Nicolaus Tideman : The Structure of the Election-Generating Universe
- Presentations ( 114 Views )This paper reports the results of using two sets of ranking data, one from actual elections and the other from surveys of voters, to examine whether the outcomes of three-candidate vote-casting processes follow a discernible pattern. Six statistical models that make different assumptions about such a pattern are evaluated. Both data sets suggest that a spatial model describes an observable pattern much better than any of the other five models. The results imply that any conclusions about the probability of voting events reached on the basis of models other than the spatial modelfor example, on the basis of the impartial anonymous cultureare suspect. (Joint work with Florenz Plassmann)
Dmitry Khavinson : From the Fundamental Theorem of Algebra to Astrophysics: a Harmonious Path
- Geometry and Topology ( 114 Views )The Fundamental Theorem of Algebra first rigorously proved by Gauss states that each complex polynomial of degree n has precisely n complex roots. In recent years various extensions of this celebrated result have been considered. We shall discuss the extension of the FTA to harmonic polynomials of degree n. In particular, the 2003 theorem of D. Khavinson and G. Swiatek that shows that the harmonic polynomial z − p(z); deg p = n > 1 has at most 3n − 2 zeros as was conjectured in the early 90's by T. Sheil-Small and A. Wilmshurst. More recently L. Geyer was able to show that the result is sharp for all n.
In 2004 G. Neumann and D. Khavinson showed that the maximal number of zeros of rational harmonic functions z − r(z); deg r = n > 1 is 5n − 5. It turned out that this result conrfimed several consecutive conjectures made by astrophysicists S. Mao, A. Petters, H. Witt and, in its final form, the conjecture of S. H. Rhie that were dealing with the estimate of the maximal number of images of a star if the light from it is deflected by n co-planar masses. The first non-trivial case of one mass was already investigated by A. Einstein around 1912.
We shall also discuss the problem of gravitational lensing of a point source of light, e.g., a star, by an elliptic galaxy, more precisely the problem of the maximal number of images that one can observe. Under some more or less "natural" assumptions on the mass distribution within the galaxy one can prove that the number of visible images can never be more than four in some cases and six in the other. Interestingly, the former situation can actually occur and has been observed by astronomers. Still there are much more open questions than there are answers.
Tomasz Mrowka : Using instantons to show that Khovanov homology detects the unknot.
- Geometry and Topology ( 114 Views )A couple of years ago Kronheimer and I revisited our work on instantons with codimension two singularities. This leads to knot invariants associated to each partial flag manifold. There invariants are related to Khovanov homology for the case of $\IP1$ and Khovanov-Rozansky homology fo $\IP^n$. We have recently understood how to use the $\IP1$ case to prove that Khovanov homology detects the unknot.
Dave Rose : Why I love cats, and you should too
- Graduate/Faculty Seminar ( 107 Views )Category theory can be described as a general mathematical theory of structures and of systems of structures. Originally developed in the 40's by Saunders Mac Lane and Samuel Eilenberg in the context of algebraic topology, category theory has since grown to serve as both an organizational tool in many areas of mathematics and as a deep theory connecting these areas. The aims of this talk are 3-fold: first, to introduce the basic notions of category theory and to give a wide range of examples; second, to show how abstract results in category theory can influence the way we think about mathematics; finally, to show how a knowledge of some general results in category theory can save us time and effort in our day to day mathematical work. Since I will be starting with the basics, this talk should be accessible to a wide audience. Students who are considering working in algebra, geometry, or topology are particularly encouraged to attend, as are any students who have ever wondered why I love covering the chalkboards of 274F with crazy-looking diagrams or why the word `natural' is the fifth most used word in my vocabulary.
Sourav Chatterjee : Superconcentration
- Probability ( 106 Views )We introduce the term `superconcentration' to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens, the field becomes chaotic under small perturbations and a `multiple valley picture' emerges. Conversely, chaos implies superconcentration. While a few notable examples of superconcentrated functions already exist, e.g. the largest eigenvalue of a GUE matrix, we show that the phenomenon is widespread in physical models; for example, superconcentration is present in the Sherrington-Kirkpatrick model of spin glasses, directed polymers in random environment, the Gaussian free field and the Kauffman-Levin model of evolutionary biology. As a consequence we resolve the long-standing physics conjectures of disorder-chaos and multiple valleys in the Sherrington-Kirkpatrick model, which is one of the focal points of this talk.
Michael Jenista : Global dynamics of switching networks in biology
- Graduate/Faculty Seminar ( 111 Views )The study of biological networks is an increasingly popular area of mathematical research. Many different approaches are applied to answer many different kinds of questions. We ask, "what kinds of behavior are observed in biological switching networks, and how can we produce this behavior?" This is therefore a question of modelling. We start with two different frameworks: boolean and continuous. Both are frequently used to model genetic transcription networks which are examples of switching networks. We then explore several principles of global dynamics that are true in both frameworks. We finish with some current research conjectures and sketches of proposed proofs.
Lan-Hsuan Huang : Constant mean curvature foliations for isolated systems in general relativity
- Geometry and Topology ( 122 Views )We will discuss the existence and uniqueness of the foliation by stable spheres with constant mean curvature for asymptotically flat manifolds satisfying the Regge-Teitelboim condition at infinity. This work generalizes the earlier results of Huisken/Yau, Ye, and Metzger. We will also discuss the concept of the center of mass in general relativity.
Mark Stern : Frommers guide to vector bundles
- Graduate/Faculty Seminar ( 143 Views )I will give an introduction to the analysis, geometry, and topology of vector bundles for a general (i.e. nongeometric) audience. I will range from how vector bundles arise in Math 103 to how we can use partial differential equation techniques to extract interesting physical, algebraic, and topological information from them.
Arthur Szlam : A Total Variation-based Graph Clustering Algorithm for Cheeger Ratio Cuts
- Undergraduate Seminars ( 244 Views )I will discuss a continuous relaxation of the Cheeger cut problem on a weighted graph, and show how the relaxation is actually equivalent to the original problem. Then I will introduce an algorithm which experimentally is very efficient at approximating the solution to this problem on some clustering benchmarks. I will also give a heuristic variant of the algorithm which is faster but often gives just as accurate clustering results. This is joint work with Xavier Bresson, inspired by recent papers of Buhler and Hein, and Goldstein and Osher, and by an older paper of Strang.
Sreekanth Pannala : Multiscale/Multiphysics simulation strategy for gas-solids flow reactors
- Nonlinear and Complex Systems ( 142 Views )Gas-solids chemically reacting flows are omnipresent in many multiphase flow reactors in various industries like Chemical, Fossil and Nuclear. The challenging aspect of modeling these reacting flows are the wide range of both temporal and spatial scales encountered in these systems. The challenge is to accurately account and bridge (as seamlessly as possible) the length and time scales involved in the problem. First, the problem is introduced using biomass gasifier/pyrolyser and nuclear fuel coater with sample results as examples and provide an overview of the various models currently used at the different scales. In particular, the critical role of the granular dynamics in the overall performance of the reactors will be highlighted. The ongoing development of a multiphysics and multiscale mathematics framework for coupling various modeling methods over a range of scales will be presented. The development of a general wavelet-based multiscale methodology called compound wavelet matrix (CWM) for bridging spatial and temporal scales will be reported. Finally, the steps needed to generalize the current methodology for arbitrary heterogeneous chemically reacting flows or other applications involving multiscale/multiphysics coupling will be elucidated. The challenges and opportunities of employing these models for rapid deployment of clean energy solutions based on multiphase flow reactors to the market place will be discussed.
Richard Durrett : Life and Death on a Random Graph
- Undergraduate Seminars ( 240 Views )At the turn of this century it was realized that social and communication networks were best modeled by graphs that were "small worlds" and/or had power law degree distributions. I will discuss two examples. The first is a situation where physicist's mean field arguments give the wrong answer about the spread of epidemic. The second, inspired by a gypsy moth outbreak in the late 1980s in NY leads to chaotic behavior. I will concentrate on what is true rather than why, so the talk should be accessible to a wide audience.
Daniella E. Raveh : Nonlinear Dynamics of Aeroelastic Airfoil Systems in Buffeting Flows
- Nonlinear and Complex Systems ( 142 Views )Transonic flows over airfoils at certain combinations of Mach numbers and steady mean angle of attack exhibit buffet; a phenomenon of large shock-wave oscillations due to flow separation and vortex shedding at a characteristic flow frequency. Buffet may occur even when the airfoil does not move. The seminar will present two recent studies of numerical simulations of an airfoil that a) undergoes prescribed harmonic oscillations, and b) is suspended by a spring in transonic buffeting flows. Both studies focus on the nonlinear interaction between the two oscillatory systems, namely the buffeting flow and the oscillating airfoil. Flow simulations of prescribed airfoil motions (using a Navier-Stokes turbulent flow solver) reveal a lock-in phenomenon. Certain combinations of amplitude and frequency of a prescribed airfoil oscillatory motion caused the buffet flow oscillations to lock into the prescribed frequency. The combinations of prescribed frequencies and amplitudes that cause lock-in present an .Arnold tongue. structure. There is a broad analogy between this flow phenomenon and the flow field of the Von Karman vortex street found behind a cylinder with the cylinder undergoing a prescribed oscillation. Flow simulations of an airfoil that is suspended on a spring reveal three distinct response characteristics, depending on the relationship of the elastic system.s natural frequency to the buffet frequency, and on the system.s mass ratio (the structural to fluid mass ratio). Elastic systems with natural frequencies that are lower than the buffet frequency exhibit a single-frequency response, with a frequency that is shifted form the buffet frequency towards the elastic natural frequency as the mass ratio is decreased (and the magnitude of the elastic response increases). On the other hand, an elastic system with a natural frequency that is the same as the buffet frequency exhibits resonance. Finally, elastic systems with natural frequencies that are higher than the buffet frequency exhibit a response with two distinct frequencies, that of the buffet and that of the elastic natural frequency. As long as the pitch amplitudes are small, the response is mostly at the buffet frequency. As the pitch amplitudes increase there is more power in the elastic natural frequency, and less in the buffet frequency. As the pitch amplitudes further grow, the response is in the elastic natural frequency solely, and the buffet frequency vanishes. To the best of the authors. knowledge the nonlinear dynamics of elastic systems in buffeting flows has not been reported previously. The authors are interested to learn whether similar phenomena are known in other research communities.
Mark Jackson : Superstring Cosmology: New Physics in the Sky
- String Theory ( 203 Views )Striking advances in observational cosmology over the past two decades have ushered in a golden era in cosmology, where our focus has turned from what the universe is made of to why it has the form we observe. The leading theory capable of answering such a question, Superstring Theory, does not appear capable of being tested using conventional accelerator-based experiments, forcing us to be more creative in our goal to verify or dismiss it. Focusing on brane inflation as a string theory-inspired model of inflationary cosmology, I will review how the cosmic microwave background (CMB) will provide a deluge of high- precision data into otherwise inaccessible energy scales. These data include possible "Transplanckian" signatures in the power spectrum, indications of variable sound speed or extra dimensions in non- Gaussianity, or constraining the inflation model parameter space using polarization. I then describe how the production of cosmic (super)strings in brane inflation would provide an additional means to verify superstring theory, and which would yield much detailed information about the underlying theory parameters.
Arend Bayer : Stability conditions on the local P2 revisited
- Algebraic Geometry ( 127 Views )We will give a description of the space of Bridgeland stability conditions on the derived category of sheaves on P2 sitting inside a compact Calabi-Yau threefold. We will discuss its fractal-like boundary, its relation with the group of auto-equivalences, with mirror symmetry, and with counting invariants for both P2 and the quotient stack [C3/Z_3]. This is joint work with E. Macri.
Mauro Maggioni : Parametrizations of manifolds via Laplacian eigenfunctions and heat kernels
- Geometry and Topology ( 97 Views )We present recent results that show that for any portion of a compact manifold that admits a bi-Lipschitz parametrization by a Euclidean ball one may find a well-chosen set of eigenfunctions of the Laplacian that gives a bi-Lipschitz parametrization almost as good as the best possible. A similar, and in some respect stronger result holds by replacing eigenfunctions with heat kernels. These constructions are motivated by applications to the analysis of the geometry of data sets embedded in high-dimensional spaces, that are assumed to lie on, or close to, a low-dimensional manifold. This is joint work with P.W. Jones and R. Schul.