Suppose ball-shaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data (i.e. Cech complexes) as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends on more than just the fibrewise homotopy type of the region covered by sensors. In the setting of planar sensors that also measure weak rotation information, we provide necessary and sufficient conditions for the existence of an evasion path, and we pose an open question concerning Cech and alpha complexes. Joint with Gunnar Carlsson.

We will discuss recent work on computing angle preserving mappings (a.k.a. conformal mappings) using linear methods. We will begin with an intro/reminder on what these mappings are, and why would one to compute them. Then we will discuss the results themselves which show that when choosing a good target domain, computation of angle preserving mappings can be made linear in the sense that (i) They are a solution of a linear PDE (ii) They can be approximated by solving a finite dimensional linear system and (iii) the approximates are themselves homeomorphisms and "discrete conformal".

This will be a wide ranging discussion on how geometric analysis may play an important role in understanding dark matter and the structure of galaxies.

Is it true that two polygons in the plane have the same area if and only if they can be decomposed into congruent polygons? What about in three and higher dimensions? And what about the analogous questions for polygons in the hyperbolic plane and polyhedra in higher dimensional hyperbolic spaces? Some aspects of the subject are elementary, while others involve Abel's dilogarithm and arcane subjects, such as algebraic K-theory. This talk will be largely elementary, and fun.

Mapping class groups are topological objects which can be used to describe the continuous symmetries of a surface. On the other hand, every compact orientable surface has a moduli space, a complex variety whose points parametrize all of its inequivalent complex structures. These concepts turn out to be closely related. In this talk, we'll cover the basics of both mapping class groups and moduli of Riemann surfaces, as well as explore their relationship.

THIS JUST IN - An Abstract: I plan to speak about the good old Calabi Conjecture, and its beautiful solution by Yau, that gave gim the Fields Medal. I will start with some background material, and see how far we can get into the proof!

The ear from the outside in. Eardrum, middle ear, cochlea, 8th nerve, brainstem, cortex. What happens anyway when you listen to Mozart or Van Halen? How do pressure waves become electrical signals? What happens next? Is there deep mathematics in the auditory system? And what are those carteliginous things doing flapping in the breeze on the side of your head? Who says an abstract has to have declarative sentences? Will some of these questions be answered? Come and see!

Simply put, complex Hadamard matrices are scaled unitary matrices with entries drawn from unit complex numbers. They appear as an essential ingredient in quantum information theory and their real members have deep connections to finite geometry and number theory. For us, in this talk, they will be the fixed points of both discrete and continuous dynamical systems. We begin by introducing complex Hadamards and some essential preliminaries. We then discuss a discrete-time dynamical system which can be used to generate complex Hadamards as well as closely related objects known as mutually unbiased bases. Finally, we construct a continuous system whose fixed points are complex Hadamards and exploit classical results from dynamical systems theory to study local structure in spaces of complex Hadamards.

Contact geometry is the study of geometric structures on odd dimensional smooth manifolds given by a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability; these hyperplane fields are called contact structures. The associated one form is called a contact form and uniquely determines a vector field called the Reeb vector field on the manifold. Contact and symplectic geometry are closely intertwined and we explain how one can make use of J-holomorphic curves to obtain contact invariants. This talk will have lots of examples, cool pictures, and animations illustrating these fascinating concepts in contact geometry.

TBA

The immune system in vertebrates is composed of individual cells called lymphocytes which work together to combat antigens such as bacteria and viruses. Upon detecting foreign molecules these immune cells secrete soluble factors that attract other immune cells to the site of the infection. The soluble factors are governed by a system of reaction-diffusion equations with sources that are centered on the cells. The motion of the cells is inherently stochastic, but biased toward the gradient of the soluble factors. I will discuss a numerical method for solving the reaction-diffusion stochastic system based on a first order splitting scheme. This method makes use of known first order schemes for solving the diffusion, the reaction and the stochastic differential equations separately.

In geometry, there are certain structures that are "rigid" (like Riemannian manifolds) and others that are "flexible" (like topological manifolds). Symplectic geometry lies in between these two extremes and incorporates some attractive features of both. One consequence is that symplectic techniques have recently been used, to great effect, to give combinatorial approaches to questions in topology that previously required difficult gauge-theoretic and analytic techniques. I will introduce symplectic structures and describe some recent developments linking them to the study of three-dimensional manifolds and knots. No real background will be assumed.

Many simple card tricks rely on mathematical principles and logic. I will be talking about some of these tricks, and the interesting ideas behind them. Hopefully I will have time to show you two or three tricks, and the proof to how they work. I will be using work from Mathematical Magic by Diaconis and Graham. The exciting thing about these tricks is they do not rely upon sleight of hand, and come out looking stunning nonetheless.

Skeletal structures, such as medial axis and curve skeleton, are a particular class of shape descriptors. They have numerous applications in shape recognition, shape retrieval, animation, morphing, registration, and virtual navigation. This talk will give a brief overview of the medial axis and the curve skeleton. The focus will be on the properties of the two objects crucial to applications. We shall show that the rigorous mathematical definition of the medial axis has allowed for an extensive and successful study of such properties. The curve skeleton, on the other hand, is typically defined by the set of properties it has to possess. As a result, numerous methods for computing the curve skeleton have been proposed, each providing mostly experimental verification of the required properties. If time permits, I will mention my work on defining shape skeleta via persistent homology, thus providing a powerful platform for investigating their properties.

Surface hopping is a very popular approach in theoretical chemistry for mixed quantum-classical dynamics. Yes, the above sentence looks scary. Let us start over again ... We will examine from a mathematical point of view how stochastic trajectories can be used to approximate solutions to a Schrodinger equation (which is different from what Feynman did). Besides some applications in chemistry, this is a nice topic since it combines ideas from asymptotic analysis, applied probability, and applied harmonic analysis. The only background assumed in this talk is "separation of variables" (and of course some PDEs where separation of variables is applied to).

What does it mean for a problem to be in P, or NP? What is NP completeness? These are questions, among others, that I hope to answer in my talk on computational complexity. Computational complexity is a branch of theoretical computer science dealing with analysis of algorithms. I hope to make it as accessible as possible, with no prior knowledge of algorithms and running times.