A K3 surface is a 4-dimensional shape that appears in many areas of geometry and mathematical physics. I will define it and explore many of its amazing properties.

The prevailing numerical methods for solving the reaction-diffusion systems in models of cardiac electrical activity currently use second-order adaptive mesh refinement, refining the spatial and temporal meshes only near the traveling wavefront(s). However, in two and three spatial dimensions under biologically relevant initial conditions and forcing, these wavefronts can constitute a relatively high percentage of the computational domain, limiting the effectiveness of the scheme. In this talk, I will present a numerical scheme based on higher order finite elements and spectral deferred correction designed to improve the efficiency in computing for domains of cardiac cells.

At first glance, knot theory and representation theory seem to be unrelated fields of mathematics. In fact, this is not the case: in the early 90's, Reshetikhin and Turaev proved that knot invariants (and 3-manifold invariants) can be derived via the representation theory of quantum groups. The key link (no pun intended) between these areas is the observation that both the category of tangles and the category of representations share many similar structural features. In this talk we will explore these ideas, and if time permits, their categorified counterparts. If things like categories scare you, fear not; as the title suggests, all categories (and constructions on them) we encounter will have pictorial descriptions. In fact, no knowledge of category theory or representation theory is assumed. At the same time, if you have indeed taken Math 253, then this talk will provide context for the material in that course.

Although originally formed as an esoteric field of study, in the last few decades Algebraic Topology has emerged as a vastly applicable field. In this talk we will discuss the basics of Computational Topology and an application to one such coverage problem in sensor networks which even involves a little probability. This talk will be accessible to anyone who enjoys doing math via lots and lots of pictures.

Beyond Krull dimension, rings and modules have various "dimensions": depth, height, projective/injective dimensions, flat dimension, global dimension, weak dimension, among others. These notions are defined homologically in terms of lengths of resolutions, and Ext and Tor functors provide a way to measure them. I'll talk about how they are related with each other. I'll start from geometic interpretation of Krull dimension and height, followed by regular sequences and depth. Then I'll introduce Cohen-Macaulay modules and Gorenstein modules as modules having particularly nice homological properties. Just as in the case of completion where analysis is introduced to algebra to prove some highly nontrivial results, homological techniques have proved to be very powerful in modern commutative algebra, producing such surprising results as homological characterization of regular rings (Serre, Auslander, Buchsbaum). I'll briefly introduce the notion of canonical modules and the question of finiteness of injective resolution. Finally, I'll talk about how these notions can be globalized to scheme and sheaves, which makes geometry "kind of equivalent" to algebra.

Kick off this year's flu season with a better understanding of within-host influenza dynamics. Influenza A is a rapidly-evolving RNA virus that typically escapes herd immunity through the generation of new antigenic variants every 3 to 8 years. An important part of this antigenic evolution is believed to occur at the intrahost level. I will present two competing models of intrahost dynamics and compare their predictions to empirical observations.

We will give an update on our last talk on a new connection between differential geometry and astrophysics which involves a model for dark matter and a possible explanation for barred spiral patterns in galaxies. We will also briefly discuss the Tully-Fisher relation, a mysterious experimental fact relating the visible mass of a galaxy to the speed of the stars in the galaxy, which to this point defies a convincing theoretical explanation.

The Positive Mass Theorem in general relativity states that a spacelike hypersurface of a spacetime satisfying the dominant energy condition must have nonnegative total mass. In the special case in which the hypersurface is totally geodesic, local energy density coincides with scalar curvature, and the above theorem becomes a purely geometric statement about complete, asymptotically flat Riemannian manifolds. I will try to present the necessary background for one to understand the statement of the theorem. I will also discuss attempts to better understand the relationship between scalar curvature and total mass. Note that this talk is especially geared towards early graduate students and people specializing in other fields, and thus I will assume no previous knowledge of smooth manifolds, Riemannian geometry or general relativity.

We consider a biological learning model composed of coupled stochastic neural ensembles obeying a nonlinear gradient dynamics. The dynamics optimize a simple error criterion involving noisy observations provided by the environment, leading to a function that can be used to make decisions in the future. The uncertainty of the resulting decision function is characterized, and shown to be controlled in large part by trading off coupling strength (and/or network topology) against the ambient neuronal noise. Further connections with classical regularization notions in statistical learning theory will also be explored.

This talk will include some PDE, some probability, and some asymptotic analysis. The FKPP equation is a nonlinear partial differential equation that admits traveling wave solutions. It has been used as a simple model for many phenomena involving a stable state invading an unstable state (for example, an advantageous gene spreading through a population). Experiments and numerical simulations show that the speed at which the wave moves is much slower than what is predicted by the deterministic, continuum equation. One way to resolve this discrepancy is to account for the role of noise in the model by adding a stochastic term in the equation (i.e. a stochastic partial differential equation). Analysis of the SPDE has shown that very small noise in the equation introduces a very large correction to the speed associated with the deterministic model. I will explain the basics of the deterministic and stochastic equations, and I will explain some ideas about the asymptotic analysis of the stochastic waves. I hope to have time to explain some open problems related to this topic.

Many machine learning algorithms are based upon estimating eigenvalues and eigenfunctions of certain integral operators. In practice, we have only finitely many randomly drawn points. How close are the eigenvalues and eigenfunctions of the finite dimensional matrix we construct in comparison to the infinite dimensional integral operator? In what way can we say these two are close if they do not even operate on the same spaces? To answer these questions, I will be showing some results from a paper "On Learning with Integral Operators" by Rosasco, Belkin, and De Vito.

The resolution of Hilbert's tenth problem yields the following unsolvability result: there is no algorithm for determining whether or not a given polynomial equation p(x_1,...,x_n) = 0 with integer coefficients will admit an integer solution. After a few definitions and examples, I will discuss another well-known unsolvable problem: the word problem for finitely presented groups. It can be shown that there is no algorithm for determining when an arbitrary word in a finitely presented group is trivial. This has many remarkable topological consequences, including the result that there is no algorithm that will determine when two given manifolds are homeomorphic (provided the dimension is at least four). The unsolvability theorem also has significant geometric applications, allowing one to prove that certain manifolds admit an infinite number of contractible closed geodesics (regardless of the Riemannian structure).

Persistent Homology is an emerging field of Computational Topology that is developing tools to discover the underlying structure in high-dimensional data sets. I will discuss the origins and main concepts involved in Persistent Homology in an accessible way, with illustrations and comprehensive examples. If time allows, I will also describe some current, as well as, future applications of Persistent Homology.

I will survey some modern questions in geometry that were solved or that could be solved using tools of gauge theory. This talk should be accessible to first year grad students, and of interest to anyone who is curious about what happens in the field of geometry.

Categorification can be viewed as the process of lifting scalar and polynomial invariants to homology theories having those invariants as (graded) Euler characteristics. In this talk, we will discuss categorification in general and as manifested in specific examples (ie Khovanov homology and knot Floer homology). Examples will be given showing how the categorified invariants are stronger and often more useful than the original invariants. I will motivate categorification using familiar constructions from (very basic) topology. It is my hope that this will make the discussion accessible to a wide audience. No prior knowledge of knot theory or category theory needed!

Models for dynamical systems that include short-time or abrupt forcing can be written as impulsive differential equations. Applications include mechanical systems with impacts and models for electro-chemical spiking signals in neurons. We consider a model for spiking in neurons given by a nonlinear ordinary differential equation that includes a Dirac delta function. Ambiguities in how to interpret such equations can be resolved via perturbation methods and asymptotic analysis of delta sequences.

This talk will be a fun discussion of the mathematical aspects of preferential ballot elections (in which voters are allowed to express their rankings of all of the candidates). After describing how single vote ballots can lead to an institutionalized two party system by discouraging third party candidates, we will then discuss the various vote counting methods for preferential ballot elections and the characteristics, both good and bad, that these various methods have. We will also touch on Arrow's Paradox, one of the most over-rated "paradoxes" in mathematics, and explain how it is much less relevant to discussions of vote counting methods than is sometimes believed.

The world is in dire need of more formalisms with which to do differential geometry. Just kidding! More seriously, over the years I have encountered some complicated formulae in differential geometry and have, successfully I believe, used canonical objects sitting on frame bundles to simplify them and their proofs. I will give at least one example of such a formula, namely Simon's calculation of the Laplacian of the second fundamental form of a minimal submanifold and show how the formalism makes it and its proof simple.

In this talk I will present a ODE model Dr. Schaeffer and I have developed in collaboration with Dr. Magwene of Duke's Department of Biology. The cAMP-PKA pathway is a key signal transduction pathway through which Yeast makes developmental decisions in response to environmental cues. A novel feature of our model is that for a wide range of parameters approach to steady state includes decaying oscillations. I aim to make this talk accessible to everyone and will give an overview of all relevant biology.

In 1917, Soichi Kakeya posed the question: What is the smallest amount of area required to continuously rotate a unit line segment in the plane by a full rotation? Inpsired by this, what is the smallest measure of a set in $\mathbb{R}^n$ that contains a unit line segment in every direction? Such sets are called Kakeya sets, and can be shown to have arbitrarily small measure w.r.t. n-dimensional Lebesgue measure [and in fact, measure zero]. The Kakeya conjecture asserts that the Hausdorff and Minkowski dimension of these sets in $\mathbb{R}^n$ is $n$. In this talk, I will introduce at a very elementary level the machinery necessary to understand what the Kakeya conjecture is asking, and how the Kakeya conjecture has consequences for fields diverse as multidimensional Fourier summation, wave equations, Dirichlet series in analytic number theory, and random number generation. I'll also touch on how tools from various mathematical disciplines from additive combinatorics and algebraic geometry to multiscale analysis and heat flow can be used to obtain partial results to this problem. The talk will be geared towards a general audience.

This will be a wide ranging discussion on how geometric analysis may play an important role in understanding dark matter and the structure of galaxies.

Category theory can be described as a general mathematical theory of structures and of systems of structures. Originally developed in the 40's by Saunders Mac Lane and Samuel Eilenberg in the context of algebraic topology, category theory has since grown to serve as both an organizational tool in many areas of mathematics and as a deep theory connecting these areas. The aims of this talk are 3-fold: first, to introduce the basic notions of category theory and to give a wide range of examples; second, to show how abstract results in category theory can influence the way we think about mathematics; finally, to show how a knowledge of some general results in category theory can save us time and effort in our day to day mathematical work. Since I will be starting with the basics, this talk should be accessible to a wide audience. Students who are considering working in algebra, geometry, or topology are particularly encouraged to attend, as are any students who have ever wondered why I love covering the chalkboards of 274F with crazy-looking diagrams or why the word `natural' is the fifth most used word in my vocabulary.

The study of biological networks is an increasingly popular area of mathematical research. Many different approaches are applied to answer many different kinds of questions. We ask, "what kinds of behavior are observed in biological switching networks, and how can we produce this behavior?" This is therefore a question of modelling. We start with two different frameworks: boolean and continuous. Both are frequently used to model genetic transcription networks which are examples of switching networks. We then explore several principles of global dynamics that are true in both frameworks. We finish with some current research conjectures and sketches of proposed proofs.

I will give an introduction to the analysis, geometry, and topology of vector bundles for a general (i.e. nongeometric) audience. I will range from how vector bundles arise in Math 103 to how we can use partial differential equation techniques to extract interesting physical, algebraic, and topological information from them.

The Conley Index is a topological tool in dynamical systems which makes heavy use of basic homological techniques, particularly exact sequence. I will present a short primer on relative homology, describe the homological version of the Conley Index, and then present a diagram chase that contains information about the unstable manifolds of invariant sets in a dynamical system. This talk is aimed at students in the Algebraic Topology I course as well as anyone interested in dynamical systems.