Mondays 214 Fine Hall 4:30 pm
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Fall 2011 Collapse/Expand/Print
| ate: September 26 |
| Speaker: Leonidas Guibas, Stanford University |
| Title: Understanding 3D Shapes Jointly |
| Abstract: The use of 3D models in our economy and life is becoming more prevalent, in applications ranging from design and custom manufacturing, to prosthetics and rehabilitation, to games and entertainment. Although the large-scale creation of 3D content remains a challenging problem, there has been much recent progress in design software tools, like Google SketchUp for buildings or Spore for creatures, or in low cost 3D acquisition hardware, like the Microsoft Kinect scanner. As a result, large commercial 3D shape libraries, such as the Google 3D Warehouse, already contain millions of models. These libraries, however, can be unwieldy, when the need arises to efficiently incorporate models into various workflows. Mathematical formulations, efficient algorithms, and software tools are required to support navigation and search over 3D model repositories. In this talk we examine the problem of facilitating these navigation and search tasks by automatically extracting relationships between shapes in a collection and understanding their common or shared structure. By effectively organizing the collection into (possibly overlapping) groups of related shapes, by separating what is common from what is variable within each group and across groups, and by understanding the main axes of variability, we can facilitate a whole slew of operations that make large 3D repositories much more navigable, searchable, compressible, and visualizable. We will present a quick summary of tools for efficiently computing informative shape descriptors as well as structure preserving maps between shapes at different levels of resolution. The main part of the talk, however, is aimed beyond pairwise relationships, to the study and analysis of many shapes jointly, looking at networks of maps between shapes in order to extract joint structure, derive consistent segmentations, infer phenotypic relationships, etc. This is preliminary work on what we believe to be a large open area for research -- the joint understanding of collections of related geometric data sets. |
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| Date: October 3 |
| Speaker: Don Saari, University of California - Irvine |
| Title: Complexity theory applied to voting theory |
| Abstract: As it will be shown with results and examples, the paradoxes associated with standard voting rules are surprisingly likely and are so complex that one must worry about the legitimacy of election outcomes. To extract an understanding of what can happen and why, it is shown how lessons from complexity theory, where complicated behavior is due to a combination of simple interactions, explain many mysteries both in this area and for related topics such as nonparametric statistics, etc. Indeed, all paradoxes of standard rules, including Arrow's seminal "Impossibility Theorem," reflect simple but hidden symmetry structures connecting the preferences of voters. |
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| Date: October 10 |
| Speaker: Eitan Tadmor, University of Maryland |
| Title: A new model for self-organized dynamics: From particle to hydrodynamic descriptions |
| Abstract: Self-organized dynamics is driven by ârules of engagement", which describe how each agent interacts with its neighbors. They consist of long-term attraction, mid-range alignment and short-range repulsion. Many self-propelled models are driven by the balance between these three forces, which yield emerging structures of interest. Examples range from consensus of voters and traffic flows to the formation of flocks of birds or school of fish, tumor growth etc. We introduce a new particle-based model, driven by self-alignment, which addresses several drawbacks of existing models for self-organized dynamics. The model is independent of the number of agents: only their geometry in phase space is involved. We will explain the emerging flocking behavior of the proposed model in the presence of non-symmetric interactions which decay sufficiently slow, and discuss the difficulties of tracing graph connectivity otherwise. The methodology is based on the new notion of active sets, which carries over from particle to kinetic and hydrodynamic descriptions, and we discuss the unconditional flocking at the level of hydrodynamic description. |
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| Date: October 17 |
| Speaker: Michael L. Overton, Courant Institute of Mathematical Sciences, NYU |
| Title: Optimization of Polynomial Roots, Eigenvalues and Pseudospectra |
| Abstract: The root radius and root abscissa of a monic polynomial are respectively the maximum modulus and the maximum real part of its roots; both these functions are nonconvex and are non-Lipschitz near polynomials with multiple roots. We begin the talk by giving constructive methods for efficiently minimizing these nonconvex functions in the case that there is just one affine constraint on the polynomial's coefficients. We then turn to the spectral radius and spectral abscissa functions of a matrix, which are analogously defined in terms of eigenvalues. We explain how to use nonsmooth optimization methods to find local minimizers and how to use nonsmooth analysis to study local optimality conditions for these nonconvex, non-Lipschitz functions. Finally, the pseudospectral radius and abscissa of a matrix $A$ are respectively the maximum modulus or maximum real part of elements of its pseudospectrum (the union of eigenvalues of all matrices within a specified distance of $A$). These functions are also nonconvex but, it turns out, locally Lipschitz, although the pseudospectrum itself is not a Lipschitz set-valued map. We discuss applications from control and from Markov chain Monte Carlo as examples throughout the talk. Coauthors of relevant papers include Vincent Blondel, Jim Burke, Kranthi Gade, Mert Gurbuzbalaban, Adrian Lewis and Alexandre Megretski. |
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| Date: October 24 |
| Speaker: Charles Epstein, UPenn |
| Title: Existence and regularity for a class of degenerate diffusions arising in population genetics |
| Abstract: Infinite population limits of standard Markov chain models lead to Markov processes on polyhedral domains that are formally generated by degenerate elliptic operators. These operators are characterized, in part, by the first order vanishing, along the boundary, of the coefficient of the second normal derivative term. This fact places these operators beyond those which have thus far been successfully analyzed using methods of geometric analysis. I will present an approach to these operators, which I have been pursuing with Rafe Mazzeo, based on an-isotropic Holder spaces, which leads to a rather complete existence, uniqueness and regularity theory. |
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| Date: November 7 |
| Speaker: Arjen Doelman, University of Leiden / Lorentz Center |
| Title: The mathematics of desertification: searching for early warning signals |
| Abstract: The process of desertification can be modeled by systems of reaction-diffusion equations. Numerical simulations of these models agree remarkably well with field observations: both show that `vegetation patterns' -- i.e. regions in which the vegetation only survives in localized `patches' -- naturally appear as the transition between a healthy homogeneously vegetated state and the (non-vegetated) desert state. Desertification is a catastrophic and non-reversible event during which huge patterned vegetation areas `collapse' into the desert state at a fast time scale -- for instance as a consequence of a slow decrease of yearly rainfall, or through an increased grazing pressure. It is crucial to be able to recognize whether a patterned state is close to collapse (or not), ecologists are thus searching for `early warning signals'. In this talk, we will translate the issues raised by the desertification process into mathematical terms and relate these to recent developments in the field of pattern formation. It will be shown that the process of desertification poses fundamental mathematical questions and that it already has led to the development of novel mathematical theory. |
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| Date: November 14 |
| Speaker: Vladimir Rokhlin, Yale University |
| Title: Accurate Randomized Algorithms of Numerical Analysis |
| Abstract: Randomized algorithms are ubiqutous in computer science and computer engineering. Many problems that are intractable when viewed deterministi- cally can be effectively solved with probabilistic techniques. Perhaps the most important aspect of most randomized procedures in current use is the fact that they produce the correct result with (practically speaking) 100% reliabil- ity, and with (essentially) machine precision. Historically, randomized techniques have been less popular in numerical analysis. Most of them trade accuracy for speed, and in many numerical environments one does not want to add yet another source of inaccuracy to the calculation that is already sufficiently inaccurate. One could say that in numerical analysis probabilistic methods are an approach of last resort. I will discuss several probabilistic algorithms of numerical linear algebra that are never less accurate than their deterministic counterparts, and in fact tend to produce better accuracy. In many situations, the new schemes have lower CPU time requirements than existing methods, both asymptotically and in terms of actual timings. I will illustrate the approach with several numerical examples, and discuss possible extensions. |
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| Date: November 21 |
| Speaker: Frederik Simons, Princeton University |
| Title: Prolates on the sphere, Extensions and Applications: Slepian functions for geophysical and cosmological signal estimation and spectral analysis |
| Abstract: Functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. In cosmology we may wish to compute the power spectral density of the cosmic microwave background radiation without the contaminating effect of the galactic plane. Analyzing and representing scientific data of this kind is facilitated in a basis of functions that are spatiospectrally concentrated, i.e. localized in both domains at the same time. Here, we give a theoretical overview of the approach to this concentration problem that was proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere. "Spherical Slepian functions" are now widely applied to the study of inverse problems with (potential-field) satellite data, including such problems whose solutions are linear (source estimation), and quadratic (spectral estimation), in the data. Among the applications that I will discuss are the analysis of the time-variable gravity field for the recovery of coseismic gravity perturbations, the sparse analysis and representation of the lithospheric magnetic field, the recovery of the power spectral density of the cosmic microwave background radiation, and so on. |
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| Date: November 28 |
| Speaker: Andrea Montanari, Stanford University |
| Title: Sharp Thresholds in Statistical Estimation |
| Abstract: Sharp thresholds are ubiquitous high-dimensional combinatorial structures. The oldest example is probably the sudden emergence of the giant component in random graphs, first discovered by Erdos an Renyi. More recently, threshold phenomena have started to play an important role in some statistical learning and statistical signal processing problems, in part because of the interest in 'compressed sensing'. The basic setting is one in which a large number of noisy observations of a high-dimensional object are made. As the ratio of the number of observations to the number of `hidden dimensions' crosses a threshold, our ability to reconstruct the object increases dramatically. I will discuss several examples of this phenomenon, and some algorithmic and mathematical ideas that allow to characterize these threshold phenomena. [based on joint work with Mohsen Bayati, David Donoho, Iain Johnstone, Arian Maleki] |
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| Date: December 5 |
| Speaker: Peter Constantin, Princeton University |
| Title: Nonlocal Evolution Equations |
| Abstract: Nonlocal evolution equations have been around for a long time, but in recent years there have been some nice new developments. The presence of nonlocal terms might originate from modeling physical, biological or social phenomena (incompressibility, Ekman pumping, chemotaxis, micro-micro interactions in complex fluids, collective behavior in social aggregation) or simply from inverting local operators in the analysis of systems of PDE. I will brifly present some regularity results for hydrodynamic models with singular constitutive laws. The main part of the talk will present a nonlinear maximum principle for linear nonlocal dissipative operators and applications. |
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Spring 2012 Collapse/Expand/Print

