# Calendar

## Tomorrow

### CCAM Seminar, Prof. Zhilin Li, N. Carolina State University, REC 113

Monday, Aug 19 4:30 pm - 5:00 pm

**Title: Some Progress on fluid flow/porous media coupling and Anisotropic Interface Problems****Abstract:** In this talk, I would like to outline new strategies for solving two challenging problems. The first one is about the coupling between a fluid flow modelled by Stokes or Navier-Stokes equations and porous media modelled by Darcy’s law. The problem is challenging because of different governing equations and complicated interface conditions such as BJ and BJS relations. Our strategy is to transform the coupling system to several Poisson equations with single and double layer source distributions. The source strengths are part of unknowns that defined only along the interface and should be chosen such that the original BJS and other interface conditions are satisfied. We have shown that the original and transformed systems are equivalent. The proposed new method has also been utilized to simulate different flow/porous media setting with complicated interfaces which leads to some interesting simulations results such as effect of corners, orientation effect etc.

The second topic is about anisotropic elliptic interface problems. There is limited literature on numerical methods based on structured meshes. Finite element methods are often preferred but the average error estimates cannot guarantee accuracy of the solution near or at the interface. Using a finite difference discretization, the coefficient matrix of the resulting linear system of equations is neither an SPD nor an M-matrix. In this work, we combine a finite element discretization at regular grid points whose coefficient part matrix is an SPD, while at irregular grid points, a finite difference discretization based on the maximum principle preserving method whose coefficient part matrix is an M-matrix. A multigrid method based a nine-point stencil is employed to solve the linear system of equations. A scaling strategy along the interface is proposed along with the discretization. Error analysis and numerical examples will also be presented.

## Next Week

### Operator Algebras Seminar, Ben Hayes, University of Virginia, MATH 731

Tuesday, Aug 20 1:30 pm - 2:20 pm

**Title: CAT(0)-spaces associated to II_{1}-factors.****Abstract:** I will discuss recent joint work with Lewis Bowen and Frank Lin. In it, we construct a natural metric satisfying the CAT(0) condition (a certain natural negative curvature condition) on a space of operators affiliated to a tracial von Neumann algebra. We also investigate how the geometric properties of this CAT(0) space reflect algebraic/analytic properties of the underlying von Neumann algebra.

### Probability Seminar, Marta Lewicka, University of PittsburgUNIV 103

Wednesday, Aug 21 1:30 pm - 2:20 pm

**Title: The Robin mean value equation: A random walk approach to the third boundary value problem.****Abstract:** We study the family of integral equations, called the Robin mean value equations (RMV), that are local averaged approximations to the Robin-Laplace boundary value problem. We interpret (RMV) as the dynamic programming principle along a suitable discrete stochastic process, interpolating between the reflected and the stopped-at-exit Brownian walks. We prove existence, uniqueness, equiboundedness and the comparison principle for their solutions.

For bounded Borel right hand sides, we show that these solutions converge uniformly, in the limit of the vanishing radius of averaging, to the unique $W^{2,p}$ solution of the continuum problem.

Along the way, we also show the asymptotic Holder equicontinuity of solutions to (RMV): Lipschitz in the interior and $\mathcal{C}^{0,\alpha}$ up to the boundary for any $\alpha\in (0,1)$.

For the proofs, we use martingale techniques involving various couplings of random walks and yielding estimates on the probabilistic representations. As an application, we further show the optimal lower bound on solutions to (RMV).

**Title: How to have a Plan in Graduate School****Abstract: **You're a young graduate student, mostly focused on getting through your coursework and qualifying exams, but what happens after that? Come join a group of experienced graduate students discuss their graduate school experience and career plans. We encourage you to attend to listen to our discussion, ask any questions that you may have, and build your local community!

Light refreshments will be provided.

## Two Weeks

**Title: Scaling limits of symmetric exclusion****Abstract:** In this talk, we review the concept of hydrodynamic limit of an interacting particle system and some results on the symmetric exclusion process. From symmetric exclusion it is possible to obtain heat equations, with or without boundary conditions, reaction-diffusion equations (when spin flip dynamics is superposed to exclusion) and motion by mean curvature (when the spin flips occur at a much higher rate than the exclusion jumps). If time allows, we will state the central limit theorem for the density of particles and use it to find the scaling limit of a random walk driven by the exclusion process.

**Title: Concentration phenomenon for compressible Navier-Stokes equations****Abstract: **We will discuss the concentration phenomenon of the kinetic energy, $\rho|u|^2$, associated to isentropic compressible Navier-Stokes equations, in $\mathbb R^n$ with $n=2,3$ and the adiabatic constant $\gamma \in [1,\frac {n}{2}]$. Except in a space-time set with Hausdorff dimension less than or equal to $\Gamma(n) + 1$, no concentration phenomenon occurs. Some recent developments will be also discussed.

## Three Weeks

### Math Colloquium, Maria Gordina, University of Connecticut, MATH 175

Tuesday, Sep 3 3:30 pm - 4:20 pm

**Title: Stochastic analysis and geometric functional inequalities.****Abstract:** We will survey different methods of proving functional inequalities for hypoelliptic diffusions and the corresponding heat kernels. Some of these methods rely on geometric methods such as curvature-dimension inequalities (due to Baudoin-Garofalo), and some are probabilistic such as coupling, and finally some use structure theory and a Fourier transform on Lie groups. For nilpotent groups, the latter can be interpreted as an eigenfunction expansion of the hypoelliptic heat kernel. This is based on joint work with M. Asaad, F. Baudoin, B. Driver, T. Melcher, Ph. Mariano.

## September

### CCAM Seminar, Dr. Juntao Huang, Michigan State University of Alabama, REC 113

Monday, Sep 9 4:30 pm - 5:00 pm

TBA

### Probability Seminar, Wai-Tong Fan, University of Connecticut, UNIV 103

Wednesday, Sep 11 1:30 pm - 2:20 pm

**Title: Joint distribution of Busemann functions in the corner growth model.**

**Abstract: **The 1+1 dimensional corner growth model with exponential weights is a centrally important exactly solvable model in the Kardar-Parisi-Zhang class of statistical mechanical models. While significant progress has been made on the fluctuations of the growing random shape, understanding of the optimal paths, or geodesics, is less developed. The Busemann function is a useful analytical tool for studying geodesics. We present the joint distribution of the Busemann functions, simultaneously in all directions of growth, in terms of mappings that represent FIFO (first-in-first-out) queues.

As applications of this description we derive a marked point process representation for the Busemann function across a single lattice edge and point out its implication on structure of semi-infinite geodesics. This is joint work with Timo Seppäläinen.

### Probability Seminar, Zoran Vondracek, University of Zagreb, Croatia, UNIV 103

Wednesday, Sep 25 1:30 pm - 2:20 pm

### CCAM Colloquium, Prof. Roland Glowinski, University of Houston, REC 113

Monday, Sep 30 4:30 pm - 5:30 pm

TBA

## October

### Probability Seminar, Mihai Gradinaru, Universite de Rennes 1, UNIV 103

Wednesday, Oct 2 1:30 pm - 2:20 pm

### CCAM Seminar, Prof. A.J. Meir, Southern Methodist University, REC 113

Monday, Oct 14 4:30 pm - 5:30 pm

### Probability Seminar, Michael Perlmutter, Michigan State University, UNIV 103

Wednesday, Oct 16 1:30 pm - 2:20 pm

### CCAM Seminar, Prof. Constantin Bacuta, University of Delaware, REC 113 REC 113

Monday, Oct 28 4:30 pm - 5:30 pm

TBA

## November

### Probability Seminar, Steve Lalley, University of Chicago, UNIV 103

Wednesday, Nov 6 1:30 pm - 2:20 pm