This talk will introduce parts of algebraic geometry and complex analytic geometry which are closely related to each other. These are important areas of pure mathematics. The presentation will start from scratch and hopefully reach the statement of a famous conjecture by the end. The first half of the talk should be accessible to incoming graduate students who have worked an exercise or two on projective space. Familiarity with complex analysis at the undergraduate level would be helpful. The second half of the talk will make use of cohomology. Students who have taken a semester of algebraic topology and are now taking a second semester together with a course on Riemann surfaces should be able to follow many of the details. Those who have completed a course on Riemann Surfaces and on Algebraic Geometry should be able to follow all the details. Professional mathematicians working in this general area will likely find new insights few and far between if present at all.

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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.

Our brains are composed of networks of cells, including neurons and glial cells. While the significance of neurons has been established by biologists, the role of glial cells is less understood. One hypothesis is that glial cells facilitate neural communication in nearby neurons, while suppressing communication among more distant neurons via a reaction-diffusion process. I consider this proposed mechanism using partial and ordinary differential equation models. By analyzing the ordinary differential equation model, I can determine conditions for this hypothesis to hold. I then compare the results of this analysis with simulations of the partial differential equation model and discuss the biological implications.

We will begin with the idea of General Relativity, which Einstein called his "happiest thought," and then proceed with a qualitative and quantitative discussion of the curvature of space-time. We will describe the central role of differential geometry in the subject and the important role that mathematicians have played proving the conjectures of the physicists, as well as making a few conjectures of our own. Finally, we will describe the geometry of black holes and their relationship to soap bubbles.

What do vector arrangements, discrete graphs, and perfect matchings have in common? These seemingly unrelated objects (and many others) have a very similar underlying structure, known as a matroid. As a result, studying matroids allows you to simultaneously study many different objects from all over mathematics. In addition, many properties and constructions from these various objects, such as loops, duals, bases, cycles, rank, polynomial invariants, and minors (subgraphs), generalize naturally to matroids. In this talk, we will give a general definition of a matroid, and motivate their study by examining some of these constructions in detail. The only prerequisite for this talk is basic linear algebra.

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We consider the whole-space decaying solution of second-order elliptic PDE in divergence form with space dimension d=3, where the coefficient field is a realization of a stationary, uniformly elliptic, unit range ensemble of random field, and the right-hand-side is deterministic and compactly supported in a ball of size \ell. Given the coefficient field in a large box of size L much larger than \ell, we are interested in an algorithm to compute the gradient of the solution with the "best" artificial boundary condition on the domain of size L which describes the correct long-range multipole behavior. We want to show that, with high probability, our algorithm reaches the CLT-type lower bound of error. Joint work with Jianfeng Lu and Felix Otto.

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.

In this talk I will introduce two stochastic voting systems and results we proved. (i) Axelrod's model generalizes the voter model in which individuals have one of Q possible opinions about each of F issues and neighbors interact at a rate proportional to the fraction of opinions they share. We proved that on large two-dimensional torus if Q/F is small, then there is a giant component of individuals who share at least one opinion and consensus develops on this percolating cluster. (ii) The latent voter model allows a latent period after each site flips its opinion. We will present Shirshendu's result on a random r-regular graph with n vertices that as the rate of exponential latent period $\lambda \gg \log n$, dynamics converge to coexistence behavior with quasi-stationary density = 1/2 at $O(\lambda)$ times. Using different technologies one can generalize it to the varying degree case, a.k.a. the configuration models. Joint work with Rick Durrett and Shirshendu Chatterjee

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).

The study of surfaces with constant mean curvature (CMC) goes back to 1841 when Delaunay classified all CMC surfaces of revolution. There has been consistent work on finding CMC hypersurfaces in various ambient manifolds. In this talk, we will discuss some nice properties of CMC surfaces, and then the existence of CMC surfaces in the Schwarzschild, and in general, asymptotically flat manifolds.

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.

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I'll be talking about how to construct the moduli space for genus g Riemann surfaces with r boundary components. I'll draw lots of pictures and focus a lot of attention on genus 1 Riemann surfaces with 1 boundary component. As an application, I'll probably talk about H^1(SL2(Z)) with coefficients in various representations--and the correspondence to modular forms (briefly, and without a whole lot of proofs).

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Biology has given mathematicians many new problems to work on in the last half century and the role of mathematics in biology research is only increasing. Through a series of examples, ranging from coat pattern formation to the evolution of RNA viruses, I will illustrate the insight that a mathematical treatment can give to problems in biology and will also discuss the difficulties involved in doing mathematical biology.

A sequence of functions $f = (f_i)$ ($-\infty < i < \infty$) on a surface $S$ is said to be \emph{equi-areal} (or sometimes, \emph{equi-Poisson}) if it satisfies the relations $$ df_{i-1}\wedge df_i = df_i\wedge df_{i+1}\ (\not=0) $$ for all $i$. In other words, the successive pairs $(f_i,f_{i+1})$ are local coordinates on $S$ that induce the same area form on $S$, independent of $i$. One says that $f$ is \emph{$n$-periodic} if $f_i = f_{i+n}$ for all $i$. The $n$-periodic equi-areal sequences for low values of $n$ turn out to have close connections with interesting problems in both dynamical systems and in the theory of cluster algebras. In this talk, I will explain what is known about the classification (up to a natural notion of equivalence) of such sequences and their surprising relationships with differential geometry, cluster algebras, and the theory of overdetermined differential equations. I won’t assume that the audience knows much differential geometry, just basic multi-variable calculus, and the emphasis will be on describing the interesting results rather than on technical details.

There are many different ways to make a compact 2-manifold of genus g into a Riemann surface. In fact, there is an entire space of dimension 3g-3 (when g>1) of possible holomorphic structures. This space is called the moduli space of Riemann surfaces of genus g. We will give a definition of moduli spaces and briefly talk about their construction, starting with the "easy" examples of g=0 and g=1. We will also talk about mapping class groups, which play an important part in the construction of moduli spaces.

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.

Tropical Geometry is a relatively recent field, that uses combinatorics to study algebraic geometry. In this talk, I'll introduce the tropical semiring, and the idea of tropical monomials and polynomials. We'll look at some examples of tropical hypersurfaces, and if there is time I will introduce the idea of a tropical variety, and it's relationship to the traditional algebraic variety of a curve.

Approximating the permanent of a matrix with nonnegative entries is a well studied problem. The most successful approach to date uses Markov chains, and Jerrum, Sinclair, and Vigoda developed such a method that runs in polynomial time O(n^7 (log n)^4). We present a very different approach using self-reducible acceptance/rejection, and show that for a class of dense problems, our method has an O(n^4 log n) expected running time. Also, we extend our approach to approximate the number of perfect matchings in non-bipartite graphs and general matchings in general graphs.

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.

A toric variety is a special class of algebraic variety, which can be expressed by a picture that makes some of it's properties much easier to analyze. These varieties are an area of ongoing research in algebraic geometry, due in part to their applications in many different fields (such as mathematical physics). In this talk I will discuss how we can use a fan to construct a toric variety as a quotient. This talk will be accessible to all graduate students and will focus on examples, to illustrate the usefulness of this method.

A body of literature has developed concerning “cloaking by anomalous localized resonance”. Most analytical work in this area has relied on separation of variables, and has therefore been restricted to radial geometries. In this talk, we will discuss a new approach based on a pair of dual variational principles, and apply it to some non-radial examples. In our examples, as in the radial setting, the spatial location of the source plays a crucial role in determining whether or not resonance occurs. The talk assumes minimal background knowledge.