Chris Wiggins : Data science @ the new york times
- Presentations ( 200 Views )Data science @ the new york times - what is data science? - where did it come from? - what does it mean at NYT? - what does it mean at Columbia? This is a public lecture targeted at a general audience. Undergraduates are particularly encouraged to come.
Wolfgang Gaim : Semiclassical approximations to quantum mechanical equilibrium distributions
- Presentations ( 213 Views )In his 1932 paper, Eugene Wigner introduced the now famous Wigner function in order to compute quantum corrections to classical equilibrium distributions. We show how to extend this program and compute semiclassical approximations to quantum mechanical equilibrium distributions for slow, semiclassical degrees of freedom coupled to fast, quantum mechanical degrees of freedom. The main examples are molecules and electrons in crystalline solids. Where we will focus on the thermodynamics of the Hofstadter model as an application of the general results. The semiclassical formulas contain, in addition to quantum corrections similar to those of Wigner, also modifications of the classical Hamiltonian system used in the approximation: The classical energy and the Liouville measure on classical phase space turn out to have non-trivial-expansions in the semiclassical parameter. This talk is based on joint work with Stefan Teufel.
Dr. Andrew Barnes : Risk Measurement and Capital Allocation for large loan portfolios
- Presentations ( 198 Views )Calculation of portfolio loss distributions is an important part of credit risk management in all large banking institutions. Mathematically, this calculation is tantamount to efficiently computing the probability distribution of the sum of a very large number of correlated random variables. Typical Monte Carlo aggregation models apply brute force computation to this problem and suffer from two main drawbacks: lack of speed and lack of transparency for further credit risk analysis. I will describe an attempt to ameliorate these drawbacks via an asymptotic probabilistic method based on the Central Limit Theorem. I will next describe capital allocation, a process of attributing risk to individual transactions or subportfolios of a given portfolio. In so doing, I will state axioms for coherent risk measures. These axioms place the notion of risk measurement and diversification on a firm mathematical foundation. I will then describe axioms for capital allocation via coherent risk measures, and illustrate the ideas with efficient computational formulae for allocating capital based on a couple of commonly used risk measures. In the course of this talk, which will be geared towards graduate students, I will attempt to give a flavor of industrial research and role of applied mathematics in industry.
Lan-Hsuan Huang : Positive mass theorems and scalar curvature problems
- Presentations ( 207 Views )More than 30 years ago, Schoen-Yau and later Witten made major breakthroughs in proving the positive mass theorem. It has become one of the most important theorems in general relativity and differential geometry. In the first part of the talk, I will introduce the positive mass theorem and present our recent work that extends the classical three-dimensional results to higher dimensions. In the second part, I will discuss how the observation from general relativity enables us to solve classical geometric problems related to the scalar curvature.
James Bremer : Improved methods for discretizing integral operators
- Presentations ( 203 Views )Integral equation methods are frequently used in the numerical solution of elliptic boundary value problems. After giving a brief overview of the advantages and disadvantages of such methods vis-a-vis more direct techniques like finite element methods, I will discuss two problems which arise in integral equation methods. In both cases, I take a contrarian position. The first is the discretization of integral operators on singular domains (e.g., surfaces with edges and curves with corners). The consensus opinion holds that integral equations given on such domains are exceedingly difficult to discretize and that sophisticated analysis, often specific to a particular boundary value problem, is required. I will explain that, in fact, the efficient solution of a broad class of such problems can be effected using an elementary approach. Exterior scattering problems given on planar domains with tens of thousands of corner points can be solved to 12 digit accuracy on my two year old desktop computer in a matter of hours. The second problem I will discuss is the evaluation of the singular integrals which arise form the discretization of weakly singular integral operators given on surfaces. Exponentially convergent algorithms for evaluating these integrals have been described in the literature and it is widely regarded as a "solved" problem. I will explain why this is not so and describe an approach which yields only algebraic convergence, but nonetheless performs better in practice than standard exponentially convergent methods.
Jayce Getz : An approach to nonsolvable base change
- Presentations ( 196 Views )In the 1970's, inspired by the work of Saito and Shintani, Langlands gave a definitive treatment of base change for automorphic representations of the general linear group in two variables along prime degree extensions of number fields. To give some idea of the depth and utility of his work, one need only remark that some consequences of it were crucial in Wiles' proof of Fermat's last theorem. In this talk we will report on work in progress on base change for automorphic representations of GL(2) along nonsolvable Galois extensions of number fields. We will attempt to explain this assuming only a little algebraic number theory.
Ellen Eischen : L-functions, congruences, and applications
- Presentations ( 166 Views )L-functions, certain meromorphic functions that include the Riemann zeta-function, encode important number-theoretic information. The first part of this talk will focus on some striking properties of special values of the Riemann zeta-function and certain other L-functions (namely, congruences modulo powers of a prime number). In the second part of the talk, I will introduce tools that are useful for studying these congruences. These tools have applications not only to number theory, but also to homotopy theory. This will be a colloquium style talk, intended for a broad audience.
Robert Lipshitz : The Jones polynomial as Euler characteristic
- Presentations ( 189 Views )We will start by defining the Jones polynomial of a knot, and discussing some of its applications. We will then explain a refinement of the Jones polynomial, called Khovanov homology, and give some applications of this refinement. We will conclude by discussing a further refinement, called a Khovanov homotopy type; this part is joint work with Sucharit Sarkar.
Pengzi Miao : Remarks on a Scalar Curvature Rigidity Theorem of Brendle and Marques
- Presentations ( 238 Views )In a recent paper, Brendle and Marques proved that on certain geodesic balls in the standard hemisphere, there does not exist small metric deformations of the standard metric which increase the scalar curvature in the interior and the mean curvature on the boundary. Such a result was motivated by the Euclidean and Hyperbolic positive mass theorems. More interestingly, this result is false on the hemisphere itself, which is shown by Brendle-Marques-Neves' remarkable counter example to the Min-Oo's conjecture. In this talk, we provide a few remarks to Brendle and Marques' theorem. We show that their theorem remains valid on slightly larger geodesic balls; it also holds on certain convex domains; moreover, with a volume constraint imposed, a variation of their theorem holds on the hemisphere. This is a joint work with Luen-Fai Tam.
Jason Mireles-James : Adaptive Set-Oriented Algorithms for Conservative Systems
- Presentations ( 129 Views )We describe an automatic chaos verification scheme based on set oriented numerical methods, which is especially well suited to the study of area and volume preserving diffeomorphisms. The novel feature of the scheme is an iterative algorithm for approximating connecting orbits between collections of hyperbolic fixed and periodic points with greater and greater accuracy. The algorithm is geometric rather than graph theoretic in nature and, unlike existing methods, does not require the computation of chain recurrent sets. We give several example computations in dimension two and three.
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)
Margaret Beck : Using global invariant manifolds to understand metastability in Burgers equation with small viscosity.
- Presentations ( 107 Views )Finding globally stable states can provide useful information about the behavior of solutions to PDEs: for any initial condition, the solution will eventually approach such a state. However, in some cases, the solution can exhibit long transients in its approach to the state. If the transient is long enough, it may be this behavior, rather than the limiting behavior, that is observed in numerical simulations or experiments. This is referred to as "metastability" and has been found, for example, in the 2D Navier-Stokes equations with small viscosity. A similar phenomenon has been seen in Burgers equation, which can be explained using global invariant manifolds. More precisely, it is shown that in terms of similarity, or scaling, variables there exists a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold. Metastability corresponds to a fast transient in which solutions approach this 'metastable' manifold, followed by a slow decay along this manifold, and, finally, convergence to the globally stable state.
Patrick Brosnan : Essential dimension and algebraic stacks
- Presentations ( 128 Views )Essential dimension is an invariant introduced by Buhler and Reichstein to measure how many parameters are needed to define an algebraic object such as a field extension or an algebraic curve over a field. I will describe joint work with Vistoli and Reichstein which studies essential dimension in the case where the algebraic objects are represented by a stack. I will also give examples of applications in the theory of quadratic forms.
Allan Sly : Mixing in Time and Space
- Presentations ( 117 Views )For Markov random fields temporal mixing, the time it takes for the Glauber dynamics to approach it's stationary distribution, is closely related to phase transitions in the spatial mixing properties of the measure such as uniqueness and the reconstruction problem. Such questions connect ideas from probability, statistical physics and theoretical computer science. I will survey some recent progress in understanding the mixing time of the Glauber dynamics as well as related results on spatial mixing. Partially based on joint work with Elchanan Mossel
David Speyer : Matroids and Grassmannians
- Presentations ( 130 Views )Matroids are combinatorial devices designed to encoded the combinatorial structure of hyperplane arrangements. Combinatorialists have developed many invariants of matroids. I will explain that there is reason to believe that most of these invariants are related to computations in the K-theory of the Grassmannian. In particular, I will explain work of mine limiting the complexity of Hacking, Keel and Tevelev's "very stable pairs", which compactify the moduli of hyperplane arrangements. This talk should be understandable both to those who don't know matroids, and to those who don't know K-theory.
Thomas Lam : Total positivity, Toeplitz matrices, and loop groups
- Presentations ( 123 Views )A real matrix is totally nonnegative if every minor in it is nonnegative. The classical Edrei-Thoma theorem classifies totally nonnegative infinite Toeplitz matrices, and is related to problems in representation theory, combinatorics and probability. I will discuss progress towards two variations on this theorem to block-Toeplitz matrices, and to finite Toeplitz matrices. Both of these variations connect the classical theory to loop groups.
Yunliang Yu : FDS
- Presentations ( 148 Views )FDS (fds.duke.edu) is a content management system (CMS) widely used across Duke for schools and departments to effectively maintain their faculty research and teaching related web pages and reports. In this talk we'll cover some fundamentals of FDS and give a short tutorial on the FDS templates. We hope this talk will help everyone (either webmasters, web developers and designers, and FDS group managers, or interested faculty/staff members) to use FDS better.
Andrei Caldararu : The Pfaffian-Grassmannian derived equivalence
- Presentations ( 153 Views )We argue that there exists a derived equivalence between Calabi-Yau threefolds obtained by taking hyperplane sections (of the appropriate codimension) of the Grassmannian G(2,7) and the Pfaffian Pf(7). The existence of such an equivalence has been conjectured in physics for almost ten years, as the two families of Calabi-Yau threefolds are believed to have the same mirror. It is the first example of a derived equivalence between Calabi-Yau threefolds which are provably non-birational.
Dragos Oprea : Theta divisors on moduli spaces of bundles over curves
- Presentations ( 159 Views )The Jacobian of any compact Riemann surface carries a natural theta divisor, which can be defined as the zero locus of an explicit function, the Riemann theta function. I will describe a generalization of this idea, which starts by replacing the Jacobian with the moduli space of higher rank bundles. These moduli spaces also carry theta divisors, described via "generalized" theta functions. In this talk, I will describe recent progress in the study of generalized theta functions.