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.

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.....

TBA

A body of literature has developed concerning Â?cloaking by anomalous localized resonanceÂ?. Most analytical work in this area has relied on separation of variables, and has therefore been restricted to radial geometries. In this talk, we will discuss a new approach based on a pair of dual variational principles, and apply it to some non-radial examples. In our examples, as in the radial setting, the spatial location of the source plays a crucial role in determining whether or not resonance occurs. The talk assumes minimal background knowledge.

Cartan's theorem on maximal tori in compact Lie groups can be thought of as a generalization of the spectral theorem for unitary matrices. The goal of this talk will be to sketch the `topological' proof of this theorem, based on the Lefschetz fixed point theorem. Along the way, we'll encounter the flag variety, an interesting object whose geometry encodes the representation theory of the Lie group. Those who don't specialize in geometry or topology fear not, we will give examples showing that these concepts are very concrete objects familiar from linear algebra.

Einstein's General Relativity is a geometric theory of space, time, and gravitation. In some sense, it is the successor of Newton's famous theory of gravitation -- the theory Newton is said to have come up with when an apple fell onto his head. But although Einstein's theory is much better at predicting gravitational effects in our universe, Newton's theory is not at all outdated or even obsolete. In fact, many astrophysical measurements and simulations still heavily rely on Newtonian intuitions, calculations, and concepts. In the talk, I will explain how and to what extent this usage of Newtonian theory in astrophysics and related fields is motivated and mathematically justified. This will lead us to the notion of Newtonian limit. We will also see some examples for the behavior of relativistic quantities like mass and center of mass under this Newtonian limit.

I will begin by recounting how I got involved in an industrial collaboration with a company that makes glass tempering furnaces and how younger grad students can seek such opportunities. I will then describe the mathematical model I developed for the company while highlighting challenges that arose due to differences in culture and priorities between academia and industry.

In this talk, I will use my recent project (a fast algorithm for calculating the energy of a many-body quantum system in the random phase approximation) as an outline to present two cool techniques in applied math and show their actual applications in the project. First, we'll see that the trapezoid rule you teach in Calculus, when applied to periodic functions, is far more impressive than you thought. We'll also get a taste of the surprisingly nice properties that come from combining matrix decompositions with randomized algorithms. Finally, as an added bonus, we'll see how Cauchy's integral formula can be used (in this project) to sum N^2 things in O(N) time.

Graphics processing units (GPUs) are powerful accelerators that can launch many processes in parallel. Over the past decade, they have been utilized for scientific computation, including molecular dynamics, fluid mechanics, machine learning, and stochastic differential equations. Although dependent on the algorithm, GPUs may execute code faster than CPUs by several orders of magnitude. The mathematics department at Duke hosts 4 older generation GPUs on two nodes that are available for department use. In this seminar I will briefly introduce how GPUs are different than CPUs; the bulk of my time will be a tutorial on how to code CUDA so that attendees may begin to take advantage of these departmental resources for their research. Depending on the attendance, it may be a hands-on tutorial so bring your laptop.