- Speaker: Alexander Duncan, University of Michigan
Time and Place: Friday, January 23 at 3:30 pm in LC 412.
Title: Essential dimension of finite groups
Abstract: Essential dimension measures the minimal number of parameters required to describe an algebraic object. I will discuss the essential dimension of finite groups and its connections to constructive inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations. Techniques from birational geometry will play a prominent role in the talk.
- Speaker: Alexandra Seceleanu, University of Nebraska-Lincoln
Time and Place: Monday, January 26 at 3:30pm in LC 412.
Title: Symbolic versus ordinary powers of ideals for points
Abstract: The problem of describing the set of hypersurfaces passing through a finite set of points with given multiplicity leads to challenging mathematical questions. For example, one can ask what the minimum degree of such a hypersurface is or how many independent hypersurfaces there are of any given degree. The most general forms of these questions are still open and have given rise to longstanding conjectures in algebraic geometry. Searching for structural reasons to explain some of the these conjectures, Harbourne and Huneke proposed an approach based on comparisons between the set of all polynomials vanishing at the points to a prescribed order, which is called a symbolic power ideal, and algebraically better understood counterparts, namely the ordinary powers of the ideal of base points. Two questions will be shown to be related: How tight can this comparison be made? Which arrangements of lines in the plane have no points where only two lines meet? I will answer these and many more questions while considering some special arrangements of lines with unexpected combinatorial and algebraic properties.
- Speaker: Nicholas Addington, Duke University
Time and Place: Friday, Februrary 6 at 3:30pm in LC 412.
Title: Recent developments in rationality of cubic 4-folds
Abstract: The question of which cubic 4-folds are rational is one of the foremost open problems in algebraic geometry. I'll start by explaining what this means and why it's interesting; then I'll discuss three approaches to solving it (including one developed in the last year), my own work relating the three approaches to one another, and the troubles that have befallen each approach.
- Speaker: Johann A. Makowsky, Technion-Israel Institute of Technology
Time and Place: Friday, Februrary 19 at 4:30pm in LC 412.
Title: On the Location of Roots of Graph Polynomials
Abstract: Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph. (Joint work with E. Ravve and N. Blanchard)
- Speaker: Wolfgang Dahmen, Institut fur Geometrie und Praktische Mathematik, RWTH Aachen
Time and Place: Friday, Februrary 25 at 4:15pm in LC 412.
Title: Models and Data - Challenges and Shortcuts
Abstract: The prize for increasing the resolution of the "mathematical microscope“ in science and engineering is to accept more and more refined and hence more complex mathematical models. The complexity stems, for instance, from the range of involved scales but also from the (spatial) dimensionality of the objects to be recovered, very often rendering established numerical simulation techniques infeasible. These obstructions are particularly severe when facing inversion tasks and assimilating data into model-based approaches. Some illustrating examples indicates that even a significantly increased computing power is not a remedy. Instead mathematical "shortcuts“ are called for. We highlight some promising strategies and sketch the underlying ideas, in particular, concerning problems in high spatial dimensions.
For more information, please visit http://www.math.sc.edu/colloquium-models-and-data-challenges-and-shortcuts-wolfgang-dahmen
- Speaker: Yong Lin, Renmin University of China
Time and Place: Wednesday, March 4 at 4:30pm in LC 412.
Title: Ricci curvature and geometric analysis on graphs
Abstract: Ricci curvature lower bound play very important rule for geometric analysis on Riemannian manifold. So it is interesting to introduce similar concept on discrete setting especially on graphs. We will talk about the Ricci curvature lower bound on graphs where the original idea comes from the Bochner formula on Riemannian geometry. Given the Ricci curvature lower bound on graphs, we will imply some classic results from Riemannian geometry for eigenvalue estimate, Li-Yau gradient estimate, Harnack inequality and heat kernel estimate, as well as the volume doubling property and Poincare inequality.
- Speaker: Cynthia Vinzant, North Carolina State University
Time and Place: Thursday, April 2 in LC 412
Title: Determinants, hyperbolicity, and interlacing
Abstract: Writing a multivariate polynomial as the determinant of a matrix of linear forms is a classical problem in algebraic geometry and complexity theory. Requiring that this matrix is Hermitian and positive definite at some point puts topological and algebraic restrictions on the polynomials that appear as the determinant and its minors. In particular the real zero sets of these polynomials are hyperbolic (or real stable) and interlace. I’ll talk about the beautiful geometry behind these determinants and its connection to optimization and combinatorics.
- Speaker: Robert Lipshitz, Columbia University
Time and Place: Thursday, April 16 at 4:30pm in LC 412.
Title: The Jones polynomial and friends.
Abstract: We will start by introducing knot theory and some of its motivating questions. We will then introduce the Jones polynomial and two refinements of it, Khovanov homology and a Khovanov stable homotopy type, and discuss how they can be used to answer some of the questions. The new content is joint with Tyler Lawson and Sucharit Sarkar
- Speaker: Michael Lacey, Georgia Institute of Technology
Time and Place: Thursday, October 29th, at 4:30pm in LC 412
title: One Bit Measurements in Signal Processing
- Speaker: Eric Carlen, Rutgers University.
Time and Place: Thursday, November 5th at 4:30pm in LC 412.
Title: Remainder terms for some classical inequalities and their uses
- Speaker: Stephan Wagner, Stellenbosch University.
Time and Place: Thursday, November 19th at 4:30pm in LC 412.
Title: Loop Models on a Fractal