SWiM 2017 Lectures

Ingrid Daubechies

June 21 (Wednesday)

Mathematicians helping Art Historians and Art Conservators

In recent years, mathematical algorithms have helped art historians and art conservators putting together the thousands of fragments into which an unfortunate WWII bombing destroyed world famous frescos by Mantegna, decide that certain paintings by masters were "roll mates" (their canvases were cut from the same bolt), virtually remove artifacts in preparation for a restoration campaign, get more insight into paintings hidden underneath a visible one. The presentation will review these applications, and give a glimpse into the mathematical aspects that make this possible.

Radmila Sazdanovic

June 22 (Thursday)

Catalan numbers, Chebyshev polynomials, and categorifications

What does it mean to "compute with diagrams"? We will construct a "diagrammatic algebra" based on properties of Catalan numbers and use it to recover some well-known facts about the Chebyshev polynomials. This novel approach, called categorification, provides insight into more complicated algebraic structures as well as inspiration for visual artists.

Ezra Miller

June 23 (Friday)

Topology for statistical analysis of brain artery images

Statistics looks for trends in data. Topology quantifies geometric features that don't change when shapes are squished, stretched, or bent continuously. What does one have to do with the other? When data objects are already geometric, such as magnetic resonance images of branching arteries, topology can isolate information of statistical relevance. This talk explains what we have learned about the geometry of blood vessels in aging human brains using topological methods in statistics. The main results are joint with Paul Bendich, Steve Marron, Alex Pieloch, and Sean Skwerer (at the time, a Math postdoc, Stat faculty, Math undergrad, and Operations Research grad student).

Hubert Bray

June 26 (Monday)

Gravity and the Curvature of Spacetime

Einstein's Theory of General Relativity explains gravity more accurately than any other theory by modeling the universe as a four dimensional curved spacetime manifold. We'll discuss the mathematics behind this amazing picture of the universe.

Nancy Rodriguez-Bunn

June 27 (Tuesday)

What calculus can tell us about life

In this talk I will discuss how we can use calculus to gain insight into complex social, ecological, and biological phenomena. We will focus on modeling urban crime and explore various important mathematical questions from the point of view of their applications.

Robert Bryant

June 28 (Wednesday)

The Idea of Holonomy

The notion of `holonomy' in mechanical systems has been around for more than a century and gives insight into daily operations as mundane as steering and parallel parking and in understanding the behavior of balls (or more general objects) rolling on a surface with friction. A sample question is this: What is the best way to roll a ball over a flat surface, without twisting or slipping, so that it arrives at at given point with a given orientation? In geometry and physics, holonomy has turned up in many surprising ways and continues to be explored as a fundamental property of geometric structures. In this talk, I will illustrate the fundamental ideas in the theory of holonomy using familiar physical objects and explain how it is also related to group theory and symmetries of basic geometric objects.

Colleen Robles

June 29 (Thursday)

The Gauss-Bonnet Theorem

The Gauss-Bonnet Theorem is the "crown jewel" of surface geometry. I will explain this beautiful result, what makes it so remarkable, and how one can prove/verify it using calculus.