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.
Steven Sam : Noetherianity in representation theory
- Presentations ( 219 Views )Abstract: Representation stability is an exciting new area that combines ideas from commutative algebra and representation theory. The meta-idea is to combine a sequence of objects together using some newly defined algebraic structure, and then to translate abstract properties about this structure to concrete properties about the original object of study. Finite generation is a particularly important property, which translates to the existence of bounds on algebraic invariants, or some predictable behavior. I'll discuss some examples coming from topology (configuration spaces) and algebraic geometry (secant varieties).
Joseph Rabinoff : From Diophantine equations to p-adic analytic geometry
- Presentations ( 230 Views )A Diophantine equation is a polynomial equation in several variables, generally with integer coefficients, like x3 + y3 = z3. Provably finding all integer solutions of a Diophantine equation is a storied mathematical problem that is easy to state and notoriously difficult to solve. The method of Chabauty--Coleman is one particularly successful technique for ruling out extraneous solutions of a certain class of Diophantine equations. The method is p-adic in nature, and involves producing p-adic analytic functions that vanish on all integer-valued solutions. I will discuss work with Katz and Zureick-Brown on finding uniform bounds on the number of rational points on a curve of fixed genus, defined over a number field, subject to a (conjecturally weak) restriction on its Jacobian. The same technique also makes progress on the uniform Manin-Mumford conjecture on the size of torsion packets on curves of fixed genus.
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.
Jessica Fintzen : Frontiers in Mathematics Lecture 1: Representations of p-adic groups
- Presentations ( 272 Views )The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of certain matrix groups, called p-adic groups. In my talk I will introduce p-adic groups and provide an overview of our understanding of their representations, with an emphasis on recent progress. I will also briefly discuss applications to other areas, e.g. to automorphic forms and the global Langlands program.
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.
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)
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
Roman Vershynin : Randomness in functional analysis: towards universality
- Presentations ( 155 Views )The probabilistic method has redefined functional analysis in high dimensions. Random spaces and operators are to analysis what random graphs are to combinatorics. They provide a wealth of examples that are otherwise hard to construct, suggest what situations we should view as typical, and they have far-reaching applications, most notably in convex geometry and computer science. With the increase of our knowledge about random structures we begin to wonder about their universality. Is there a limiting picture as the dimension increases to infinity? Is this picture unique and independent of the distribution? What are deterministic implications of probabilistic methods? This talk will survey progress on some of these problems, in particular a proof of the conjecture of Von Neumann and Goldstine on random operators and connections to the Littlewood-Offord problem in additive combinatorics.
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.
Eugenia Cheng : PLUM Lecture: How to Bake Pi, Making abstract mathematics palatable
- Presentations ( 248 Views )Mathematics can be tasty! Its a way of thinking, and not just about numbers. Through unexpectedly connected examples from music, juggling, and baking, I will show that math can be made fun and intriguing for all, through hands-on activities, examples that everyone can relate to, and funny stories. I'll present surprisingly high-level mathematics, including some advanced abstract algebra usually only seen by math majors and graduate students. There will be a distinct emphasis on edible examples.
Laura DeMarco : Complex dynamics and potential theory
- Presentations ( 148 Views )I will begin with the basics of the two subjects, with the goal of explaining how each has been used as a method to obtain results in the other. The first half will be devoted to foundational results, dating to the 1930s for potential theory and the 1980s for complex dynamics. The second half will be devoted to more recent developments.
James Nolen : Reaction-Diffusion Fronts in Heterogeneous Media
- Presentations ( 141 Views )Reaction-diffusion equations are used in mathematical models of many physical and biological phenomena involving front propagation and pulse propagation. How do variations in the environment effect these phenomena? In this seminar, I will describe recent progress in understanding how fronts propagate in heterogeneous media. In particular, I will describe properties of generalized traveling waves for one-dimensional reaction-diffusion equations with variable excitation. I also will discuss multi-dimensional fronts in stationary random media, a model relevant to premixed-turbulent combustion. Along the way, I plan to describe interesting topics for future research.
Nicholas Eriksson : Combinatorial methods in evolutionary biology
- Presentations ( 149 Views )My research focuses in three areas of evolutionary biology: the structure of viral populations, the evolution of drug resistance, and phylogenetics. Knowledge of the diversity of viral populations is important for understanding disease progression, vaccine design, and drug resistance, yet it is poorly understood. New technologies (pyrosequencing) allow us to read short, error-prone DNA sequences from an entire population at once. I will show how to assemble the reads into genomes using graph theory, allowing us to determine the population structure. Next, I will describe a new class of graphical models inspired by poset theory that describe the accumulation of (genetic) events with constraints on the order of occurrence. Applications of these models include calculating the risk of drug resistance in HIV and understanding cancer progression. Finally, I'll describe a polyhedral method for determining the sensitivity of phylogenetic algorithms to changes in the parameters. We will analyze several datasets where small changes in parameters lead to completely different trees and see how discrete geometry can be used to average out the uncertainty in parameter choice.
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.
Troy Schaudt : Mathematica 11 in Education and Research
- Presentations ( 271 Views )This technical talk will show live calculations in Mathematica 11 and other Wolfram technologies relevant to courses and research. Specific topics include: * Enter calculations in everyday English, or using the flexible Wolfram Language * Visualize data, functions, surfaces, and more in 2D or 3D * Store and share documents locally or in the Wolfram Cloud * Use the Predictive Interface to get suggestions for the next useful calculation or function options * Access trillions of bits of on-demand data * Use semantic import to enrich your data using Wolfram curated data * Easily turn static examples into mouse-driven, dynamic applications * Access 10,000 free course-ready applications * Utilize the Wolfram Language's wide scope of built-in functions, or create your own * Get deep support for specialized areas including machine learning, time series, image processing, parallelization, and control systems, with no add-ons required Current users will benefit from seeing the many improvements and new features of Mathematica 11 (https://www.wolfram.com/mathematica/new-in-11/), but prior knowledge of Mathematica is not required.