|Thursday, September 3||Ilia Itenberg, University of Strasbourg and MSRI|
|ABSTRACT||The purpose of the talk is to make an introduction to tropical geometry, a new mathematical domain which has undergone a spectacular progress during the last nine years. In tropical geometry, algebro-geometric objects are replaced with piecewise-linear ones. For example, the tropical curves in the plane
are certain rectilinear graphs. We will present the basic tropical notions and the first results in tropical geometry limiting ourselves to planar tropical curves.
|Thursday, September 17||Kirsten Eisentraeger, Penn State|
|2:30pm||Using elliptic and hyperelliptic curves in pairing-based cryptography|
|ABSTRACT||Over the past years, many new and exciting cryptographic schemes based on pairings have been suggested, including one-round three-way key establishment, identity-based encryption, and short signatures. Originally, the Weil and Tate pairings on supersingular elliptic curves were proposed for such applications, providing non-degenerate bilinear maps that are efficient to evaluate. As an alternative to elliptic curve groups, Koblitz suggested hyperelliptic curves for use in cryptography. In this talk we will explain how elliptic and hyperelliptic curves are used in these applications, and we will explain some of the techniques to optimize computations of elliptic and hyperelliptic pairings.|
|Thursday, September 24||Matthias Weber, Indiana University|
|2:30pm||An Invitation to Minimal Surfaces|
|ABSTRACT||Minimal Surfaces have attracted mathematicians for over 300 years. While much progress has been made,
the key question remains: What are the possible topological types of complete, embedded minimal surfaces. This talk will be very elementary with many pictures.
|Thursday, October 1||Mark Levi, Penn State|
|2:30pm||Discovering and proving theorems by physical reasoning|
|ABSTRACT||Physics often provides mathematics not only with a problem, but also with the idea of a solution. Some calculus problems can be solved more quickly without calculus, by using physics instead. A few examples of such problems will be given in a part of this talk.
In addition to these problems, quite a few theorems which may seem somewhat mysterious become completely obvious when given a proper physical incarnation. This is the case for both ``elementary" theorems (the Pythagorean theorem, Pappus' theorems, some trig identities, and many, many more) and the less elementary ones: Noether's theorem, the preservation of Poincare's integral invariants, the Gauss-Bonnet theorem, the Riemann Mapping Theorem, Green's theorem, Moser's theorem on uniformization of density, etc. (no familiarity with any of these is assumed).
I will describe a miscellaneous sampling from the above list, according to the audience's interest. No background beyond calculus and basic mechanics will be assumed in this talk.
|Thursday, October 22||Luca Capogna, University of Arkansas|
|2:30pm||Minimal surfaces in sub-Riemannian geometry|
|Thursday, October 29||Jon Chaika, N/A|
|2:30pm||[0,1] is not a minimality detector of [0,1]^2|
|ABSTRACT||This talk will introduce the notion of minimal sequences. It will then show that [0,1]^2 can detect the minimality of sequences in any compact metric space. However, [0,1] can not detect the minimality of sequences in [0,1]^2.|
|Thursday, November 5||Svetlana Katok, Penn State|
|2:30pm||Everything you wanted to know about 2x2 matrices|
|ABSTRACT||The group SL(2,R) is at the junction of number theory, representation theory, topology, geometry and dynamics. This seemingly simple object is both a source of deep questions and a proving ground for a variety of methods.
We will reveal some surprising connections between arithmetic, geometry and dynamics that arise from the study of this interesting group.
|Thursday, November 12||Robert Ghrist, University of Pennsylvania|
|2:30pm||Topological Network Topology|
|ABSTRACT||Networks are ubiquitous: communications networks, social networks, sensor networks, biological networks, etc., abound. "Network topology" is, usually, a misnomer, connoting graph theory. This talk will, via simple examples, argue for a topological interpretation of networks that, via algebraic topology, reveals classes of information hidden within networks around us.|
|Thursday, December 3||S. Weinberger, University of Chocago|
|2:30pm||Topology and Social Choice|
|ABSTRACT||Often one is in the situation when there are many agents or voters who each have an idea of what should be done, and we must find some way to combine their preferences and decide what "society wants". (This might even be what happens in an individual, where various subroutines in the brain each calculate a preference based on the function they are designed to "want" to optimize.) It turns out that this is rarely possible, but that one can have a lot of geometric and topological fun thinking about and manipulating such elections.|
|Thursday, October 8||Anton Petrunin, Penn State|
|2:30pm||Two problems in combinatorial plane geometry|
|ABSTRACT||It is a story, how I was making exercises for school students and what happened after it.|
|Thursday, October 15||Frederick Cohen, University of Rochester|
|2:30pm||Braid groups and their applications|
|ABSTRACT||Braids are easy to picture. The purpose of this talk is to describe Borromean braids as well as what these 'measure'.|