Past Colloquia

Title: Transverse Instability of Rogue Waves 

Video 

Title: Cascade Feedback Linearization and Two-Legged Copepod Motion

Video

Title: Efficient Training of Infinite-Depth Neural Networks via Jacobian-Free Backpropagation

Video

Title: Data-Efficient Deep Learning using Physics-Informed Neural Networks

Video

Slides

Title: Ideal Extensions and Directly Infinite Algebras

Video

Title: Analytical study of Floquet topological insulators 

Video

Title: The fractal geometry of Schramm-Loewner Evolution (SLE)

Title: Turning A Coin Instead of Tossing It

Title: The Bak-Sneppen Model of Biological Evolution and Related Models

Title: The Toda lattice and semiclassical orthogonal polynomials

Title: Partition Identities and Quiver Representations

Title: Pairing between zeros and critical points of random polynomials

Title: Should you quit your job and start working on the
Twin Prime Conjecture?

Title: Modeling grain boundaries in metals with optimal transportation theory, calculus of variations, and the phase field method.

Title: Numerical Solutions of the Painlevé Equations

Title: Representation Theory and Random Walks

Title: Exact Solutions of Integrable Nonlinear Evolution Equations.

Title: Mathematical Problems that Cannot be Solved

Title: Painlevé Functions in Statistical Physics

Title: Markov Chains, Metastability and Sampling

Title: Maximal Subgroup Growth of some Groups

Patient Specific Modeling of Cardiovascular System Dynamics

The Dirichelt Problem for Elliptic Systems in the Upper-Half Plane

Initial Value Problems and Initial-Boundary Value Problems for Nonlinear Evolution Equations

Coupling for Brownian Motion with Redistribution

Arithmetic Sets in Groups

Hodge-de Rham Operator on (some) Classical and Quantum Spheres

The Geometry of Graphs

Adiabatic Perturbation Theory for Vector NLS and Application in BECs

Mesoscopic Models for Flow in Spatial Networks

The dynamics of flows in spatial networks, such as the pressure-driven blood flow in the human arterial network or the flow of cars in a traffic network, is most suitably described by PDE-based 'macroscopic' models. To cope with the computational complexity, often simplified models are employed, including at the level of individual particle tracking, usually called 'microscopic' models. Here we describe a mathematical model for blood flow in vascular networks, and compare numerical solutions of the underlying system of PDEs with those of a simplified models, based on pulse-tracking arguments (mesoscopic models). We then use these models to study flow optimization task, for variable size and/or topology of the network. Physiologically realistic control mechanisms are tested in the context of these simplified models.

The ubiquity of the Fibonacci Sequence: It comes up in the study of Leavitt path algebras too!

The majority of this talk should be quite accessible to math majors, to graduate students, even to math faculty: indeed, to anyone who has heard of the Fibonacci sequence ... Since its origin (more than eight centuries ago) as a puzzle about the number of rabbits in a (fantasmagorically expanding) colony, the Fibonacci Sequence 1,1,2,3,5,8,13,... has arguably become the most well-known of numerical lists, due in part to its simple recursion formula, as well as to the numerous connections it enjoys with many branches of mathematics and science. Since its origin (less than ten years ago), the study of Leavitt path algebras (a type of algebraic object which arises from directed graphs) has been the focus of much research energy throughout the mathematical world (well, at least throughout the ring-theory world), especially here at UCCS. In this talk we'll show how Fibonacci's sequence is naturally connected to data associated with the Leavitt path algebras of a natural collection of directed graphs. No prior knowledge about Leavitt path algebras will be required. [But in fact we will show how to compute the Grothendieck group $K_0(L(E))$ of the Leavitt path algebra $L(E)$ for a directed graph $E$, by considering only elementary-level properties of the graph. Those properties will lead us directly to Fibonacci. Plenty of easy-to-see examples will be given.] This is joint work with Gonzalo Aranda Pino of the University of Malaga (Spain). Many of you have met Gonzalo: he is a very frequent visitor to UCCS.

Integrable Curve Flows: the solitary travels of a vortex filament

The Vortex Filament Equation, describing the self-induced motion of a vortex filament in an ideal fluid, is a simple but important example of integrable curve dynamics. Its connection with the cubing focusing Nonlinear Schrodinger equation through the well-known Hasimoto map allows the use of many of the tools of soliton theory to study properties of its solutions. I will discuss the construction of knotted solutions, their dynamics, and their stability properties.

Poster (PDF)

Video Lecture (archived)

The center of rings associated to directed graphs

In 2005 Abrams and Aranda Pino began a program studying rings constructed from directed graphs. These rings, called Leavitt Path algebras, generalized the rings without invariant basis number introduced by Leavitt in the 1950's. Leavitt path algebras are the algebraic analogues of the graph C*-algebras and have provided a bridge for communication between ring theorists and operator algebraists. Many of the properties of Leavitt path algebras can be inferred from properties of the graph, and for this reason provide a convenient way to construct examples of algebras with a particular set of attributes. In this talk we will explore how central elements of the algebra can be read from the graph.

Poster (PDF)

Video Lecture (archived)

Game theory and the mathematics of altruism

Game theory is a branch of mathematics that deals with strategy and decision making and is applied in economics, computer science, biology, and many other disciplines as well. We will discuss some of the basic points of game theory and discuss the so-called iterated prisoner dilemma, a game that is of central importance in the study of cooperation between individuals. We will then describe various strategies to this game and explain why altruism is something that can evolve naturally.

Poster (PDF)

Video Lecture (archived)

Myopic Models of Population Dynamics on Infinite Networks

Population models In mathematical biology often use equations blending diffusion (for movement) with local descriptions of population growth and multispecies interactions (reaction diffusion models). A modern problem is how to make sense of such models on gigantic networks such as the human population or the World Wide Web. One approach is to work in a space of functions which 'look flat' at 'infinity'. A correct formulation of this idea supports a theory of reaction-diffusion models on infinite networks where the network is compactified by adding points at infinity, diffusive effects vanish at infinity, and finite dimensional approximations can be described http://cmes.uccs.edu/temp/colloquium110614.mov.

Nonlinear magnetization dynamics in nanoparticles and thin films

Even the simplest magnetic system can undergo unusual nonlinear dynamics. In this talk I will discuss two magnetic systems that display unexpected nonlinear phenomena. Firstly, the magnetization dynamics in a nanoparticle will be detailed. It is found that the transient dynamics in this system can be made to persist for extremely long times when the nanoparticle is driven by oscillating magnetic fields at a very particular frequency and strength. [1] Secondly, thin magnetic films will be discussed and a perturbative expansion of nonlinear dynamic terms will be presented. In thin films, the threshold above which the system is driven nonlinear depends sensitively on the thickness of the film. [2] Connections to experiments will briefly be mentioned. [1] M.G. Phelps, K.L. Livesey and R.E. Camley, in preparation (2014). [2] K.L. Livesey, M.P. Kostylev and R.L. Stamps, Phys. Rev. B. 75, 174427 (2007).

Experiments on Solitons, Dispersive Shock Waves, and Their Interactions

A soliton is a localized traveling wave solution to a special class of partial differential equations (integrable equations). A defining property of solitons is their interaction behavior. In his seminal work of 1968 introducing a notion of integrability (the Lax pair), Peter Lax also proved that the Korteweg-de Vries (KdV) equation admits two soliton solutions whose interaction behavior is quite remarkable. Two solitons interact elastically, i.e., each soliton maintains the same speed and shape post-interaction as they had pre-interaction. Moreover, Lax classified the interaction geometry into three categories depending on the soliton amplitude ratio. This talk will present a physical medium (corn syrup and water) modeled by the KdV equation in the weakly nonlinear regime that supports approximate solitons. Numerical analysis and laboratory experiments will be used to show that the three Lax categories persist into the strongly nonlinear regime, beyond the applicability of the KdV model. Additionally, a wavetrain of solitons called a dispersive shock wave in this medium will be described and investigated using a nonlinear wave averaging technique (Whitham theory) and experiment. Interactions of dispersive shock waves and solitons reveal remarkable behavior including soliton refraction, soliton absorption, and two-phase dynamics.

A unified approach to boundary value problems:

Over the last fifteen years, A unified approach has recently been developed to solve boundary value problems (BVPs) for integrable nonlinear partial differential equations (PDEs). The approach is a generalization of the inverse scattering transform (IST), which was originally introduced in the 1970's to solve initial value problems for such PDEs. Interestingly, this approach also provides a novel and powerful way to solve BVPs for linear PDEs. This talk will discuss the application of this method for linear PDEs. Specifically, we will look in detail at the solution of BVPs on the half line (0<x<infty) for linear evolution PDEs in 1 spatial and 1 temporal dimension. Time permitting, two-point BVPs, multi-dimensional PDEs and BVPs for linear elliptic PDEs will also be discussed.

Distribution of Runs in Gambler's Ruin

Sharp existence and Liouville type theorems for a class of weighted integral equations

The Leavitt path algebras of arbitrary graphs over a field

Using results from dynamical systems to classify algebras and C*-algebras

Evaluating Polynomials on Matrices:

A classical theorem of Shoda from 1936 says that over any field K (of characteristic 0), every matrix with trace 0 can be expressed as a commutator AB-BA, or stated another way, that evaluating the polynomial f(x,y)=xy-yx on matrices over K gives precisely all the matrices having trace 0. I will describe various attempts over the years to generalize this result.

Large equilibrium configurations of two-dimensional fluid vortices

The point-vortex equations, a discretization of the Euler equations, describe the motion of collections of two-dimensional fluid vortices. The poles and zeros of rational solutions to the Painleve II equation describe equilibrium configurations of vortices of the same strength and mixed rotation directions. There is an infinite sequence of such rational solutions with an increasing number of poles and zeros. In joint work with P. Miller (Michigan), we compute detailed asymptotic behavior of these rational functions with error estimates. Our results include the limiting density of vortices for these configurations. We will also describe how knowledge of the asymptotic behavior of the rational Painleve II functions is useful in understanding critical phenomena in the solution of nonlinear wave equations.

Solving non-linear dispersive equations by the method of inverse scattering:

The celebrated Korteweg-de Vries (KdV) equation and the nonlinear Schrodinger (NLS) equations are partial differential equation that describe the motion of weakly nonlinear long waves in a narrow channel. They predict "solitary waves" which do not disperse, which have been observed in nature, and used in many applications. In this lecture we'll talk about the "KdV miracle" of complete integrability that explains the solitary waves and establishes a remarkable connection between these equation and quantum mechanics. We will also discuss work in progress involving generalizations of the KdV and NLS equations to two space dimensions that describe surface waves and, like their one-dimensional counterpart, are completely integrable.

Secrets From Deep Human History

Dispersive shock waves and shallow ocean-wave line-soliton interactions:

Many physical phenomena are understood and modeled with nonlinear partial differential equations (PDEs). A special subclass of these nonlinear PDEs has stable localized waves -- called solitons -- with important applications in engineering and physics. I'll talk about two such applications: dispersive shock waves and shallow ocean-wave line-soliton interactions.

Dispersive shock waves (DSWs) occur in systems dominated by weak dispersion and weak nonlinearity. The Korteweg de Vries (KdV) equation is the universal model for phenomena with weak dispersion and weak quadratic nonlinearity. I'll show that the long-time asymptotic solution of the KdV equation for general step-like data is a single-phase DSW; the boundary data determine its form and the initial data determine its position. I find this asymptotic solution using the inverse scattering transform (IST) and matched-asymptotic expansions.

Ocean waves are complex and often turbulent. While most ocean-wave interactions are essentially linear, sometimes two or more waves interact in a nonlinear way. For example, two or more waves can interact and yield waves that are much taller than the sum of the original wave heights. Most of these nonlinear interactions look like an X or a Y or an H from above; much less frequently, several lines appear on each side of the interaction region. It was thought that such nonlinear interactions are rare events: they are not. I'll show photographs and videos of such interactions, which occur every day,close to low tide, on two flat beaches that are about 2,000 km apart. These interactions are related to the analytic, soliton solutions of the Kadomtsev Petviashvili equation, which extended the KdV equation to include transverse effects. On a much larger scale, tsunami waves can merge in similar ways.

Weight Problems in Harmonic Analysis, Especially the Fourier Transform:

Three important operators in harmonic analysis include the maximal operator, the singular integral operator, and the Fourier transform. A recurring problem in studying these operators is measuring the ''size" of an output function given some knowledge of the size of the input function--that is, finding the mapping properties of the operator. A further complication is introduced by using weighted measures of size. Determining whether an operator maps a weighted space into another weighted space is sometimes referred to as a ''weight problem" for the operator.

The weight problem is completely solved for the maximal operator,mostly solved for the singular integral operator, but unsolved for the Fourier transform. This is peculiar, since the Fourier transform is, in fact, the most widely used and oldest of the operators. In this talk I will review weight problems, their solutions, and focus especially on recent progress on the weight problem for the Fourier transform.

Large Deviations in The Reinforced Random Walk Model on Trees

In this talk, we consider the linearly reinforced and the once-reinforced random walk models in the transient phase on trees. We show the large deviations for the upper tails for both models. We also show the exponential decay for the lower tail in the once-reinforced random walk model. However,the lower tail is in polynomial decay for the linearly reinforced random walk model.

Population Persistence in River Networks

Evolution Problems in Materials with Memory & Free Energy Functionals

The Linear Diophantine Frobenius Problem: An Elementary Introduction to Numerical Monoids

Discrete Integrable Systems

TBA

Graph algebras: from analysis to algebra and back

Improving Numerical Accuracy for Solving Evolutionary PDEs in the Presence of Corner Singularities

An Independent Axiom System for the Real Numbers

Mathematics and the Ocean

Convective flows under strong rotational constraints

Convective flows under strong rotational constraints

Extremal Laws the Real Ginibre Ensemble

Incidence Algebras for Everyone

KP Solitons in shallow water: Mach reflection and tsunami

Psudospectral Methods for Optimal Control

Nelder-Mead Method and Applications

Atomic Decomposition of Some Banach Spaces and Applications

Development of Governing Equations for Unsaturated Porous Media and An Overview of Hybrid Mixture Theory

Alternative Perspectives in Module Theory

Zeta Functions and L-Functions in Number Theory

Inverse Theorems in Additive Combinatorics

Quadratures for the sphere, MRIs and radiation transport, what they have in common

Patterns induced by nucleation and growth in biological and atmospheric systems

Vanishing viscosity on networks

Primitivity in Leavitt Path Algebras

Critical Behavior for percolation in the 2D high-temperature Ising model

2011 Distinguished Math Lecture: Random Planar Maps

After the Explosion: An Analytical Look at Boundary Problems for Continuous Time Markov Chains

Symbolic Computation of Conservation Laws of Nonlinear Particle Differential Equations

Parameterization of Invariant Manifolds for Lagrangian Systems with Long-range Interactions

Solitons, boundary value problems and a nonlinear method of images

Luminescent solar concentrators, photon transport, and affordable solar harvesting

Transport and Mixing in Time-Dependent Flows

Faithful torsion modules and rings

Evans functions and the stability of viscous shock and detonation waves

On the Evolution of Perturbations to Solutions of the KP Equation using the Benney-Luke Equation

Local properties of Schubert varieties

Stochastisity and structure in biological systems: from the evolution of gene regulation to sleep-wake dynamics

Jonsson Modules

Baecklund transformations, Recursion Techniques and Noncommutative soliton solutions

A Numerical Methodology for the Painlevé equations.

Dispersive shock waves and Painleve' Trascendents

Tsunamis

Various

Approximating permutations and automorphisms

Annealed bounds for the return probability of Delayed Random Walk on finite critical percolation clusters

Unified description of Bloch envelope dynamics in the 2D nonlinear periodic lattices

A New Proof of Carleson's Theorem

The Graph Menagerie

2010 Distinguished Math Lecture: Computational Modeling of Neurons

An interacting branching particle model

The Jacobi polynomials, their Sobolev orthogonality, and self-adjoint operators

Dynamics of a Driven Spin

Sequential Selection and the Symmetric Group

Representations, quivers and periodicity

Theorems of Cohen and Kaplansky: from commutative to noncommutative algebra

Conjugation of injections by permutations

Supersonic Dispersive Fluid Dynamics

Soliton Dynamics in Inhomogeneous Media

Convex combinations of harmonic mappings

Excited Bose-Einstein Condensates: Quadrupole Oscillations and Dark Solitons

High Field Limits for magnetized plasmas

Nonconservative Transmission Line Networks, or Jordan normal form for some differential equations

"Factorizations of rational matrix functions with applications to discrete integrable systems and discrete Painlevé equations"

Harmonic Analysis for Star Graphs and the Spherical Coordinate Trapezoidal Rule

Extraordinary Waves: From Beaches to Lasers

Modeling the Electrical Activity of the Heart

Synchronization of Oscillators with Noisy Frequency Adaptation

What do stock market and positive L1 operators have in common?

Almost sure scaling limit for monotone interacting particles systems in one dimension.

Non-homogeneous random walks systems on Z

Excitation of chaotic spin waves through three-wave and four-wave interactions

Using Special Ideals to Illustrate a Research Philosophy in Ring Theory

Critical exponents for Brownian motion and random walk

Integrable Systems, Inverse Scattering Transform and Solitons

Emergent Time Scales in Ultracold Molecules in Optical Lattices [Joint Math/Physics Colloquium]

How Statistics Explain What Cancer Is

Bi-directional wave propagation in the human arterial tree

The uncanny resemblance between Leavitt path algebras and graph C*-algebras

Linear Algebra and Random Triangles

K-theory for Leavitt path algebras

Hunting for Eigenvalues of Quantum Graphs

On the Soliton Resolution Conjecture

Limit Theorems for Maximum Flows on a Lattice

The Classification Question for Leavitt Path Algebras

Bringing Matlab into Introductory Differential Equations

Parametric representations for Eisenstein series from Ramanujan's differential equations

Classification Theorems for Acyclic Leavitt Path Algebras

Sudoku and Orthogonality

Feynman diagrams, integrals, and special functions

On Leavitt path algebras over infinite graphs

Spatial self-organization in cyclic particle systems

The Multiple Personality of Schizophrenia

Diffusion Approximations to Stability and Control Problems for Stochastic Networks in Heavy Traffic

Mathematics and Biology in the 21st Century (joint math and biology colloquium)

Geometrical Multiscale Models of the Cardiovascular System