely related to perturbation theory, and we will look at one perturbation theorem, called the Kato-Rosenblum theorem, which states that the Wave Operators exist if the potential, V, in the Schrödinger Operator is of trace class. The scattering operator and scattering matrix will be introduced, but further discussion will take place in the Analysis Seminar on 4/20.
- April 27, Ping Zhong (University of Wyoming)
Title: Brown measures of random matrices and free Brownian motions
Abstract:
Take your favorite probability distribution that has means zero and variance one (standard Gaussian distribution for example), and consider an n x n random matrix X_n with i.i.d. entries with no symmetry and each entry has the same distribution as your favorite distribution. It is known by the Circular Law that the eigenvalue distribution of \displaystyle {\frac {1}{{\sqrt {n}}}}X_{n} converges in distribution to the uniform measure on the unit disk. Let A_n be a sequence of Hermitian, deterministic, matrices which converges to a probability distribution on the real line. What is the limit distribution of A_n +\displaystyle {\frac {1}{{\sqrt {n}}}}X_{n} ? You will find the answer in this talk.
The main tools are from free probability, a probability theory studying noncommutative random variables with highest degree that was introduced by Dan Voiculescu about forty years ago. Free probability is a suitable framework to study random matrices of large size. The Brown measure of operators was initially introduced in the context of von Neumann algebras. It is an important tool to study the distributions of free random variables, which are regarded as limits of empirical spectral distributions (ESD) of random matrices that are not necessarily Hermitian. I will introduce the topics of Brown measure and its connection with certain random matrix models. More specifically, I will discuss random matrices from Brownian motions on matrix Lie groups (such as general linear group) and, their limit objects, free circular and multiplicative Brownian motions.
This is an introductory talk on this topic prepared to be accessible for any graduate students. It will have many pictures on simulations of random matrices.
Based on joint work with Ching Wei Ho.
- May 4, Zhuang Niu (University of Wyoming)
Title: The stable rank of transformation group C*-algebras
Abstract:
The (topological) stable rank of a C*-algebra is introduced by M. Rieffel as the topological version of the Bass stable rank of a ring, and its value can be any natural number, including infinity. In this talk, let us consider an arbitrary free and minimal action of the group Z^d on a compact Hausdorff spaces X. It turns out that the stable rank of the transformation group C*-algebra is always one. The talk is based on a joint work with Chunguang Li.
Organizer: Dr. Ping Zhong