Pertussis, commonly known as whooping cough, is caused by the bacterial pathogen Bordetella pertussis. Completely susceptible individuals experience severe disease, with the hallmark whooping cough, but those with partial immunity have milder, if any symptoms. Immunity following natural infection (or immunization) may wane, increasing susceptibility with time since exposure. In this talk, we begin by examining…

Students often take precalculus or college algebra as a terminal math course, leaving them with the impression that mathematics lacks real meaning. Due to the increasingly interdisciplinary nature of the mathematical sciences, we are well-poised to intervene and design inspiring general education courses that reveal the utility of mathematics. In this talk, I will share…

A circle packing is the mathematical name for a collection of circles. I am interested in circle packings with a fixed pattern of tangencies between the circles. Given a tangency pattern, one might ask questions like, "Can I find a circle packing with that tangency pattern?" and "How many such circle packings can I find?"…

In this demo, we will discover some interesting properties about symmetry by starting with some special transformation matrices. This talk will be a combination of interactive work with the material and some discussion of teaching strategies.

How could you design a stadium so that a rectangular playing field looks the same size to every spectator? What about for a circular wrestling ring? In this talk, we study these and related questions, which can all be viewed as generalizations of Thales' Theorem---that a line segment L in the plane "looks the same…

Experience suggests that uncertainties often play an important role in quantifying the performance of complex systems. Therefore, uncertainty needs to be treated as a core element in the modeling, simulation, and optimization of complex systems. In this talk, I will first present a review of the novel UQ techniques I developed to conduct stochastic simulations…

Principal component analysis (PCA) is a useful tool when trying to uncover factor models from historical asset returns. For the implied volatilities of U.S. equities there is a PCA-based model with a principal eigenportfolio whose return time series lies close to that of an overarching market factor. Specifically, this market factor is the index resulting…

Persistent homology is a key method in topological data analysis, a young but rapidly growing field at the intersection of computational geometry and algebraic topology. Persistence is used to describe the shape of data in a way that generalizes clustering: besides considering what connected components (clusters) are present in the data, it also describes their…

Many estimates in analysis  have certain “common structures” which makes it possible to obtain them with  what is now called Bellman function method. Originally the method appeared in control theory (stochastic or deterministic), however its systematic use  in harmonic analysis or probability started only recently  in works of Burkholder where he obtained the sharp constants…

Free-boundary problems are partial differential equations in which the unknown function and its domain must be simultaneously determined. They arise ubiquitously as mathematical models for phenomena in many fields, most notably in physics, biology and finance. Free boundary problems are typically highly nonlinear and nonlocal in nature, making their analysis challenging. I will discuss two fundamental…

A central question in machine learning theory is whether an algorithm enjoys good generalization, which is the ability to correctly predict new examples from prior observations. While classical wisdom advocates for methods with fewer parameters than data points in order to avoid overfitting, modern machine learning algorithms are severely over-parameterized and perfectly fit training data.…

Thin liquid films flowing down vertical fibers exhibit complex and interesting interfacial dynamics, including droplet formation and traveling wave patterns. Such dynamics play a crucial role in the design of heat and mass exchangers for many engineering applications, including cooling and desalination systems. Recent experiments present a wealth of new dynamics that illustrate the need…

Computerized tomography techniques, which are used for visualizing the interior structure of an object of interest in a non-invasive manner, have played a central role in medical imaging, industrial non-destructive testing, geophysics, astronomy, and other fields. Yet, the quest for more sensitive, reliable, robust, safer, and cheaper imaging methods is ongoing and has intensified in…

We focus on analysis and data-driven algorithms for rare events such as essential conformational transitions in biochemical reactions which are modeled by Langevin dynamics on manifolds. We first reinterpret the observed transition paths from the stochastic optimal control viewpoint, which realizes the transitions almost surely. Then based on collected high dimensional point clouds and nonlinear…

Energy-driven pattern formation is ubiquitous in nature; the character and dynamics of such patterns is selected as local minimizers and gradient flows, respectively, of non-convex, and often, non-local energies with multiple spatio-temporal scales. Analysis of such patterns sheds valuable insight upon their origins, and from the viewpoint of applications, is necessary for their control. In this talk, after introducing a…

Sam Hopkins will present a heuristic for finding special families of partially ordered sets. The heuristic says that the posets with order polynomial product formulas are the same as the posets with good dynamical behavior. Here the order polynomial is a certain enumerative invariant of the poset. Meanwhile, the dynamics includes promotion of linear extensions,…

A fundamental problem from population biology is finding conditions under which interacting species coexist or go extinct. I present results that lay the foundation for a general theory of stochastic coexistence. This theory extends and makes rigorous Modern Coexistence Theory and leads to resolving a number of conjectures due to Chesson, Ellner, and Palis. I…

Many natural and social phenomena involve individual agents coming together to create group dynamics, whether they are cells in a skin pattern, voters in an election, or pedestrians in a crowded room. Here I will focus on the specific example of pattern formation in zebrafish, which are named for the dark and light stripes that…

In algebraic combinatorics, the Robinson-Schensted-Knuth algorithm is a fundamental correspondence between words and pairs of semistandard tableaux illustrating identities of dimensions of irreducible representations of several groups. In this talk, I will present a generalization of the Robinson-Schensted-Knuth algorithm to the insertion of two-row arrays of multisets. This generalization leads to new enumerative results that have…

I will give a tour of the origins of Heegaard Floer homology and its applications in topology and representation theory, highlighting recent work that relates Heegaard Floer homology with a tensor product operation for higher representations as well as with new geometric constructions. https://sites.google.com/usc.edu/manion/home