Department of Mathematics Special Colloquium
Conjecture: As with many areas of topology and geometry, a starting point in the study of 3-manifolds is to try to understand codimension one objects in them, namely embedded surfaces. A particularly useful class of surfaces are the "incompressible" ones which are topologically essential; a 3-manifold containing such a surface is called a Haken manifold. There are many 3-manifolds which are not Haken, but if we ask about immersed, rather than embedded, surfaces the situation becomes much more mysterious. A closely related question is this: Suppose M is a 3-manifold with infinite fundamental group, does M have a finite cover which is Haken? The virtual Haken Conjecture posits that the answer to this question is yes.
This talk will survey some recent results in this area, focusing on my work with (variously) William Thurston, Dylan Thurston, and Frank Calegari. From the point of view of Thurston's Geometrization Conjecture, this is really a question about hyperbolic 3-manifolds, that is, lattices in PSL(2, C). This opens the door to a rich array of tools that might seem quite surprising in light of the purely topological description of the problem above. Indeed, some unlikely-sounding terms that I will probably mention in my talk are "the Classification of Finite Simple Groups" and "the Langlands Conjecture", as well as such topological oddities as "random 3-manifolds"!
Brown University Center for Statistical Sciences
Lectureship Series
Department of Biostatistical Science, Harvard School of Public Health, Dana Farber Cancer Institute | |
(Refreshments beginning at 3:15 p.m.) |
Abstract: The availability of commercial genome tiling microarrays enabled the genome-wide study of transcription and epigenetic factors. However, it also poses computational challenges for data analysis. We have developed a number of algorithms to facilitate the adoption of genome tiling microarray applications to model global transcription regulation.
The first suite of algorithms aims to analyze ChIP-chip data on Affymetrix genome tiling microarrays. This includes MBR for array image quality control, xMAN for fast probe mapping and filtering, MAT for binding location prediction, and CEAS for comprehensive annotation of the genome-wide ChIP-regions. Application of these algorithms to estrogen receptor (ER) and androgen receptor (AR) ChIP-chip not only identified their respective cistrome, but also uncovered novel mechanisms of ER and AR regulation.
The second suite of algorithms adopts signal processing techniques to locate positioned nucleosomes in mammalian promoters tiled on high resolution NimbleGen microarrays. We found that expressed genes and genes containing pre-initiation complexes are characteristically nucleosome-free at transcription start sites, and transcription factor oncogene MITF binds predominantly in nucleosome-free regions. We also found variations in nucleosome positioning at promoters to be correlated with lineage-specific gene expression patterns.
Sponsored by The Charles K. Colver Lectureship and Publication Fund
Center for Fluid Mechanics Seminar
And
The Fluids, Thermal and Chemical Processes Group
Of
The Division Of Engineering
Seminar Series
Technion -- Israel Institute of Technology, Israel | |
Abstract: When a stationary bubble is exposed to an external temperature gradient, Marangoni stresses at the bubble surface result in fluid motion. A straight-forward attempt to calculate the influence of this thermocapillary flow upon the temperature distribution fails to provide a well-behaved solution [Balasubramaniam & Subramanian, Phys. Fluids 16, 3131 (2004)]. This problem is revisited here using a regularization procedure which exploits the qualitative disparity in the long-range flow fields generated by a stationary bubble and a moving one. The regularization parameter is an (exponentially small) artificial bubble velocity, which reflects the inability of any asymptotic expansion to satisfy the condition of exact bubble equilibrium. The solution is obtained using asymptotic matching of two separate Reynolds-number expansions: an inner expansion, valid at the bubble neighborhood, and a remote outer expansion, valid far beyond the familiar Oseen region. This procedure provides a well-behaved solution, which is subsequently used to evaluate the convection-induced correction to the hydrodynamic force exerted on the bubble. The independence of that correction upon the artificial velocity confirms the adequacy of the regularization procedure to describe the stationary-bubble case. The ratio of the calculated force to that pertaining to the classical pure-conduction limit [Young, Goldstein & Block, J. Fluid Mech. 6, 350 (1959)] is given by 1 - Ma/8+o(Ma), where Ma is a radius-based Marangoni number.
Department of Mathematics Special Colloquium
Abstract: The theme of group actions in arithmetic has been especially prominent in recent years, thanks to the realization that powerful techniques from ergodic theory can often yield stronger theorems with cleaner proofs than can classical analytic techniques (e.g., the work of Furstenberg on Szemeredi's theorem, Vatsal on nonvanishing, Margulis on the Oppenheim conjecture...). A key idea is that the group action or group structure involved is not always immediately apparent (e.g., Bhargava's recent "higher composition laws" generalizing classical Gauss composition). I will explain how this theme applies to the classical problem of representing one positive definite quadratic form by another. By relating this problem to the action of a p-adic group on a homogeneous space, we answer in the affirmative (most of) the old conjecture that a "large enough" quadratic form which is locally represented by some form Q is in fact globally represented by Q.
Brown University Division of Applied Mathematics
Transatlantic Seminar
Abstract:
Outer billiards is a basic dynamical system based on a planar
convex shape, B.H. Neumann introduced outer billiards in the
1950s and J. Moser popularized it in the 1970's, noting
connections between it and celestial mechanics. All along,
one of the central questions has been:
Does there exist an outer billiards system with an unbounded orbit.
In this talk I will show that outer billiards for the Penrose kite
has an unbounded orbit. The Penrose kite is the convex quadrilateral
that arises in connection with the famous Penrose tilings. My proof
relates the problem to self-similar tilings, polygon exchange maps,
and arithmetic dynamics.
The second part of the talk will provide deeper insight into polygon exchange maps and arithmetic dynamics.
Applied Mathematics Colloquium
Delayed Thanksgiving Talk
Abstract: Referring both to pure and applied mathematical practices, the speaker will describe and comment on a paradox of the infinite which has heretofore received little publicity as such.
Scientific Computing Seminar
Columbia University, New York, NY | |
Abstract: Optimal complexity algebraic preconditioners, such as multigrid/multilevel preconditioners for PDE-governed problems, keep the time spent in dominant algebraic kernels close to linear as the applications scale out to the limit of currently available parallel computers (e.g., BlueGene/L with 131,072 (2**17) processors). Krylov accelerators and Jacobian-free variants of Newton's method, as appropriate, are wrapped outside to deliver robustness in multirate, multiscale coupled systems, which are solved either implicitly or in more traditional forms of operator splitting. The Towards Optimal Petascale Simulations (TOPS) project, directed by the speaker, is sponsored by the U.S. Department of Energy to research and deploy a collection of open-source, scalable (or semi-scalable) solver software components (PETSc, Hypre, SuperLU, etc.) for discrete problems arising in several large-scale applications, including fusion reactor modeling and design.
We outline the TOPS research agenda and illustrate with a selection of five applications in DOE's magnetically confined fusion energy portfolio, ranging from swapping software through a standard interface to prototyping entirely new codes for the high-end. We also talk through the "Jardin-Keyes Roadmap" for fusion energy simulation featured in the September 2006 issued SIAM News. The noteworthy messages are: (1) that relatively straightforward insertion of available techniques has paid great dividends so far, but (2) that the challenges of the next generation of codes will require fresh contributions from mathematicians. As the fusion community gears up for U.S. participation in the International Thermonuclear Experimental Reactor (ITER) consortium, the ultimate goal of which is abundant energy production outside of the planetary carbon cycle, simulations on petascale hardware are expected to play an essential role.
PDE Seminar
Department of Mathematics Special Colloquium
Abstract: We consider the question: "How bad can the deformation space of an object be?" (Alternatively: "What singularities can appear on a moduli space?") The answer seems to be: "Unless there is some a priori reason otherwise, the deformation space can be arbitrarily bad." We show this for a number of important moduli spaces. More precisely, up to smooth parameters, every singularity that can be described by equations with integer coefficients appears on moduli spaces parameterizing: smooth projective surfaces (or higher-dimensional manifolds); smooth curves in projective space (the space of stable maps, or the Hilbert scheme); plane curves with nodes and cusps; stable sheaves; isolated threefold singularities; and more. The objects themselves are not pathological, and are in fact as nice as can be. This justifies Mumford's philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. I will begin by telling you what "moduli spaces" and "deformation spaces" are. The complex-minded listener can work in the holomorphic category; the arithmetic listener can think in mixed or positive characteristic. This talk is intended to be (mostly) comprehensible to a broad audience.
<--- 2007 Index