David Shea Vela-Vick : Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links
- Geometry and Topology ( 98 Views )To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two- component link is the degree of its associated Gauss map from the 2- torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.
Francesco Zamponi : Jamming and hard sphere glasses
- Nonlinear and Complex Systems ( 107 Views )I will review a theory of amorphous packings of hard spheres based on the assumption that these packings are the infinite pressure limit of long-lived metastable glassy states. Technically, the theory makes use of the replica method and of standard liquid theory; it gives predictions on both the structure and the thermodynamics of amorphous states. In dimensions between two and six these predictions can be successfully compared with numerical simulations. I will finally discuss the limit of large dimension, that is relevant for information theory problems, where an exact solution is possible. Ref: G.Parisi and F.Zamponi, J.Chem.Phys. 123, 144504 (2005); arXiv:0802.2180 (to appear on Rev.Mod.Phys.)
Kevin Gonzales : Modeling mutant phenotypes and oscillatory dynamics in the cAMP-PKA pathway in Yeast Cells
- Graduate/Faculty Seminar ( 103 Views )In this talk I will present a ODE model Dr. Schaeffer and I have developed in collaboration with Dr. Magwene of Duke's Department of Biology. The cAMP-PKA pathway is a key signal transduction pathway through which Yeast makes developmental decisions in response to environmental cues. A novel feature of our model is that for a wide range of parameters approach to steady state includes decaying oscillations. I aim to make this talk accessible to everyone and will give an overview of all relevant biology.
Andrea Bertozzi : Geometry based image processing - a survey of recent results
- Applied Math and Analysis ( 101 Views )I will present a survey of recent results on geometry-based image processing. The topics will include wavelet-based diffuse interface methods, pan sharpening and hyperspectral sharpening, and sparse image representation.
Mauro Maggioni : On estimating intrinsic dimensionality of noisy high-dimensional data sets
- Geometry and Topology ( 97 Views )We discuss recent and ongoing work, joint with A. Little, on estimating the intrinsic dimensionality of data sets assumed to be sampled from a low-dimensional manifold in high dimensions and perturbed by high-dimensional noise. This work is motivated by several applications, including machine learning and dynamical systems, and by the limitations of existing algorithms. Our approach is based on a simple tool such as principal component analysis, used in a multiscale fashion, a strategy which has its roots in geometric measure theory. The theoretical analysis of the algorithm uses tools from random matrix theory and exploits concentration of measure phenomena in high-dimensions. The talk will have a tutorial flavour: no previous knowledge of what mentioned above will be required, and several toy examples to build intuition about some measure-geometric phenomena in high-dimensions will be presented.
Kash Balachandran : The Kakeya Conjecture
- Graduate/Faculty Seminar ( 134 Views )In 1917, Soichi Kakeya posed the question: What is the smallest amount of area required to continuously rotate a unit line segment in the plane by a full rotation? Inpsired by this, what is the smallest measure of a set in $\mathbb{R}^n$ that contains a unit line segment in every direction? Such sets are called Kakeya sets, and can be shown to have arbitrarily small measure w.r.t. n-dimensional Lebesgue measure [and in fact, measure zero]. The Kakeya conjecture asserts that the Hausdorff and Minkowski dimension of these sets in $\mathbb{R}^n$ is $n$. In this talk, I will introduce at a very elementary level the machinery necessary to understand what the Kakeya conjecture is asking, and how the Kakeya conjecture has consequences for fields diverse as multidimensional Fourier summation, wave equations, Dirichlet series in analytic number theory, and random number generation. I'll also touch on how tools from various mathematical disciplines from additive combinatorics and algebraic geometry to multiscale analysis and heat flow can be used to obtain partial results to this problem. The talk will be geared towards a general audience.
Nicolaus Tideman : The Structure of the Election-Generating Universe
- Presentations ( 115 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 ( 115 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 ( 115 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 ( 108 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 ( 107 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 ( 112 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 ( 125 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 ( 144 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 ( 249 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.