Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2014-2015Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2014-2015Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2014-2015Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2014-2015Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2014-2015Graduate Student Seminar, Department of Mathematics, University of Notre Dame, 2014-2015
Notre Dame Math Graduate Student Seminar, 2015-2016
Min-max methods have been used recently to solve several long-standing problems in differential geometry by Fernando Coda Marques and Andre Neves. I will explain the basic idea of the method and I will give a sketch of some of these applications.
September 14, 2015
Speaker
Alexander Diaz
Title
Peak Sets of Classical Coxeter Groups
Abstract
Given a permutation (or more generally a signed permutation) we can "graph" it and study its "peaks." The combinatorial study of peaks of permutations is a topic that has caught the attention of mathematicians in the past 20 years. For example, it has been shown that the set of sums of permutations with a given peak set is a subalgebra of the group algebra. Extending the notion of peaks to signed permutations, we can generalize some of the results for usual permutations, while some others do not admit a generalization. In this talk I will survey some of the most relevant and beautiful results in this area, including some of my work in collaboration with Jose Pastrana and many others.
September 28, 2015
Speaker
Jeremy Mann
Title
Differential Cohomology
Abstract
A differential cohomology theory produces invariants of manifolds. Like a generalized cohomology theory, these invariants are in some sense “locally determined,” and give global measurements of shape. However, unlike a “regular” cohomology theory, a differential cohomology theory is not homotopy invariant. Thus, these theories can “see” more refined geometric properties of manifolds, such as the curvature of a connection. In this talk, I will present some of the basics aspects of differential cohomology theories, their applications to physics, and their modern formulation
October 12, 2015
Speaker
Alan Liddell
Title
A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems
Abstract
Exceptional sets of a parameterized polynomial system are the sets in parameter space where the fiber has higher dimension than at a generic point. Such sets are arise in kinematics, for example, in designing mechanisms which move when the generic case is rigid. In 2008, Sommese and Wampler showed that one can use fiber products of bounded order to compute exceptional sets since they become irreducible components of larger systems. We propose an alternative approach using rank constraints on Macaulay matrices. This hybrid symbolic-numerical approach first symbolically constructs the appropriate matrices and then uses numerical algebraic geometry to solve the rank-constraint problem. We demonstrate the method on several examples, including exceptional RR dyads, lines on surfaces in C^3, and exceptional planar pentads.