Posted January 16, 2024

Last modified February 28, 2024

Mathematical Physics and Representation Theory Seminar

2:30 pm – 3:20 pm 233 Lockett
Ben McCarty, University of Memphis

A new approach to the four color theorem

The Penrose polynomial of a graph, originally defined by Roger Penrose in an important 1971 paper, shares many similarities with Kauffman’s bracket and the Jones polynomial. In order to capitalize on these similarities, we first modify the definition of the Penrose polynomial to obtain a related family of polynomials, called the n-color polynomials. Each of the n-color polynomials may be thought of as an analog of the Jones polynomial, and is the graded Euler characteristic of a bigraded homology theory (analogous to Khovanov homology). We then show how to define a spectral sequence leading to a filtered homology theory (analogous to Lee homology) where coloring information becomes apparent. We will then discuss several applications of the theory to graph coloring and the four color problem. This is joint work with Scott Baldridge.

Posted February 20, 2024

Pasquale Porcelli Lecture Series Special Lecture Series

3:30 pm – 4:30 pm Atchafalaya Room, LSU Student Union
R. Tyrrell Rockafellar, University of Washington

Risk and Uncertainty in Optimization

Abstract: New mathematics is on the forefront in many emerging areas of technology, and its methods for sorting out ideas and testing for truth and shortcomings are as vital as ever. This talk aims to explain how that has worked in confronting “risk”. Problems of optimization are concerned with deciding things “optimally”. In many situations in management, finance, and engineering design, however, plans have to be fixed in the present without knowing fully how they will play out in the future. A future cost or hazard may depend on random variables with probability distributions that a present decision can only influence in a limited way. Should optimization then rely on average outcomes? Worst-case outcomes? High-probability avoidance of dangerous outcomes? Or what? This is a subject with a history of competing approaches that reached a turning point with the axiomatic development of a powerful theory of risk. The mathematical concepts and results from that have been overturning tradition in one important area of application after another.

Posted February 20, 2024

Pasquale Porcelli Lecture Series Special Lecture Series

2:30 pm – 3:30 pm Hill Memorial Library
R. Tyrrell Rockafellar, University of Washington

Variational Analysis and Geometry

The theory needed for problems of optimization has required vast developments of a kind of alternative calculus in which, for instance, discontinuous functions that might take on infinite values nonetheless have “subgradients” which are highly useful. In maximizing and minimizing, variables are often required to be nonnegative, or to have values not too high or too low. This leads to a fascinatingly different nonclassical kind of geometry.

Posted November 13, 2023

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Edmund Yik-Man Chiang, The Hong Kong University of Science and Technology

TBA

Posted January 18, 2024

Computational Mathematics Seminar

3:30 pm – 4:20 pm Digital Media Center: Room 1034
Soeren Bartels, University of Freiburg, Germany

Babuska's paradox in linear and nonlinear bending theories

The plate bending or Babuska paradox refers to the failure of convergence when a linear bending problem with simple support boundary conditions is approximated using polygonal domain approximations. We provide an explanation based on a variational viewpoint and identify sufficient conditions that avoid the paradox and which show that boundary conditions have to be suitably modified. We show that the paradox also matters in nonlinear thin-sheet folding problems and devise approximations that correctly converge to the original problem.

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Justin Murray, Louisiana State University

TBA

Posted February 20, 2024

Pasquale Porcelli Lecture Series Special Lecture Series

2:30 pm – 3:30 pm Hill Memorial Library
R. Tyrrell Rockafellar, University of Washington

Variational Convexity and Local Optimality

For necessary and sufficient conditions for local optimality, the inherited ideal has been for them to be as close to each other as possible. In optimization, however, what’s more important is sufficient conditions that identify key common features in a problem which support algorithmic developments. Variational convexity, although only recently identified as such a condition, appears to be fundamentally important.

Posted January 9, 2024

Last modified February 27, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Jone Lopez de Gamiz Zearra, Vanderbilt University

On subgroups of right-angled Artin groups

In this talk we will discuss subgroups of right-angled Artin groups (RAAGs for short). Although, in general, subgroups of RAAGs are known to have a wild structure and bad algorithmic behaviour, we will show that under certain conditions they have a tame structure. Firstly, we will discuss finitely generated normal subgroups of RAAGs and show that they are co-(virtually abelian). As a consequence, we deduce that they have decidable algorithmic problems. Secondly, we will recall results of Baumslag-Roseblade and Bridson-Howie-Miller-Short on subgroups of direct products of free groups and explain how they generalize to other classes of RAAGs.

Posted September 29, 2023

Last modified January 29, 2024

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Jacob Rasmussen, University of Illinois Urbana-Champaign

The L-space conjecture for 3-manifolds

The L-space conjecture of Boyer-Gordon-Watson and Juhasz relates three very different properties that a closed 3-manifold M can possess. One of these properties is algebraic: is \pi_1(M) left orderable? The second is geometric: does the M admit a coorientable taut foliation? The third is analytic: is the Heegaard Floer homology M as simple as it can be, given the size of H_1(M). If the conjecture is true, it would reveal the existence of a striking dichotomy for rational homology 3-spheres. In this talk, I'll explain what each of the three conditions appearing in the L-space conjecture mean, and then discuss efforts to prove and disprove it, and why we should care.