Quantized higher Teichmüller theory, as described by Fock and Goncharov, assigns an algebra and its representation to a surface and a Lie group. This assignment is equivariant with respect to the action of the mapping class group of the surface, and is conjectured to give an analog of a modular functor, that is it should respect the operation…
The cellular cytoskeleton is essential in proper cell function as well as in organism development. These filaments represent the roads along which most protein transport occurs inside cells. I will discuss examples where questions about filament-motor protein interactions require the development of novel mathematical modeling, analysis, and simulation. In the development of egg cells into…
Fusion categories are rich mathematical structures generalizing the representation categories of finite groups. They arise in many areas of mathematics and physics. Most strikingly, they have emerged as models for particle-like excitations with exotic exchange statistics in low dimensional quantum field theories. We will provide an introduction to these ideas and discuss recent results on…
A hypersurface in a Riemannian manifold is called minimal if its mean curvature vanishes identically. Minimal surfaces have fascinated mathematicians since the time of Euler, and tremendous progress has been made in understanding the structure of the space of embedded minimal surfaces in various ambient manifolds in the last 50 years. After surveying the subject, I will…
This talk will cover two approaches for modeling blood flow in the human body. The first approach describes blood transport in elastic vessels and requires the numerical solution of a nonlinear hyperbolic system on branching vessel networks. I will discuss some mathematical properties of these equations that seem to be useful for analysis of numerical schemes,…
What can we tell about the interior structure of our planet, if we observe the travel time of a large number of earthquakes? This is the time it takes for a seismic wave to travel from the epicenter of the earthquake to the seismic sensor. In the geometric literature, the boundary rigidity problem on a compact Riemannian manifold…
Many materials are built from a grid of flexible but nearly inextensible rods that behaves as a shell-like structure. Everyday examples range from fabrics made of 1000s of interwoven yarns; to kitchen strainers made of 100s of plastically deforming wires; to architectural gridshells or medical stents made of 10s of elastically deforming rods. In this…
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…