Kirsten Wickelgren : An arithmetic count of the lines on a cubic surface
- Presentations ( 292 Views )A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. By contrast, over the real numbers, the number of real lines depends on the surface. A classification was obtained by Segre, but it is a recent observation of Benedetti-Silhol, Finashin-Kharlamov, Horev-Solomon and Okonek-Teleman that a certain signed count of lines is always 3. We extend this count to an arbitrary field k using an Euler number in A1-homotopy theory. The resulting count is valued in the Grothendieck-Witt group of non-degenerate symmetric bilinear forms. (No knowledge of A1-homotopy theory will be assumed in the talk.) This is joint work with Jesse Kass.
Troy Schaudt : Mathematica 11 in Education and Research
- Presentations ( 292 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.
Florian Naef : A real description of brackets and cobrackets in string topology
- Presentations ( 289 Views )Let M be a manifold with non-vanishing vectorfield. The homology of the space of loops in M carries a natural Lie bialgebra structure described by Sullivan as string topology operations. If M is a surface, these operations where originally defined by Goldman and Turaev. We study formal descriptions of these Lie bialgebras. More precisely, for surfaces these Lie bialgebras are formal in the sense that they are isomorphic (after completion) to their algebraic analogues (Schedler's necklace Lie bialgebras) built from the homology of the surface. For higher dimensional manifolds we give a similar description that turns out to depend on the Chern-Simons partition function.
This talk is based on joint work with A. Alekseev, N. Kawazumi, Y. Kuno and T. Willwacher.
David Shea Vela-Vick : The equivalence of transverse link invariants in knot Floer homology
- Presentations ( 259 Views )The Heegaard Floer package provides a robust tool for studying contact 3-manifolds and their subspaces. Within the sphere of Heegaard Floer homology, several invariants of Legendrian and transverse knots have been defined. The first such invariant, constructed by Ozsvath, Szabo and Thurston, was defined combinatorially using grid diagrams. The second invariant was obtained by geometric means using open book decompositions by Lisca, Ozsvath, Stipsicz and Szabo. We show that these two previously defined invariant agree. Along the way, we define a third, equivalent Legendrian/transverse invariant which arises naturally when studying transverse knots which are braided with respect to an open book decomposition.
Pengzi Miao : Remarks on a Scalar Curvature Rigidity Theorem of Brendle and Marques
- Presentations ( 256 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.
Joseph Rabinoff : From Diophantine equations to p-adic analytic geometry
- Presentations ( 254 Views )A Diophantine equation is a polynomial equation in several variables, generally with integer coefficients, like x^{3} + y^{3} = z^{3}. 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.
Wolfgang Gaim : Semiclassical approximations to quantum mechanical equilibrium distributions
- Presentations ( 227 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.
Robert Ghrist : Sheaves and Sensors
- Presentations ( 224 Views )This work is motivated by a fundamental problem in sensor networks -- the need to aggregate redundant sensor data across a network. We focus on a simple problem of enumerating targets with a network of sensors that can detect nearby targets, but cannot identify or localize them. We show a clear, clean relationship between this problem and the topology of constructable sheaves. In particular, an integration theory from sheaf theory that uses Euler characteristic as a measure provides a computable, robust, and powerful tool for data aggregation.
James Bremer : Improved methods for discretizing integral operators
- Presentations ( 218 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.
Robert Lipshitz : The Jones polynomial as Euler characteristic
- Presentations ( 206 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.
Andrei Caldararu : The Pfaffian-Grassmannian derived equivalence
- Presentations ( 175 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.
Ben Weinkove : Symplectic forms, Kahler metrics and the Calabi-Yau equation
- Presentations ( 171 Views )Yau's theorem on Kahler manifolds states that there exists a unique Kahler metric in every Kahler class with prescribed volume form. This has many applications in complex geometry. I will discuss symplectic manifolds. In a different direction, I will talk about the problem of existence of constant scalar curvature Kahler metrics, which can also be considered a generalization of Yau's theorem.
Nicholas Eriksson : Combinatorial methods in evolutionary biology
- Presentations ( 166 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.
Laura DeMarco : Complex dynamics and potential theory
- Presentations ( 165 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.
Yunliang Yu : FDS
- Presentations ( 163 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.
James Nolen : Reaction-Diffusion Fronts in Heterogeneous Media
- Presentations ( 160 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.
Patrick Brosnan : Essential dimension and algebraic stacks
- Presentations ( 142 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.
Henrique Bursztyn : Reduction in generalized complex geometry
- Presentations ( 139 Views )Generalized complex structures, introduced by Hitchin in 2003, interpolate between symplectic and complex structures. In this talk, I will discuss a reduction procedure unifying symplectic reduction and holomorphic quotients. In general, this reduction procedure presents new features that will be illustrated in examples (e.g., the generalized reduction of a symplectic structure can be complex, and 3-form twists can appear in the quotient). If time permits, I will also discuss a super-geometric viewpoint to generalized reduction.
Thomas Lam : Total positivity, Toeplitz matrices, and loop groups
- Presentations ( 138 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.
Allan Sly : Mixing in Time and Space
- Presentations ( 131 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