Colloquium

In 1859 Bernhard Riemann published a groundbreaking paper in which he introduced the zeta function that came to bear his name. But some of Riemann's most important results had been discovered more than a hundred years earlier by Leonhard Euler, who was known to his peers as "analysis incarnate". 

This talk will showcase three amazing results Euler uncovered about the zeta function: his solution to the "Basel problem", the "Euler product" formula, and the functional equation. Euler's arguments are accessible to any student of calculus, but their brilliance and elegance have moved readers across the centuries. As William Dunham wrote, "No mathematician should go through a career without meeting Euler face to face."

January 26, 2024
Annalisa Calini, CofC
The Effects of Viscosity on the Linear Stability of Stokes Waves, Downshifting, and Rogue Wave Generation

This joint work with Connie Schober (UCF) and our undergraduate students Lane Ellisor and Evelyn Smith focuses on rogue wave formation in realistic models of waves dynamics in deep water. Rogue waves are high amplitude waves that appear suddenly and unexpectedly, then disappearing without a trace in a variety of water conditions: deep and shallow water, calm and wind-swept seas, and with or without currents. 

We investigate a higher-order nonlinear Schrödinger (NLS) equation recently proposed as a model of the dynamics of deep water waves in the presence of linear damping and weak viscosity, and address two interesting phenomena:

1. Experimental and numerical observation indicate how damping and viscous effects can, under certain conditions, increase rogue wave activity.
2. Small perturbations of unstable Stokes waves (single-frequency waves) in NLS models typically result in waves of greater amplitude that recur in time. When the steepness of the wave is large enough, energy is permanently transferred from the carrier wave to lower "sidebands" modes. This is known as permanent downshifting, a phenomenon that usually inhibits rogue wave formation.

Through analysis and numerical simulations, we discuss how the viscosity affects the linear stability of the Stokes wave solution, enhances rogue wave formation, and leads to permanent downshift. The novel results in this work include the analysis of the transition from the initial Benjamin-Feir instability to a predominantly oscillatory behavior, which takes place in a time interval when most rogue wave activity occurs. In addition, we propose new criteria for downshifting and characterize the time of permanent downshift in terms of the linear momentum of the system. 

February 2, 2024
Michael Levet, CofC Computer Science Department
On the Parallel and Descriptive Complexities of Group Isomorphism via Weisfeiler-Leman

The Group Isomorphism problem takes as input two finite groups and asks if G≅H. When the groups are given by their multiplication tables, Group Isomorphism is strictly easier than Graph Isomorphism under several notions of parallel reduction, including for instance AC⁰-reductions. Despite this fact and several decades of research on both problems, there are several measures of complexity, including logical definability (descriptive complexity) and the parallel (circuit) complexity of isomorphism testing, that are cornerstones of the Graph Isomorphism literature but have received minimal attention in the setting of groups. In this talk, I will discuss recent advances in both of these directions using classical color refinement techniques from Graph Isomorphism—namely, the Weisfeiler—Leman algorithm. This is joint work with Nathaniel A. Collins and Joshua A. Grochow.

February 9, 2024 ** ROOM CHANGE RSS 235 **

Presentation about the MUSC Ph.D. program in Epidemiology and Biostatistics

February 16, 2024
Paul Young, CofC
Complex and p-adic Analysis of Zeta Functions

Mathematics has two faces.  In constructive mathematics we build theories that explain patterns and relationships among objects; in destructive mathematics we use our theoretical tools to knock off lingering problems.  Of course, building theories helps us solve problems, and solving problems helps us build theories.

For our constructive part, we first consider Euler's constant γ, which is the constant Laurent coefficient of the Riemann zeta function at its pole.  We show that two classical series expressions for γ, due to Euler (1731) and to Mascheroni (1790), have natural analogues for height 1 multiple zeta functions.  We then interpret the Laurent coefficients of these height 1 multiple zeta functions as Ramanujan summations of Roman harmonic numbers.  Finally, we build a theory in which multiple zeta functions generate multiple zeta functions. 

For our destructive part, we consider the sequence of Cullen numbers m2^m+1, introduced by James Cullen in 1905, which is reputed to contain very few primes.  In 2018 the mighty tools of Diophantine analysis were engaged to demonstrate that only finitely many Cullen numbers are also generalized Fibonacci numbers; however, there still remained over 10²⁶ possibilities, which put the complete resolution beyond our computational capabilities.  Working with Attila Bérczes and István Pink of the University of Debrecen, we employed a strange brew of complex and p-adic techniques to completely resolve that eternal question, as well as a related problem, with surprisingly little computation. 

This talk represents a few of the results of my Fall 2023 sabbatical leave.

February 23, 2024

TBA

March 1, 2024
Mick Norton, CofC Emeritus
Some Personal Experiences as an Expert Witness in Statistics

Mathematics and statistics have been good to me, providing many stories during and after my career at C of C, some of which have to do with acting as an expert witness in statistics.  Four kinds of cases will be discussed - cheating on multiple-choice exams (the first such case is what got me started on expert witnessing), age and race discrimination suits, Medicare overpayment suits, and miscellany.  The statistical methods used in these cases run the gamut from techniques typically taught in freshman courses in probability and statistics to tests devised to detect a scenario for which no test previously existed. 

March 15, 2024
Vitaly Bergelson, Ohio State University
The Prime Number Theorem via Ergodic Theory

We will discuss a new type of ergodic theorem which has among its corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg and a theorem of Erdős-Delange.

The talk is based on joint work with Florian Richter and is intended for a general audience.

March 22, 2024
David Ross, University of Hawaii
Hat puzzles, Yablo's paradox, and the liar at infinity

A popular family of puzzles, often involving prisoners guessing the colors of hats, is well known to be closely connected to ideas in set theory.  This paper looks at a few such puzzles using the language of nonstandard analysis, in which context the algorithms even for highly nonconstructive solutions look entirely mechanical.  It then considers how similar methods might apply to Stephan Yablo's version of the Liar Paradox.

March 29, 2024
Dan Maroncelli, CofC
What is a solution to a differential equation?

In this talk, I will discuss various definitions that one might use when considering a solution to a given differential equation. Along the way, I will try to indicate why one might, in a natural way, be "led" into thinking about these various function spaces. 

April 5, 2024
Amy Langville and Kathryn Pedings-Behling, CofC
Reforming Business Calculus Through Curriculum

Join us for an exploration of the expansive Deconstruct Calculus Project. We will share where we are with curriculum development along with some of the incredible data we have collected from our own Business Calculus students. See if you're ready to join the Deconstruct Business Calculus teaching team! 

April 12, 2024
Stéphane Lafortune, CofC
On the Stability of Smooth Solutions to Equations describing wave breaking

The Camassa-Holm equation is a shallow water wave equation which model wave breaking. Among its many interesting mathematical properties, perhaps the most striking is the fact that it admits weak solutions - ‘peakons’- with a peaked shape corresponding to a discontinuous first derivative. Since the discovery of the Camassa-Holm equation, several ‘’peakon equations’’ with similar properties have been studied. Among them, the integrable Novikov equation, which can be regarded as a generalization to a cubic nonlinearity of the Camassa-Holm equation. Furthermore, while both the Camassa-Holm and the Novikov equations are integrable (i.e., they admit soliton solutions), each admit a generalization of peakon equations, which are not integrable.

In this talk, we study the stability of various smooth solutions to peakon equations. One of the main difficulties when dealing with the linear operators arising from peakon equation is that they often include a non-local term. An additional challenge is the fact that the localized smooth solutions admit a nonzero background, that is their asymptotic state (as the space variable x goes to infinity) is not zero.

CANCELLED - April 19, 2024 - CANCELLED

Ilya Gekhtman, Technion - Israel Institute of Technology
Invariant random subgroups and injectivity radius of hyperbolic manifolds

There is a long tradition of using probabilistic methods to solve geometric problems. I will present one such result. Namely, I will show that  if the bottom of the spectrum of the Laplacian on a hyperbolic manifold (or more generally rank one locally symmetric space) M  is equal to that of its universal cover then M has points with arbitrary large injectivity radius.   A key tool will be the theory of invariant random subgroups, which are conjugation invariant probability measures on the space of subgroups of a given group. All terms will be defined during the talk.  Joint work with Arie Levit.

If you would like to give a talk or invite someone to give a talk on our colloquium, please contact the colloquium coordinator, Dan Maroncelli (maroncellidm@cofc.edu).

For outside visitors please include visitor's name, affiliation, and the title or the area of the suggested talk.   

Also, as the sponsor, please note that you will be responsible to find an accommodation and take care of the visitor during his/her visit.

“Week of Welcome” event for our majors/math club.

Jo Boaler, Stanford University, RITA 101, 3PM

The Beauty of Teaching & Learning Multidimensional Mathematics.

What happens when mathematics is learned differently — not as a subject of short questions and right and wrong answers, but as concepts and ideas? Join Professor Jo Boaler of Stanford University’s Graduate School of Education as we explore foundational ideas in mathematics, such as arithmetic and algebra, and consider how they can be learned visually. Together we will celebrate the diverse ways in which people see mathematics, focus on the importance of learning connections between mathematical ideas, and discover how encountering the beauty of multidimensional mathematics can positively impact classroom experiences.

Derrick Niederman, Author, Mathematician, Game Designer, and former CofC Faculty

Common Sense Statistics

This will be the most elementary colloquium in the college’s history. My intent is to review an electronic text I wrote for Math 104 specifically, focusing on the challenges posed by such a construction within a highly competitive marketplace. 

The material should be easily understood by all. My only request is that attendees give some thought to the number of possible color combinations in the first row of a Wordle puzzle. 

Thomas Ivey, CofC

Constructing Solitons for an Isometric Flow on G₂-structures

G₂-structures arise naturally when a 7-dimensional Riemannian manifold M carries a distinguished positive differential 3-form 𝜑. A G₂-structure endows the manifold with an orientation and Riemannian metric g. (Thus, the pointwise automorphism group of 𝜑 is isomorphic to a compact subgroup of SO(7).) A G₂-structure is torsion-free if 𝜑 is parallel with respect to the Levi-Civita connection of g, in which case (M,𝜑) is called a G₂-manifold. Non-trivial G₂-manifolds, which have reduced holonomy, are difficult to construct. Inspired by the Ricci flow, several authors have studied evolution equations forG₂-structures, with the hope that these limit to closed or co-closed structures.

This talk concerns a flow on isometric G₂-structures which is the negative gradient flow for total torsion. (Two G₂-structures are said to be isometric if they induce the same metric g.)  The flow was recently introduced by Grigorian, and Dwivedi, Giannotis and Karigiannis recently proved long-time existence and convergence to a structure with divergence-free torsion or to a shrinking soliton. In this talk, I discuss constructing soliton structures isometric to a fixed torsion-free background, such as the canonical flatG₂-structure on R⁷, the product of a Calabi-Yau 3-fold with a circle, or a Bryant-Salamon metric. The restriction of the flow to structures like these, which have a high degree of symmetry, provides important test cases for convergence. This is joint work with Spiro Karigiannis.

Garrett Mitchener, CofC 

A biology-inspired approach to symbolic regression

Symbolic regression is a family of techniques for finding simple exact expressions that solve classification and regression problems in data science. Using my experience modeling the evolution of gene regulatory networks, I've been experimenting with adapting evolutionary dynamics for use in symbolic regression. This project is in the early stages, so I'll present the important ideas and show some preliminary computational results.

Rodrigo Trevino, University of Maryland

Title: TBA