Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted May 2, 2016

1:30 pm - 2:30 pm Lockett 233
Sean Taylor, LSU

Introduction to Algebraic Stacks Part 2

After introducing what a Grothendieck topology was last week, we will be able to proceed to actually define stacks - which are higher analogues of sheaves - and then algebraic stacks. If time provides, we will look at common examples of these beautiful and powerful geometric objects.

Applied Analysis Seminar
Questions or comments?

Posted May 2, 2016

1:30 pm - 2:30 pm Lockett 233
Kim Pham, ENSTA ParisTech

Construction of a macroscopic model of phase-transformation for the modeling of superelastic Shape Memory Alloys

Abstract: Shape Memory Alloys (SMA) e.g. NiTi display a superelastic behavior at high temperature. Initially in a stable austenite phase, SMA can transform into an oriented martensite phase under an applied mechanical loading. After an unloading, the material recovers its initial stress-free state with no residual strain. Such loading cycle leads to an hysteresis loop in the stress-strain diagram that highlights the dissipated energy for having transformed the material. In a rate-independent context, we first show how a material stability criterion allows to construct a local one-dimensional phase transformation model. Such models relies on a unique scalar internal variable related to the martensite volume fraction. Evolution problem at the structural scale is then formulated in a variational way by means of two physical principles: a stability criterion based on the local minima of the total energy and an energy balance condition. We show how such framework allows to handle softening behavior and its compatibility with a regularization based on gradient of the internal variable. We then extend such model to a more general three dimensional case by introducing a tensorial internal variable. We derive the evolution laws from the stability criterion and energy balance condition. Second order conditions are presented. Illustrations of the features of such model are shown on different examples.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted June 26, 2015

Last modified March 17, 2016

Maria Chudnovsky, Princeton University
MacArthur Foundation Fellowship recipient 2012.

Perfection and Beyond

About 10 years ago one of the central open problems in graph theory at the time, the Strong Perfect Graph Conjecture, was solved. The proof used structural graph theory methods, and spanned 155 journal pages. The speaker was part of the team of authors of this mathematical beast. In this talk we will explain the problem, describe some of the ideas of the proof (that has since been shortened somewhat), and discuss related problems that have been a subject of more recent research.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted January 22, 2016

Last modified March 17, 2016

Maria Chudnovsky, Princeton University
MacArthur Foundation Fellowship recipient 2012.

Coloring some perfect graphs

Perfect graphs are a class of graphs that behave particularly

well with respect to coloring. In the 1960's Claude Berge made two

conjectures about this class of graphs, that motivated a great deal of

research, and by now they have both been solved.

The following remained open however: design a combinatorial algorithm that

produces an optimal coloring of a perfect graph. Recently, we were able to

make progress on this question, and we will discuss it in this talk. Last

year, in joint work with Lo, Maffray, Trotignon and Vuskovic we were able

to construct such an algorithm under the additional assumption that the

input graph is square-free (contains no induced four-cycle). More

recently, together with Lagoutte, Seymour and Spirkl, we solved another

case of the problem, when the clique number of the input graph is fixed

(and not part of the input).

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted January 22, 2016

Last modified March 17, 2016

Maria Chudnovsky, Princeton University
MacArthur Foundation Fellowship recipient 2012.

Induced cycles and coloring

The Strong Perfect Graph Theorem states that graphs with no no induced odd cycle of length at least five, and no complements of one behave very well with respect to coloring. But what happens if only some induced cycles (and no complements) are excluded? Gyarfas made a number of conjectures on this topic, asserting that in many cases the chromatic number is bounded by a function of the clique number. In this talk we discuss recent progress on some of these conjectures. This is joint work with Alex Scott and Paul Seymour.