Lefschetz Center for Dynamical Systems Seminar
Abstract: The transfer operator is important in many questions of dynamical systems and statistical mechanics. It is particularly helpful for studying the mixing and statistical properties of measures, investigations of zeta functions and Fredholm determinants, piecewise monotong transformations, etc. Estimates for the essential spectral radius of the transfer operator were obtained by V. Baladi, P. Collet, S. Isola, D. Ruelle, and many others. In this talk we give an exact formula for the essential spectral radius of the matrix coefficient transfer operator.
We study Ruell's transfer operator $\mathcal{R}$ induced by a $C^{\mathbf{r}+1}$--smooth expanding map $\varphi$ of a smooth manifold and a $C^{\mathbf{r}}$-- smooth bundle automorphism $\Phi$ of a real vector bundle. We prove the following formula for the essential spectral radius of $\mathcal{R}$ on the space $\mathbf{r}$-times continuously differentiable sections of the bundle with $\alpha$-H"{o}lder $\mathbf{r}$-th derivative: \[ \text{ress} (\mathcal{R})=\exp\left(\sup_{\nu\in\text{Erg }} \{h_\nu+\lambda_\nu-(\mathbf{r}+\alpha)\chi_\nu\} \right) , \] where $\text{Erg }$ is the set of $\varphi$-ergodic measures, $h_\nu$ the entrophy of $\varphi$ with respect to $\nu$, $\lambda_\nu$ the largest Lyapunov exponent of the cocycle induced by $\Phi$, and $\chi_\nu$ the smallest Lyapunov exponent for the differential $D\varphi$.
Brown Analysis Seminar
Applied Mathematics Colloquium
Thanksgiving Lecture
Abstract: The Paradox: Though G\{"}odel's Incompleteness Theorem has been touted in some quarters as the most significant mathematical achievement of the 20th Century, it seems to be of little significance to the bulk of research mathematicians. Why is this the case?
Scientific Computing Seminar
Abstract: The methods we discuss use a Hermitian/skew-Hermitian splitting (HSS) iteration and its inexact variant, the inexact Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs inner iteration processes at each step of the outer HSS iteration. Theoretical analyses show that the HSS method converges unconditionally to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part. Numerical examples are presented to illustrate the effectiveness of both HSS and IHSS iterations. In addition, a model problem of three-dimensional convection-diffusion equation is used to illustrate the advantages of our methods.
PDE Seminar
Abstract: The wave map equation between two Riemannian manifolds, has extensively been studied by a number of mathematicians in the last decade. It is the natural Minkowskian analogue of the harmonic map equation and its solutions are the critical points of the associated Lagrangian.
In the study of well-posedness of the Cauchy problem with initial data in the Sobolev spaces seeks answers to the following questions. For what values of $s$ does the initial value problem admit a unique local solution? Does it depend continously on the initial data? For what values of $s$ does this solution extend for all time? Does the solution corresponding to smooth initial data stay smooth for all times ({\it global regularity})?
T. Tao established the global regularity for wave maps from ${\Bbb R} \times {\Bbb R}^n$ into the sphere ${\Bbb S}^m$ when $n \ge 5$; and later extended these same result to all dimensions $n \ge 2$.
In this talk we will describe recent joint work with A. Stefanov and K. Uhlenbeck where we study the Cauchy problem for wave maps from ${\Bbb R} \times {\Bbb R}^n$ into a (compact) Lie group or a Riemannian symmetric spaces $M$ and were able to establish global existence and uniqueness provided the Cauchy initial data are small in the critical Sobolev norm ${\dot H}^{n/2} \times {\dot H}^{n/2-1}$ for all $n \ge 4$. Our method combines both delicate techniques from harmonic analysis with fairly standard global gauge theoretic geometric methods, developed among others by K. Uhlenbeck in the late 70's and early 80's. Similar results were obtained by Shatah and Struwe at roughly the same time when the target is any complete Riemannian manifold with bounded curvature.
If time permits we will discuss a new result that extends our theorems to (spatial) dimension $n=3$.
Department of Mathematics Colloquium
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