Meeting 1: Special colloquium, Monday, 1/10/11, KAP 414, 3:30-4:30 PM, John Baldwin, Princeton University

Title: "Contact structures, open books, and Khovanov homology"

Abstract: I will describe an important class of geometric objects on 3-dimensional manifolds called contact structures, and will survey Giroux's celebrated correspondence between contact structures and more topological objects called open books. I will summarize progress on some open questions related to this connection, and will explain how link invariants like Khovanov homology may be used to provide further insights.

Meeting 2: Special colloquium, Wednesday, 1/12/11, KAP 414, 3:30-4:30 PM, Moon Duchin, University of Michigan

Title: "Finer coarse invariants"

Abstract: In geometric group theory and geometric topology, it is often useful to study large-scale or coarse properties of groups and metrics, especially via curvature invariants like Gromov hyperbolicity. In this vein, I will discuss some recent work on filling functions in the settings of Teichmuller theory and finitely generated groups.

Meeting 3: Colloquium, Wednesday, 1/19/11, KAP 414, 3:30-4:30 PM, Blake Mellor, Loyola Marymount University

Title: Topological symmetry of complete graphs

Abstract: Given an embedding $\Gamma$ of a graph in $S^3$, the topological symmetry group TSG$_+(\Gamma)$ is the subgroup of the automorphism group of the abstract graph which is induced by orientation-preserving homeomorphisms of the pair $(S^3, \Gamma)$.

Complete graphs are an interesting class of graphs to consider, since they have the largest possible automorphism groups. Flapan, Naimi and Tamvakis have shown that the only possible topological symmetry groups for complete graphs are finite subgroups of $SO(3)$ or $D_m \times D_m$ (with $m$ odd). However, their result does not specify which complete graphs admit embeddings realizing particular topological symmetry groups.

In this talk, we answer this question for the symmetry groups of the regular polyhedra, $A_4, S_4$ and $A_5$. For each of these groups, we will determine exactly which complete graphs have an embedding which realizes that group as its topological symmetry group. As part of our investigation, we develop an Edge Embedding Lemma which can be applied to many other groups and graphs as well. This is joint work with Erica Flapan and Ramin Naimi.

Meeting 4: Special Colloquium, Monday, 1/24/11, KAP 414, 2:00-3:00 PM, Sami Assaf, MIT

Title: Applications of dual equivalence

Abstract: A dual equivalence for an arbitrary collection of combinatorial objects endowed with a descent set is a relation for which equivalence classes group together terms according to the Schur expansion of the corresponding generating function. After outlining the definition of dual equivalence, we'll present three main applications: the Schur expansion of Macdonald polynomials, Schur positivity of k-Schur functions (joint with S. Billey), and a combinatorial rule for the Littlewood-Richardson coefficients of the Grassmannian in the special case of a Schubert polynomial times a Schur function (joint with N. Bergeron and F. Sottile).

Meeting 5: CAMS Colloquium, Monday, 1/24/11, KAP 414, 3:30-4:30 PM, Peter Smereka, University of Michigan

Title: Modeling and Computation of Strained Heteroepitaxial Growth using Kinetic Monte Carlo

Abstract: Heteroepitaxial growth is a process where crystals are grown one layer at a time using a molecular beam in a vacuum. When strain is present these systems can form three dimensional islands often called quantum dots. This is a nanoscale process and continuum models struggle to capture many of the phenomena that occur. Kinetic Monte Carlo (KMC) is an alternative approach which is quite promising but has had limited use in strained systems due to a variety computational bottle necks. In this talk, I will outline KMC models for strained epitaxial growth and how one can go about preforming simulations in an efficient manner.

Meeting 6: Colloquium, Wednesday, 1/26/11, KAP 414, 3:30-4:30 PM, Amjad Tuffaha, Petroleum Institute, Abu Dhabi

Title: Free-Boundary Problems in Fluid-Structure Interaction

Abstract: In this talk, we discuss well-posedness results for a coupled system of partial differential equations arising in Fluid-Structure interactions. The system consists of an incompressible Navier-Stokes equation and a hyperbolic Lame' equation with velocity and stress matching boundary conditions prescribed at the free moving interface in between the two dynamic domains where each of the two equations is defined. We give an overview of results known for the system and our most recent regularity result.

Meeting 7: CAMS colloquium, Monday, 1/31/11, KAP 414, 3:30-4:30 PM, Roger Temam, Indiana University

Title: Theoretical and numerical problems related to the equations of the atmosphere and the oceans in a limited domain

Abstract: In this lecture we will discuss the issue of boundary conditions for equations in a limited domains. This problem will be addressed in the context of the primitive equations of the atmosphere and the oceans, and of the shallow water equations, but it is relevant to many areas of physics and mechanics. On the theoretical side the aim is to find boundary conditions which produce a well-posed problem. On the computational side, the aim is to find boundary conditions which are transparent, that is, which let the waves move freely in and out of the domain.

Meeting 8: Colloquium, Wednesday, 2/02/11, KAP 414, 3:30-4:30 PM, Philip Wood, Stanford University

Title: Singularity of Discrete Random Matrices

Abstract: Consider an n by n square matrix where n is large. For each entry, flip a fair coin, making the entry +1 if the coin comes up heads, and -1 if the coin comes up tails. What is the probability that the matrix has determinant equal to zero? This talk will discuss work that builds on a breakthrough by Terence Tao and Van Vu in 2007 who used ideas in combinatorics, probability theory, and additive combinatorics to prove upper bounds on the probability of singularity. New results allow us to work with more general random matrices and prove better upper bounds. Joint work with Jean Bourgain and Van Vu.

Meeting 9: CAMS colloquium, Monday, 2/07/11, KAP 414, 3:30-4:30 PM, Anna Mazzucato, Penn State University

Title: Vanishing viscosity limit and boundary layers in incompressible fluids

Abstract: I will present recent results concerning the vanishing viscosity limit for incompressible, Newtonian fluids: that is, whether solutions of the Navier-Stokes equations converge in an appropriate sense to solutions of the Euler equations as viscosity vanishes. Studying this singular limit is challenging, especially in bounded domains due to the change in boundary conditions, from no-slip to slip, in the limit. This abrupt change gives rise to a boundary layer where the flow is potentially violent. I will give a detailed picture of the boundary layer for certain classes of Taylor-Couette flows in channels and pipes.

Meeting 10: Wednesday, 2/9/11, *KAP 427* (not KAP 414), 3:30-4:30 PM, Jason Fulman, USC

Title: Sperner's Lemma in fair division

Abstract: A very pretty application of topology/combinatorics to the problem of how to fairly divide a piece of cake.

Meeting 11: CAMS Colloquium, Monday, February 14, KAP 414, 3:30-4:30 PM, Peter Constantin, University of Chicago

Title: Analysis of complex fluids models

Abstract: Complex fluids are fluids carrying particulate matter. There are several models for such fluids, including stochastic models, kinetic models and closures thereof. I will discuss some open problems, isolate some of the mathematical difficulties, and illustrate some of the phenomena on simpler didactic models.

Meeting 12: Colloquium, Wednesday, February 16, KAP 414, 3:30-4:30 PM, Anatoly Libgober, University of Illinois at Chicago

Title: Elliptic genus of singular varieties

Abstract: Complex elliptic genus is an invariant which can be defined for singular varieties and which is a modular form. I will discuss constructions of such correspondence between varieties and modular forms and several applications including McKay correspondence, mirror symmetry and topology of singular real algebraic varieties.

Meeting 13: Joint CAMS and department colloquium, Wednesday, February 23, KAP 414, 3:30-4:30 PM, Qing Han, University of Notre Dame

Title: Isometric Embedding of Surfaces in Euclidean space

Abstract: It is a classical question in geometry whether a surface (2-dimensional Riemannian manifold) admits an isometric embedding in 3-dimensional Euclidean space. There are two versions of the question, local and global. In the global version, the manifold is complete and generally compact. In the local version, we are interested only in a neighborhood of a given point. Darboux pointed out that such isometric embedding can be expressed by a fully nonlinear equation of Monge-Ampere type. It turns out the Gauss curvature plays an essential role in the Darboux equation. In this talk, we will review some well-known results for both.

Meeting 14: CAMS colloquium, Monday, February 28, KAP 414, 3:30-4:30 PM, Dorit Hochbaum, USC

Title: Replacing spectral techniques for expander ratio, normalized cut and conductance by combinatorial flow algorithms

Abstract: Several challenging problem in clustering, partitioning and imaging have traditionally been solved using the ``spectral technique". These problems include the normalized cut problem, the graph expander ratio problem, the Cheeger constant problem and the conductance problem. These problems share several common features: all seek a bipartition of a set of elements; the problems are formulated as a form of ratio cut; the formulation is shown to be equivalent to a quadratic ratio, sometimes referred to as the isoperimetric or Raleigh ratio, on discrete variables and a single sum constraint which we call the balance or orthogonality constraint.

We introduce a unified framework viewing the spectral method for these problems as a relaxation of the Raleigh problem, and introducing a new, combinatorial, algorithm which involves a different relaxation of the Raleigh problem. This relaxation is shown here to be solved optimally, and in strongly polynomial time, in $O(mn\log {{n^2}/{m}})$ for a graph on n nodes and m edges. The algorithm, using HPF (Hochbaum's Pseudo-Flow) as subroutine, is efficient enough to to solve these bi-partitioning problems on millions of elements and more than 300 million edges within a couple of 10 minutes. It is also shown, via an experimental study, that the results of the combinatorial algorithm proposed often improve dramatically on the quality of the results of the spectral method.

Meeting 15: Joint CAMS and department colloquium, Wednesday, March 2, KAP 414, 3:30-4:30, George Sell, University of Minnesota

Title: Ensemble Dynamics and Bred Vectors

Abstract: The concept of a "bred vector" dates back to 1993, and Google now has over 500,000 hits in its data base. Two of these hits are found under the title of this lecture. On the other hand this concept has not (as of a week ago) made it into the AMS Math SciNet. One of the most interesing features about this concept is that it plays a central and important role in the related areas of dynamical systems and numerical studies. Nevertheless, the bred vector is a mathematical orphan.

For example, the matter of the sensitivity of model outputs to changes in the initial conditions for weather forcasting is a topic of widespread interest. While bred vectors have been accepted as useful tools in the study of sensitivity, there is a lacking of a sound theoretical basis for the applications of this concept. In this lecture we present some new developments in the theory of bred vectors.

By using a new concept of "ensemble dynamics" we obtain better insights into the part to be played by bred vectors in the theory of dynamical systems. Among other things, we will show how to use ensemble dynamics to obtain a new point of view of the fractal nature of the Lorenz attractor.

Topics that arise in this lecture include: invariant splittings, such as exponential dichotomies, and the multiplicative ergodic theorem. We will also use the Lorenz attractor to motivate one of the basic problems arising in discrete dynamics: Can one find a proof of the apparent robustness of the Lorenz attractor?

Meeting 16: Joint CAMS and department colloquium, Monday March 7, KAP 414, 3:30-4:30, Dennis Sullivan, State University of New York, Stony Brook

Title: Correlated finite energy models of Navier Stokes time evolution

Abstract: If one has an AT (Algebraic Topology) model of a system of fields and operations in Riemannian geometry, there is a natural way to construct derived models at each scale of resolution. In addition there are transition mappings between these derived models at different scales.The process of constructing derived models is based on the key idea of AT: chain homotopy equivalences between chain complexes. If a nonlinear PDE among the original system of fields and operations can be reformulated in the derived models, one can obtain a system of finite energy or finite scale models which are correlated by structure mappings. Incompressible Navier Stokes evolution in 3D can be described by the differential algebra of differential forms, the Hodge star operator and the projections of the Hodge decomposition. These objects are naturally interpreted in AT. The lecture will discuss this AT approach to deriving computational fluid models.

Meeting 17: Colloquium, Wed. March 9, KAP 414, 3:30-4:30, Patricia Hersh, North Carolina State University

Title: Combinatorics and Topology of Stratified Spaces

Abstract: For regular CW complexes, one may deduce topological structure from the combinatorics of the poset of closure relations on cells. However, two different stratified spaces with the same closure poset may have very different topological structure. I will discuss this interplay of combinatorics with topology, including the positive resolution of a conjecture regarding the topology of stratified spaces having the intervals of Bruhat order as their closure posets. A key ingredient is a new criterion for deciding if a finite CW complex is regular with respect to a choice of characteristic maps. I will review background and history of this area along the way.

Meeting 18: CAMS colloquium, Mon. March 21, KAP 414, 3:30-4:30, Donald Estep, Colorado State University

Title: A Measure Theoretic Computational Approach for Inverse Sensitivity Problems

Abstract: We consider the inverse sensitivity analysis of a map from a set of parameters and data to a quantity of interest. We are particularly interested in implicitly-defined maps, e.g. involving the solution of a differential equation. The inverse problem is to describe the random variation in the input that leads to an imposed or observed random variation in the output quantity. We formulate this as an ill-posed inverse problem for an integral equation using the Law of Total Probability. We then describe a computational method for computing solutions that has two stages. In the first part, we approximate the unique set-valued solution to the inverse of the integral equation using derivative information. In the second part, we apply basic ideas from measure theory to compute the approximate probability measure on the parameter and data space that solves the integral equation. We discuss convergence of the method, and explain how to use the method to compute the probability of events in the input (parameter) space. The talk is illustrated with a number of examples. We also discuss briefly the numerical analysis (accuracy) of the method and the consideration of multiple quantities of interest and data assimilation.

Meeting 19: Colloquium, Wed. March 23, KAP 414, 3:30-4:30, T. Kyle Petersen, DePaul University

Title: What is the gamma vector? (And what does it count?)

Abstract: The Charney-Davis conjecture is a reformulation of a conjecture of Hopf about the Euler characteristic of a nonpositively curved Riemannian manifold. Incredibly, Charney and Davis showed that their conjecture boils down to a conjecture about the combinatorics of certain simplicial complexes with the homology of sphere. The "f-vector" of a simplicial complex counts the number of simplices in the complex according to dimension (number of vertices, edges, triangles, etc.). Loosely speaking, the conjecture follows if one can come up with an adequate characterization of the f-vectors of these sorts of simplicial spheres.

I will discuss several approaches to the Charney-Davis conjecture undertaken over the years. In particular, I will describe work of Gal that transforms the question by introducing an invariant derived from the f-vector called the "gamma vector". Gal's conjecture, which implies Charney-Davis, is that this gamma vector always consists of nonnegative integers. As always when faced with a mysterious collection of nonnegative integers, the natural question for the combinatorialist is: What does the gamma vector count?

In recent work with Eran Nevo ( http://arxiv.org/abs/0909.0694) and Nevo and Bridget Tenner ( http://arxiv.org/abs/1003.2544), we propose an answer to this question.

The talk will be accessible to graduate students and others with little background in the area.

Meeting 20: CAMS colloquium, Monday, March 28, KAP 414, 3:30-4:30, Thomas Hou, Caltech

Title: Extracting trend and instantaneous frequency from multiscale data

Abstract: How to extract trend from highly nonlinear and nonstationary data is an important problem that has many practical applications ranging from bio-medical signal analysis to econometrics, finance, and geophysical fluid dynamics. We review some exisiting methodologies in defining trend in data analysis. Many of these methods use pre-determined basis and is not completely adaptive. They tend to introduce artificial harmonics in the decomposion of the data. Various attempts to preserve the temportal locality property of the data introduce problems of their own. Here we discuss how adaptive data analysis can be formulated as a nonlinear optimization problem in which we look for a sparse representation of data in some unknown basis which is derived from the physical data. We will show that this formulation has some beautiful mathematical structure and can be considered as a nonlinear version of compressed sensing.

Meeting 21: Colloquium, Wed. March 30, KAP 414, 3:30-4:30, Xianzhe Dai, University of California, Santa Barbara

Title: The Ray-Singer conjecture for singular manifolds

Abstract: The Reidemeister torsion (R-torsion) is a combinatorial invariant introduced by Reidemeister in 1935. It is a secondary invariant associated to the Euler characteristic and is the first topological invariant which distinguishes homotopy equivalent spaces. The analytic torsion is introduced by Ray and Singer in the 70's as an analytic analog of the R-torsion. The Ray-Singer conjecture, which is proven independently by Cheeger and Mueller, says that the analytic torsion equals the R-torsion for closed manifolds. Recent interesting application of the Cheeger-Mueller theorem includes detecting torsion homology classes of hyperbolic manifolds. Thus it will be both interesting and desirable to extend it to singular manifolds. We will discuss the recent understanding along this direction.

Meeting 22: WEDNESDAY, 4/06/11 ALBERT L. WHITEMAN MEMORIAL MATHEMATICS LECTURE 4:30-5:30 PM, Mudd Hall Philosophy Room 101, George Andrews, Penn State University

Title: Ramanujan, the Lost Notebook, and Related Incidents

Meeting 23: THURSDAY, 4/07/11 ALBERT L. WHITEMAN MEMORIAL MATH LECTURE 3:30-4:30 PM, KAP 414, George Andrews, Penn State University

Title: Partition Function Differences, Ehrenpreis's Problem and the Anti-Telescoping Method

Meeting 24: FRIDAY, 4/08/11 PROBABILITY & STATISTICS SEMINAR 3:30-4:30 PM, KAP 414, Ken Alexander, USC

Title: Layering transitions in the solid-on-solid model without external field

Meeting 25: Monday, April 11, CAMS colloquium, 3:30-4:30 KAP 414, Paolo Galdi, Univeristy of Pittsburgh

Title: Some Recent Results and Open Questions in the Mathematical Theory of Liquid-Solid Interaction

Abstract: Even though problems of liquid-solid interaction are more or less ubiquitous in many branches of applied science - ranging from small to large scale - a systematic mathematical treatment of some of their relevant and basic aspects has begun only a few years ago. This late start is probably due to the intrinsic difficulty of the relevant equations. In fact, the presence of the solid (rigid or elastic) affects the flow of the liquid, and this, in turn, affects the motion of the solid, so that the problem of determining the flow characteristics is highly linked, typically, through a non-local coupling. It is just this latter feature that makes any fundamental mathematical problem related to fluid-solid interaction a particularly challenging one. Objective of this talk is to give an account of certain significant problems in the mathematical theory of liquid-solid interaction, as well as to present some new results and open question, when the solid is either rigid or elastic.

Meeting 26: Wednesday, April 13, Colloquium, 3:30-4:30, KAP 414, Witold Sadowski, Warwick Mathematics Institute

Title: Lagrangian trajectories for weak solutions to the 3D Navier-Stokes equations

Abstract: The putative singular set of a suitable weak solution to the 3D Navier-Stokes equations cannot be too large. Limiting its size can be done both in terms of Hausdorff and the box-counting dimension. In the talk I will show how the smallness of the singular set can be used to prove uniqueness of individual particle trajectories corresponding to a suitable weak solution of the Navier-Stokes equations.