We will discuss techniques for approximating a point in high-dimensional Euclidean space which is close to a known low-dimensional compact submanifold when only a compressed linear sketch of the point is available. More specifically, given a point, x, close to a known submanifold of R^D, we will consider linear measurement operators, M: R^D -> R^m, which have associated nonlinear inverses, A: R^m -> R^D, so that || x - A(Mx) || is small even when m << D. Both the design of good linear operators, M, and the design of stable nonlinear inverses, A, will be discussed. An algorithmic implementation of a particular nonlinear inverse will be presented, along with related stability bounds for the approximation of manifold data.

Categorification can be viewed as the process of lifting scalar and polynomial invariants to homology theories having those invariants as (graded) Euler characteristics. In this talk, we will discuss categorification in general and as manifested in specific examples (ie Khovanov homology and knot Floer homology). Examples will be given showing how the categorified invariants are stronger and often more useful than the original invariants. I will motivate categorification using familiar constructions from (very basic) topology. It is my hope that this will make the discussion accessible to a wide audience. No prior knowledge of knot theory or category theory needed!

Langlands' beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the Langlands $L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal automorphic representation of $\GL_n(\A_F)$ where $F$ is a global field.

TBA

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.

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.

Given a metric space and a positive scale parameter, the Vietoris-Rips simplicial complex has a vertex for each point in the metric space, and contains a set of vertices as a simplex if its diameter is less than the scale parameter. I will introduce these complexes and survey how they are used: originally to define a homology theory for metric spaces, later to study hyperbolic groups, and more recently to analyze high-dimensional data. A theorem of Jean-Claude Hausmann states that if the metric space is a Riemannian manifold and if the scale parameter is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. What happens for larger scale parameters? We will see that as the scale parameter increases, the Vietoris-Rips complex of the circle obtains the homotopy type of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible.

In this talk we look at a specific problem in probability related to the stochastic versions of the Burgers' and Navier-Stokes equations, and the path taken to construct sufficient assumptions to prove the desired properties, specifically the existence of an invariant distribution. This talk covers material in Stochastic Differential Equations and Stochastic Partial Differential Equations, but also in Real Algebraic Geometry and Perturbation Theory.

In the first part of this talk I will review the basic properties of the Yamabe problem on compact manifolds and recall some well-known results of LeBrun concerning the computation of the Yamabe invariant in dimension four. In the second part I will suggest some possible generalizations for finite volume 4-manifolds. Hopefully some grad student will want to resume this work were I left it...

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on~$\Gamma_0(N)$ to a simpler function of $k$ and~$N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on~$\Gamma_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found to not be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input. (Joint work with Greg Martin.)

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.

The theory of theta functions, which are defined by certain Fourier series, was developed by great mathematicians like Jacobi and Riemann. Among the numerous applications of this theory are certain results in projective geometry pertaining to complex tori. In this talk, we will focus on the 1-dimensional case and briefly discuss the higher-dimensional story towards the end.

Oh rhythm of my heart is beating like a drum...... with the words "I love you" rolling off my tongue....... No never will I roam for I know my place is home..... where the ocean meets the sky...... I'll be sailing.....

Come learn about a whole new perspective on PDE and watch the very entertaining video "Outside In."

The human heart is a multi chambered organ using valves and compression to create pressure heads, which drive blood flow. Some organisms don't share this type of heart structure, namely, very small invertebrates and embryonic vertebrates. We begin by discussing scaling in fluid dynamics and why merely existing at such a small scales can cause difficulty in effectively transporting fluid. We then move to describing effective pumping mechanisms in valveless tubular hearts and how numerical modeling can help understand how these organisms are able to transport fluid, despite their limitations. Once we have a numerical framework to investigate these organisms we can then begin to add more biological structure in our models and begin to investigate optimal regimes of transport.

The advection-diffusion equation commonly arises in the tracking of substances that are carried by and diffuse in a fluid. My interest lies in models of biological flows, where the equation is defined on domains representing the lumen of blood vessels, tubules, or ducts, which owing to its complex geometry and deformation, has a significant impact on the underlying substance that cannot be ignored. In the talk, I will go through the basic steps involved in the development of a numerical method for the solution of the advection-diffusion equation on such domains with an emphasis on domains with moving boundaries. To handle the difficulties introduced by the boundary, the proposed method will follow the finite volume formulation.

What do eigenvalues have to do with geometry and topology? The first part of the talk will provide a few answers to that very broad question, including a discussion of the Euler characteristic from a spectral theory perspective. The second part of the talk will be a brief introduction to my research in analytic torsion, a topological invariant defined in terms of eigenvalues. In particular I'll explain some similarities and differences between analytic torsion and Euler characteristic.

Do you ever wanted to know how many lines in 3−space intersect 4 given random lines ? (Answer: 2.) One way to prove this is to do explicit computations in the cohomology of the Grassmannian of lines in the projective space. But interestingly enough, one can also use Representation Theory, or symmetric functions (Schur polynomials), to answer this question. The aim of this talk is to present the basics of Schubert Calculus, as seen from the cohomological point of view. I will define Schubert varieties in Grassmannians, and discuss about how they intersect. The final goal is to show that 2 = 1+ 1 (and I may also use Knutson’s puzzles for another proof of this).

In a chemical reaction system, the concentrations of chemical species evolve in time, governed by the polynomial differential equations of mass-action kinetics. This talk provides an introduction to chemical reaction network theory, and gives an overview of algebraic and combinatorial approaches to determining whether a chemical reaction network admits multiple steady states. In general, establishing the existence of (multiple) steady states is challenging, as it requires the solution of a large system of polynomials with unknown coefficients. However, for networks that have special structure, various easy criteria can be applied. This talk will highlight results from Deficiency Theory (due to Feinberg), criteria for multistationarity for chemical reaction systems whose steady states are defined by binomial equations (in joint work with Carsten Conradi, Mercedes Pérez Millán, and Alicia Dickenstein), and a classification of small multistationary chemical reaction networks (in joint work with Badal Joshi).

Optimization problems in material science often require quickly varying composites of materials. In this talk, basic homogenized properties of composites and laminates are discussed. The basic theory is applied to construct and optimize interesting structures, such as field rotators and conducting wheels. The talk will be accessible to everyone.

In the 19th century, people began to study mechanical systems in which motion in a configuration space was constrained by 'no slip' conditions, such as, for example, a wheel or a ball rolling on a plane without slipping. It was immediately noticed that there were many cases in which these 'rolling' constraints did not prevent one from being able to join any two points in a configuration space by an admissible path, and these situations were called 'non-holonomic'. The notion of 'holonomy' arose as a way to quantify and study these 'non-holonomic' systems, and it has turned out to be very fruitful, with many applications in differential geometry and mathematical physics as well as in practical mechanics problems (such as figuring out how to use robot hands to manipulate 3-dimensional objects). In this talk, I'll introduce the ideas that led to the development of the concept of holonomy, show how some simple examples are computed, and describe how even very simple systems, such as a convex surface rolling over another surface without slipping or twisting, can lead to some surprising and exceptional geometry. No expertise in differential geometry will be assumed; if you are comfortable with vector calculus, you can enjoy the talk.

This talk is directed to anyone who cares about anything, at all levels. In particular, it will be a soft introduction to exterior differential systems (EDS). EDS is often associated with differential geometry, but it is really just a language for understanding the solution space of differential equations. The EDS viewpoint is temporarily mind-bending, but its concise and clean description of integrability, from conservation laws to geometric invariants, justifies the initial cramps.

The law of quadratic reciprocity was conjectured by Euler and first proved in full generality by Gauss. I will not prove quadratic reciprocity but I will discuss it in the context of the general reciprocity law due to Emil Artin. I will then explain how Langlands's program is a nonabelian generalization of this. If there is time, I will indicate how my work fits into this program.

Nowadays researchers encounter high dimensional data that arises in a variety of forms such as digital images, videos, and hyperspectral images. How to efficiently and effectively modeling such data sets has become an active research topic. A common model is to approximate such data by a mixture of affine subspaces. In this talk I will present a fast and accurate algorithm that can solve this problem in full generality, addressing both theoretical and applied issues. If time permits, I will also talk about the use of nonlinear models. This is joint work with Mauro Maggioni.

Problems in computational geometry, such as parameterizing a surface or computing shape deformation under geometric constraints, pose various challenges. I will give brief overview of related problems, highlighting the link between discrete differential geometry, optimization and computer graphics. Then, we will see how convex optimization can be used to approximate a specific class of geometric problems, that include shape mapping, bending a cube (https://youtu.be/iOwPGG5-54Q) and perhaps matching lemur teeth.