Combinatorial transgressions are secondary invariants of a space admitting triangulations. They arise from subdivisions and are analogous to transgressive forms such as those in Chern-Weil theory. In this talk, I characterize transgressions that are path-independent of subdivision sequence. The result is obtained by using a cohomology on posets that is shown to be equivalent to higher derived functors of the inverse (or projective) limit over the opposite poset.

On a compact 4-manifold, every self-dual connection and every anti-self-dual connection minimizes the Yang-Mills energy. In this talk, I will answer the converse question for compact homogeneous 4-manifolds. I will also survey related stability results in other dimensions.

Let $\Omega$ and $\tilde{\Omega}$ be domains in the hyperbolic plane with smooth boundary. Assume that both domains are uniformly convex, and have the same area. We show that there exists an area-preserving, orientation-preserving diffeomorphism $f: \Omega \to \tilde{\Omega}$ such that the graph of $f$ is a minimal surface in $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, we show that the set of all such diffeomorphisms is parametrized by the circle.

"Finding and Measuring Stratified Spaces Using Persistent Intersection and Local Homology"

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.

"Numerical Approximation of Layer Potentials Along Curve Segments"

We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. Such nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry; however, despite their ubiquity, the intrinsic property of nondegeneracy has not seen much detailed study. We prove that every curve of genus $g \geq 4$ over an algebraically closed field is nondegenerate in the above sense. More generally, let $\mathcal{M}_g^{\textup{nd}}$ be the locus of nondegenerate curves inside the moduli space of curves of genus $g \geq 2$. Then we show that $\dim \mathcal{M}_g^{\textup{nd}} = \min(2g+1,3g-3)$, except for $g=7$ where $\dim \mathcal{M}_7^{\textup{nd}} = 16$; thus, a generic curve of genus $g$ is nondegenerate if and only if $g \geq 4$

The Boltzmann equation governs the dynamics of a dilute gas. In this talk, I will address the L^p-stability problem of the Boltzmann equation near vacuum and a global Maxwellian. In a close-to-vacuum regime, I will explain the nonlinear functional approach motivated by Glimm's theory in hyperbolic conservation laws. This functional approach yields the uniform L^1-stability estimate. In contrast, in a close-to-global maxwellian regime, I will present the L^2-stability theory which establishes the uniform L^2-stability of several classical solutions.

We review recent developments in the control of deterministic and stochastic nonlinear dynamics by time-delayed feedback methods [1]. We point out how to overcome the alleged odd number limitation for unstable periodic orbits, and discuss the control of complex chaotic or noise-induced space-time patterns. Our findings are applied to a selection of models ranging from semiconductor nanostructures, like resonant-tunneling diodes [2], to neural systems. [1] E. Sch{\"o}ll and H.G. Schuster (Eds.): Handbook of Chaos Control (Wiley-VCH, Weinheim, 2008), second completely revised and enlarged edition. [2] E. Sch{\"o}ll, Nonlinear spatio-temporal dynamics and chaos in semiconductors (Cambridge University Press, Cambridge, 2001).

The nephron is the functional unit of the kidney. The flow rate in each nephron is regulated, in part, by tubuloglomerular feedback, a negative feedback loop. In some parameter regimes, this feedback system can exhibit oscillations that approximate limit-cycle oscillations. However, nephron flow in spontaneously hypertensive rats (SHR) can exhibit highly irregular oscillations similar to deterministic chaos. We used a mathematical model of tubuloglomerular feedback (TGF) to investigate potential sources of the irregular oscillations and the associated complex power spectra in SHR. A bifurcation analysis of the TGF model equation was performed by finding roots of the characteristic equation, and numerical simulations of model solutions were conducted to assist in the interpretation of the analysis. Four potential sources of spectral complexity in SHR were identified: (1) bifurcations that produce qualitative changes in solution type, leading to multiple spectrum peaks and their respective harmonic peaks; (2) continuous lability in delay parameters, leading to broadening of peaks and their harmonics; (3) episodic lability in delay parameters, leading to multiple peaks and their harmonics; and (4) coupling of small numbers of nephrons, leading to broadening of peaks, multiple peaks, and their harmonics. We conclude that the complex power spectra in SHR may be explained by the inherent complexity of TGF dynamics, which may include solution bifurcations, variation in TGF parameters, and coupling between small numbers of neighboring nephrons.

I will describe some recent results that concern various aspects of the mixing properties of a strong incompressible flow acting together with a diffusion. In particular, we will discuss the short-time decay of solutions of the corresponding initial value problem, asymptotics of the principle Dirichlet eigenvalue and the behavior of the explosion threshold in the Zeldovich problem when the incompressible flow is strong. When the flow is prescribed, the "enhancement" of these characteristics comes from the geometric properties of the flow. We will also show that flows arising from the Stokes-Bousisnesq problems possess these "enhancement" features.

Class groups of cyclotomic fields have long been of central interest in number theory. We consider elements of these class groups that arise as values of a cup product pairing on cyclotomic units. These pairing values yield information on a wealth of algebraic objects, but any analytic interpretation of them was heretofore unknown. We will describe how, conjecturally, modular representations can be used to relate the pairing values to p-adic L-values of cusp forms.

Asymmetric exclusion processes, with periodic or open boundaries, have been studied extensively in the mathematics and statistical physics communities, as paradigmatic models for stochastic particle transport far from equilibrium. Though significant progress was made only recently, the original model was actually introduced decades ago to model protein synthesis1. In this talk, I will describe recent efforts to develop a comprehensive theory for protein synthesis, building on asymmetric exclusion processes with extended objects, modeling ribosomes covering multiple codons. We discuss the effects of local hopping rates and ribosome size on density profiles and particle currents. The latter translate directly into synthesis rates for the corresponding protein. Some intriguing results for real genes will be presented. 1C.T. MacDonald, J.H. Gibbs and A.C. Pipkin, Kinetics of biopolymerization on nucleic acid templates, Biopolymers,6 1 (1968); C.T. MacDonald and J.H. Gibbs, Concerning the kinetics of polypeptide synthesis on polyribosomes, Biopolymers, 7, 707, (1969).

Our numerical computation group has studied several problems in computational finance over the past decade. One that we've looked at recently is the pricing of "collateralized debt obligations" (CDOs). The market for CDOs has grown rapidly to over US$1 trillion annually in 2006, since the appearance of JP Morgan's Bistro deal, the first synthetic CDO, in December 1997. Much of the turmoil in the financial markets recently has been due to such credit derivatives. As this suggests, there are still many open problems associated with the pricing and hedging of these complex financial instruments. I will talk briefly about some work that we have done recently in this area.

3+1 dimensional black holes have spherical topology, but in higher dimensions this is no longer true. In this talk I will explain the preceding statement and show a construction, in terms of Riemannian geometry, of outermost apparent horizons with nonspherical topology.

In vertebrates, ovulation is triggered by a surge of luteinizing hormone (LH) from the pituitary. The precise mechanism by which rising estradiol (E2) from the ovaries initiates the LH surge in the human menstrual cycle remains a mystery. The mystery is due in part to the bimodal nature of estradiol feedback action on LH secretion, and in part to disagreement over the site of the feedback action.

We will describe a differential equations model in which the mysterious bimodality of estradiol action arises from the electrical connectivity of a network of folliculo-stellate cells in the pituitary. The mathematical model is based as closely as possible on current experimental data, and is being used to design and conduct new experiments. No biological background will be assumed.

J. Berge discovered a simple condition on a knot, K, in the three-sphere which ensures that Dehn surgery on K yields a lens space. It is an open conjecture, known as the Berge conjecture, that any knot on which one can perform surgery and obtain a lens space satisfies his condition. I will discuss a strategy, developed jointly with Ken Baker and Eli Grigsby by which the knot Floer homology invariants of Ozsvath, Szabo, and Rasmussen could be used to prove this conjecture.

Many classes of mathematical objects turn out to be classified in the same way --- two infinite series and 3 "exceptional" objects. These include symmetries of 3-dimensional solids, rigid singularities, certain types of Lie algebras, positive definite even intersection forms, etc. Discovering why such classes should have the same classification has led to many beautiful ideas and observations. I will give a review of some of the basic ideas (assuming very little in the way of prerequisites) and I may have time to say why string theory has been important in this context.

(joint with Mboyo Esole) String theory considerations lead to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. After giving a very sketchy idea of the physics arguments leading to this identity, I will present a Chern class identity which confirms it, generalizing it to arbitrary dimension and to varieties that are not necessarily Calabi-Yaus. The relevant loci are singular, and this plays a key role in the identity.

A Calabi-Yau orbifold is locally modeled on C^n/G with G a finite subgroup of SU(n). If the singularity is isolated, then the crepant resolution (if it exists) is an ALE manifold, for which index-type results are well known. However, most of the time the singularity is not isolated, and for the corresponding crepant resolution there is no index theorem so far. In this talk, I present the first step towards obtaining such a result: I will introduce the class of iterated cone-edge singular manifolds and the corresponding quasi-asymptotically conical spaces (of which orbifolds and their resolutions of singularities are examples), and build-up the general set-up for studying Fredholm properties of geometrical elliptic operators on these spaces. This is joint work with Rafe Mazzeo.

Coupling between free fluid flow and flow through porous media is important in many industrial applications, such as filtration, underground water flow in hydrology, oil recovery in petroleum engineering, fluid flow through body tissues in biology, to name a few.

Stokes flows appear in many applications where the fluid viscosity is high and/or the velocity and length scales are small. The flow through a porous medium can be described by Darcy's law. A region that contains both requires a careful coupling of these different systems at the interface through appropriate boundary conditions.

Our objective is to develop a method based on the boundary integral formulation for computing the fluid/porous medium problem with higher accuracy using fundamental solutions of Stokes and Darcy's equations. We regularize the kernels to remove the singularity for stability of numerical calculations and eliminate the largest error for higher accuracy.

How does the structure of a network effect how randomness propagates through it? This is a fun mix of probability, differential equations, and graph theory.

We propose the following model of a random graph on $n$ vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair ij with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by picking a random point X from F and defining a graph on n vertices whose edges are pairs ij for which X_{ij} \le p. The standard Erd\H{o}s-R\'{e}nyi model is the special case when F is uniform on the 0-1 unit cube. We determine basic properties such as the connectivity threshold for quite general distributions. We also consider cases where the X_{ij} are the edge weights in some random instance of a combinatorial optimization problem. By choosing suitable distributions, we can capture random graphs with interesting properties such as triangle-free random graphs and weighted random graphs with bounded total weight. This is joint work with Alan Frieze (CMU) and Juan Vera (Waterloo). The talk will be self-contained and no prior knowledge of random graphs is assumed.

In this series of three lectures we will describe positivity conditions on Riemannian metrics including the classical conditions of positive sectional, Ricci, and scalar curvature. We will discuss open problems and recent progress including our recent proof of the differentiable sphere theorem (joint with Simon Brendle). That proof employs the Ricci flow, so we will spend some time explaining that technique. Finally we will discuss problems related to positive scalar curvature including some high dimensional issues which occur in that theory. If time allows we will describe recent progress on black hole topologies. These lectures, especially the first two, are intended for a general audience.

In this series of three lectures we will describe positivity conditions on Riemannian metrics including the classical conditions of positive sectional, Ricci, and scalar curvature. We will discuss open problems and recent progress including our recent proof of the differentiable sphere theorem (joint with Simon Brendle). That proof employs the Ricci flow, so we will spend some time explaining that technique. Finally we will discuss problems related to positive scalar curvature including some high dimensional issues which occur in that theory. If time allows we will describe recent progress on black hole topologies. These lectures, especially the first two, are intended for a general audience.