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
approachfirst 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.
October 26, 2015
Speaker
Eric Wawerczyk
Title
Congruences of Modular Cusp Forms
Abstract
Sequences of numbers {a_n} arise everywhere. We can study properties of a sequence using different functions: Arithmetic functions, Generating functions, and L-functions. If this sequence of numbers is "modular" then we can say a lot about it. Our specimen is the Ramanujan Tau function \tau(n) which will be our prime example of a ``normalized eigen modular cusp form". We will discuss the duality of weight k modular forms with the weight k Hecke algebra to motivate the key functionalities of a "Hida Family" and their relationship to congruences of modular forms.
November 9, 2015
Speaker
Gabriela Clemente
Title
Toric manifolds and polytopes
Abstract
We talk about the correspondence between symplectic toric manifolds and Delzant polytopes.
November 23, 2015
Speaker
Michael Perlman
Title
Minimal Free Resolutions and Hilbert's Syzygy Theorem
Abstract
The idea to associate a free resolution to a finitely generated module was introduced by Hilbert in the late nineteenth century. Minimal free resolutions are invariants of modules that encode algebraic and geometric information. This talk in meant to be an introduction to the theory of minimal resolutions through the computation of several examples. We will prove the Hilbert Syzygy Theorem and provide a geometric application.