Viscous flows in thin domains are encountered in a great number of practical situations. Think, for example, of flows in gas or pipelines, in fluid film bearings, in capillaries, or in oceans and in atmosphere. It is natural to expect that in these situations the governing equations of motion are approximated by simpler, lower-dimensional models leading to solutions which are precise only asymptotically as the width of the domain tends to zero. In his seminal paper in 1886, Osborne Reynolds proposed a model for the main pressure distribution in lubrication. This model, the classical Reynolds equation, even though derived under the assumption that the inertial and curvature effects are negligible, makes a reasonable (first-order) approximation for many, but not all, problems in hydrodynamic lubrication. In our work, we derive an equation, referred to as modified nonlinear Reynolds equation, for higher-order approximation of pressure (and consequently of velocity field) in hydrodynamic lubrication. This equation is a nonlinear second-order differential equation for a scalar unknown, which approximates the first two terms in the asymptotic expansion of the pressure field, and, in contrast to the classical Reynolds' equation, takes into account both the inertial and the curvature effects. The equation is justified with optimal integral and pointwise error estimates. Up to now, this kind of optimality has only been achieved for small fluxes and in simpler geometrical situations. Here, the flux, equivalently the Reynolds number, can be as large as possible within the limits of dimension reduction and the optimal estimates result from an accurate analysis of the auxiliary divergence equation.

Bridge number for knots in the 3-sphere is a classical knot invariant introduced by Schubert more than 50 years ago. More recently it has been generalized to knots in any manifold by using a Heegaard surface for the manifold in the place of a bridge sphere. We will discuss some results about bridge surfaces that are analogous to results about Heegaard splittings as well as some of the behaviors of bridge surfaces that seem to have no Heegaard splitting analogs.

We will discuss again the Signorini problem and its optimal regularity in a more rigorous manner than we did last week. The optimal regularity follows from two important lemmas that we will discuss in detail: first that solutions to the thin obstacle problem are semi-convex in the tangential directions to the free boundary and then that the normal derivative of the solution satisfies a monotonicity formula.

is the (1D) continued fraction. The 's can be connected to geometry, via approximating a line L of slope by a sequence of bases of , with 'update matrices' : that is, so that ;. One need only use 2 such 's, to wit and . Then each is just the number of one of these that are needed before the other occurs. CF's are useful for many problems in number theory, in particular, using the ideas of 'best approximants' and 'poorly approximable' numbers lead
1) to special types of (real) numbers useful in mathematics.
2) As another example, one can decide whether 2 'Pisot' 2 by 2 matrices are conjugate over the integers; and
3) Can show that given A as in (2) there is a B in GL(z,2) such that ; has all entries of the same sign. Though mostly other authors have focused on problems of type (1) in their generalizations to ``higher dimensional continued fractions'', we propose here a theory of HDCF useful in problems of the second two types. In particular, this solves these problems in dimension 3. The fundamental construction is a cell complex homotopy equivalent to the 2 skeleton of the 3 dimensional torus, tying arithmetic to geometry.

At the center of modular forms is the theory of Hecke operators. Hecke operators are a commutative algebra of operators acting on the space of modular forms of weight . This space has a canonical basis of simultaneous eigenvectors of all the and these particular modular forms have very special Fourier coefficients, L functions, and connections to elliptic curves. We will begin with the definition of the Hecke operators and explain some key results. If time permits, we will explain how this connects to the Taniyama-Shimura-Weil Theorem proved by Wiles for Fermat’s Last Theorem, and show how to get an elliptic curve from a cusp form. There will be a recap of what we discussed last week as well. The notes from last week's talk can be found at the following link: http://www.math.utexas.edu/users/khopkins/JNTTalkModFormsFirstHalf.pdf

We will discuss the regularity theory of the Monge-Ampere equation det D^2u=f(x) where f(x) is in VMO^\psi spaces defined in terms of sections.

After reviewing last week's talk, I will finish what I started.

We consider a few modeling and computation problems that is related to incoherent imaging in highly heterogeneous (random) media. We show that such imaging problems can be treated as inverse problems related to the linear transport equations. Numerical solutions of those inverse problems will then be discussed. Reconstructions from both synthetic and experimental data will be presented.

Among the simplest of the classical polynomials are the Chebyshev polynomials of the first and second kind, and . In our normalization, the indices are allowed to be half- integers as well as integers, and the ``polynomials'' actually live in . We will show that the rational functions and are very remarkable from the point of view of Grothendieck's dessins d'enfants. The fibers of these rational functions are likewise very remarkable from the point of view of algebraic number theory. For example, for the fiber of the second function above is given by a degree 15875 polynomial in with discriminant and Galois group the entire symmetric group .

Articles: Aspinwall, "D-branes on Calabi-Yau manifolds", Progress in String Theory, TASI 2003 Lecture. Notes (2005), World Scientific. Bridgeland, "Stability conditions on triangulated categories", Ann. of Math. 2007, Vol 166, No.2. Kontsevich, "Donaldson-Thomas invariants", 2007 MPIM Mathematische Arbeitstagung, http://www.mpim-bonn.mpg.de/preprints/send?bid=3352.

In this talk will define strongly -regular rings and show how that every strongly -regular element can be written as the sum of an idempotent, a unit and a nilpotent element, all of which commute with one another. That every strongly -regular element is strongly clean is a consequence of this result.

Building convergent numerical schemes for fully nonlinear partial differential equations is a challenge. In the case of first order equations, schemes have been built based on ideas from Conservation Laws. Recently several authors have tried adapting techniques from fluid mechanics or finite elements to solve second order equations. These attempts have had limited success. We will present recent work on bulding convergent difference schemes for degenerate and fully nonlinear elliptic Partial Differential equations. These equations include: the obstacle problem, the infinity Laplacian, the equation for motion of level sets by mean curvature, the Monge-Ampere equation, the Pucci maximal and minimial equations, an equation for the convex envelope. Earlier work emphasized building convergent schemes over fast or accurate solvers. Since the resulting algebraic equations are nonlinear and non-differentiable, attempts to build implicit solvers have met with little success. As a result, the solution methods have been iterative, and consequently suffer from CFL type time step limitations. Recent work has explored technique to improve accuracy as solution speed. Remarkably, many of the fully nonlinear schemes can be reformulated as Linear Programs (LPs). We will discuss hybrid schemes for better accuracy, and implicit solvers for fully nonlinear equations.