(September 29) Marc Yor: Some remarkable properties of Gamma processes - A number of remarkable properties of the Gamma processes, i.e. : realisation of their bridges, absolute continuity relationships, realisation of a gamma process as an inverse local time, the effect of a gamma process as a time change... are gathered in this paper. Some of them are put in perspective with their Brownian counterparts.

(October 6) C. David Levermore: Fluid Dynamics from Boltzmann Equations - We survey some recent results that establish the validity of fluid dynamical systems from the Boltzmann equation. The starting point will be the DiPerna-Lions theory of global solutions to the Boltzmann equation. The target fluid systems will be incompressible and weakly compressible. A new multiple time-scale result will be presented that unifies the acoustic and Stokes limits. Open problems will be discussed.

(October 20) Ricardo Nochetto: Discrete Gradient Flows for Shape Optimization and Applications - We present a variational framework for shape optimization problems that hinges on devising energy decreasing flows based on shape differential calculus followed by suitable space and time discretizations (discrete gradient flows). A key ingredient is the flexibility in choosing appropriate descent directions by varying the scalar products, used for computation of normal velocity, on the deformable domain boundary. We discuss applications to image segmentation, optimal shape design for PDE, and surface diffusion, along with several simulations exhibiting large deformations as well as pinching and topological changes in finite time. This work is joint with E. Baensch, G. Dogan, P. Morin, and M. Verani.

(October 23) Giovanni Forni: On the speed of ergodicity for rational polygonal billiards - In the 90's Zorich derived from Veech's work a polynomial upper bound on the deviation of ergodic averages from the mean for typical interval exchange transformations (IET's). Such a result says nothing about rational billiards since they correspond to a zero measure set in the space of IET's. We describe the proof of a similar polynomial upper bound for any translation surface (hence any rational billiard). The argument is based on a 'spectral gap' result for a cocycle over the Teichmuller flow (introduced by Kontsevich and Zorich). This is joint work with J. Athreya. If time permits we will briefly discuss what can be proved about lower bounds on deviations.

(October 27) Klaus Schmidt: Mahler measure, Fuglede-Kadison determinants and entropy - Every element fof the integral group ring Z[Z^d], d >= 1, defines a measure preserving action $\alpha _f$ of Z^d by automorphisms of a compact abelian group X_f, whose entropy is the logarithm of the Mahler measure of f.

Mahler measures of multivariate polynomials f in Z[Z^d] and, in particular, certain values of L-functions, also occur as entropies of certain lattice models (especially dimer models) in statistical mechanics, and the connection between these lattice models and the Z^d-actions associated with these polynomials is still somewhat mysterious.

The connection between elements of the integral group ring $\mathbf{Z}[\Gamma ]$ of a discrete group $\Gamma $ and $\Gamma $-actions extends to arbitrary discrete (amenable) groups. The entropy of such an action is a quantity which extends the notion of logarithmic Mahler measure to elements of $\mathbf{Z}[\Gamma ]$. The last part of the lecture will discuss what little is currently known explicitly about this extension.

Much of this material is joint work with Christopher Deninger and/or Doug Lind.

(November 3) Charles Chui: Image Noise Removal based on the Variational Approach and Wavelets - The background of this work is the standard problem of minimization of some total energy functional, but with specific choices of the internal energy density functions g(x). Our interest in this study is motivated by the search of effective solutions to certain inverse problems, particularly for real-time image noise removal for digital cameras. In general, depending on the objectives of the inverse problems under investigation, such as curve fitting, image noise removal, and feature extraction, the internal energy in our study is governed by g( | Lu | ); with (Lu) (x) = u (x), (Lu) (x,y) = (Grad u) (x,y), and Lu being some wavelet transform of u in any dimension. For digital image noise removal, in particular, a suitable choice of g(x) leads to the anisotropic diffusion model, the discretization of which, in turn, is relevant to the design of certain content-dependent filters, notably the bilateral filters. A natural generalization of this approach also gives rise to the notions of diffusion maps and geometric harmonics that constitute the foundation for the recent research investigations in diffusion wavelets for analyzing complex data in high dimensions. This is a joint work with Jianzhong Wang.

(December 1) George Papanicolaou: Optimal Illumination in Array Imaging - I will discuss the mathematical problem of optimally illuminating an object for imaging by an array. In a certain regime of parameters and for a special class of objects this can be done by using spheroidal wave functions. In general situations one must use algorithms that optimally image the object by detecting its edges. Such algorithms are very different from the ones that maximize the energy of the signals received by the array so as to enhance detectability. I will analyze and compare the two types of algorithms. (Joint work with L. Borcea and C. Tsogka.)

(December 8) Jacob Sterbenz : Formation of Singularities in the Critical Harmonic-Map Wave-Flow into S² - I will discuss recent work in collaboration with Igor Rodnianski concerning the longstanding problem of finite time breakdown in the wave-map flow from (2+1) dimensional Minkowski space into the sphere S². Despite a wealth of numerical and heuristic arguments, this phenomena has only very recently been established established by us analytically for the simplified case of equivariant flows.

In the talk I will discuss some of the history of the problem, as well as our method of proof. In particular I will discuss how our method ties together the so called ``Bogomolnyi structure'' of the corresponding harmonic map equation, with techniques coming from the modulational stability theory of non-integrable solitons, for example as found in the work of M. Weinstein.

(December 15) Alexandre Chorin : Problem reduction and memory - I will present methods for the reduction of the complexity of computational problems, both time-dependent and stationary, and their connections to probability, renormalization, scaling, and statistical mechanics, together with examples. The main points, are: (i) in time dependent problems, it is not legitimate to average equations without taking into account memory effects and noise; (ii) mathematical tools developed in physics for carrying out renormalization group transformations yield effective block Monte-Carlo methods; (iii) the Mori-Zwanzig formalism, which in principle yields exact reduction methods but is often hard to use, can be tamed by approximation; and (iv) more generally, problem reduction is a search for hidden similarities. In the examples I will emphasize the "t-model" for problems where the memory (=autocorrelation of the noise) has a large support; this is important in applications to hydrodynamics.

(January 26) Michael Lacey: Fourier Series: Past, Present, Future - The focus of this talk will be on Carleson's celebrated theorem asserting the pointwise convergence of Fourier series. We will explain why the theorem is hard: Any proof of this result is necessarily multiscale. The recent efficient proof of the speaker and C. Thiele will be outlined. The talk will end with a very quick survey of the emerging theory that has developed around this Theorem. The future of the subject lies in the appropriate understanding of a quadratic Fourier analysis.

(February 2) Antoine Mellet: Homogenization of some free boundary problems - Free boundary problems arise in the modeling of many natural phenomena. They are mathematically very interesting to study, mainly because of the lack of a priori regularity of the free boundary. In this talk, I want to discuss the behavior of the free surface in heterogeneous media (periodic or random). Well-known examples include the study of geodesic curves (or minimal surfaces) for periodic metric. I will present some results concerning capillary surfaces (drops of water on a heterogeneous surface) and combustion fronts propagation. In particular, I will show that inhomogeneities typically yield hysteresis phenomena.

(February 9) Emeriti Colloquium: Early Mathematics at Maryland - This is the 2006-2007 version of the annual colloquium honoring our Emeriti Colleagues. We are all well-acquainted with the current mathematical scene in our department, but many might not be aware of the very high level mathematical activity in the early days, before roughly 1960. In an effort to inform us of that period, three of our senior emeriti--Jack Goldhaber, John Horvath, and Jim Hummel--have agreed to share their recollections of that era. In addition there will be opportunity for comments from the audience.

(February 16) Margaret Cheney : Radar Imaging - Radar imaging is a technology that has been developed, very successfully, within the engineering community during the last 50 years. Radar systems on satellites now make beautiful images of regions of our earth and of other planets such as Venus. One of the key components of this impressive technology is mathematics, and many of the open problems are mathematical ones.

This lecture will explain, from first principles, the basics of radar and the mathematics involved in producing high-resolution radar images.

(February 23) John Lott: The work of Grigory Perelman - This talk will explain (for a general audience) the work of Perelman on the Poincaré conjecture, for which he was offered the Fields Medal.

(March 16) Alex Lubotzky: Finite groups and hyperbolic manifolds - The isometry group of a closed hyperbolic n-manifold is finite. In 1974, Leon Greenberg proved the converse for n=2, i.e., every finite group is the full isometry group of some closed hyperbolic surface. Kojima (1988) extended the result to n=3. We will show the general case: For every n>1 and every finite group G there is an n-dimensional closed hyperbolic manifold whose isometry group is isomorphic to G. An interesting aspect of the proof is that it is nonconstructive; it uses a 'probabilistic method', i.e. counting results from the theory of 'subgroup growth'.
The talk won't assume any prior knowledge on the subject, and is based on joint work with M. Belolipetsky (Invent. Math. Dec. 2005).

(March 30) Walter Strauss: Steady Water Waves: Theory and Computation - Consider a classical 2D gravity wave with arbitrary vorticity. Assume such a wave is traveling at a constant speed over a flat bed. There exist huge families of such waves of small or large amplitude. I will discuss their existence and also exhibit some recent numerical computations. The maximum amplitudes of the waves and the locations of possible stagnation points are greatly affected by the vorticity.

(April 27) Dan Freed: Secondary differential-geometric invariants, generalized cohomology, and QCD - Topological invariants, such as the degree of a map, often have associated secondary geometric invariants. In differential geometry they were introduced by Chern, Simons, Cheeger, and others. In the early 1980s they appeared in physics in work of Wess, Zumino, Novikov, and Witten. Such invariants exist in any generalized cohomology theory, as developed by Hopkins and Singer. One recent application of these ideas is to the physics of pions in quantum chromodynamics.

(May 4) Emmanuel Candes: Applications of Compressive Sampling to Error Correction - ``Compressed Sensing'' or ``Compressive Sampling'' (CS) is a new sampling or sensing theory which goes somewhat against the conventional wisdom in signal acquisition. This theory allows the faithful recovery of signals and images from what appear to be highly incomplete sets of data, i.e. from far fewer data bits than traditional methods use. It is believed that this phenomenon may have significant implications. For instance, CS may come to underlie procedures for sensing and compressing data simultaneously and much faster. In this talk, we will present the basic tenets of this new sampling theory and introduce applications in the area of error correction.

(May 11) Jeff Adams : The Atlas of Lie Groups and Representations - I will discuss the goal of the Atlas project, which is to classify the unitary dual of a real Lie group by computer. I will talk about the progress we have made, and what we have learned so far about tackling a major problem in pure mathematics by computer. I will discuss our recent calculation of Kazhdan-Lusztig-Vogan polynomials for E₈, and say something about the attention this result recently got in the press.