In geometry, there are certain structures that are "rigid" (like Riemannian manifolds) and others that are "flexible" (like topological manifolds). Symplectic geometry lies in between these two extremes and incorporates some attractive features of both. One consequence is that symplectic techniques have recently been used, to great effect, to give combinatorial approaches to questions in topology that previously required difficult gauge-theoretic and analytic techniques. I will introduce symplectic structures and describe some recent developments linking them to the study of three-dimensional manifolds and knots. No real background will be assumed.

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.

The immune system in vertebrates is composed of individual cells called lymphocytes which work together to combat antigens such as bacteria and viruses. Upon detecting foreign molecules these immune cells secrete soluble factors that attract other immune cells to the site of the infection. The soluble factors are governed by a system of reaction-diffusion equations with sources that are centered on the cells. The motion of the cells is inherently stochastic, but biased toward the gradient of the soluble factors. I will discuss a numerical method for solving the reaction-diffusion stochastic system based on a first order splitting scheme. This method makes use of known first order schemes for solving the diffusion, the reaction and the stochastic differential equations separately.

TBA

Contact geometry is the study of geometric structures on odd dimensional smooth manifolds given by a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability; these hyperplane fields are called contact structures. The associated one form is called a contact form and uniquely determines a vector field called the Reeb vector field on the manifold. Contact and symplectic geometry are closely intertwined and we explain how one can make use of J-holomorphic curves to obtain contact invariants. This talk will have lots of examples, cool pictures, and animations illustrating these fascinating concepts in contact geometry.

The law of quadratic reciprocity was conjectured by Euler and first proved in full generality by Gauss. I will not prove quadratic reciprocity but I will discuss it in the context of the general reciprocity law due to Emil Artin. I will then explain how Langlands's program is a nonabelian generalization of this. If there is time, I will indicate how my work fits into this program.

This talk is directed to anyone who cares about anything, at all levels. In particular, it will be a soft introduction to exterior differential systems (EDS). EDS is often associated with differential geometry, but it is really just a language for understanding the solution space of differential equations. The EDS viewpoint is temporarily mind-bending, but its concise and clean description of integrability, from conservation laws to geometric invariants, justifies the initial cramps.

Math Slam

Simply put, complex Hadamard matrices are scaled unitary matrices with entries drawn from unit complex numbers. They appear as an essential ingredient in quantum information theory and their real members have deep connections to finite geometry and number theory. For us, in this talk, they will be the fixed points of both discrete and continuous dynamical systems. We begin by introducing complex Hadamards and some essential preliminaries. We then discuss a discrete-time dynamical system which can be used to generate complex Hadamards as well as closely related objects known as mutually unbiased bases. Finally, we construct a continuous system whose fixed points are complex Hadamards and exploit classical results from dynamical systems theory to study local structure in spaces of complex Hadamards.

Early in the history of complex analysis, it was realized that there are no continuous versions of the square root or the logarithm on the entire complex plane; instead, analysts invented multi-valued functions to deal with these strange behaviors. The "graphs" of these multi-valued functions can get very interesting, and can be interpreted topologically. In general, the space of solutions to a "nice" system of holomorphic ordinary differential equations on the non-zero complex numbers will not be made up of functions, but of multi-functions. Studying these spaces of solutions have led to several ideas in algebraic topology, especially monodromy, and the relationship between systems of ODE and possible monodromies is called the Riemann-Hilbert Correspondence.

We'll look at a random process on the integer lattice $/mathbb{Z}^2$ known as the voter model. Let's suppose that each point on the lattice represents a single household with one voter who holds one of two possible opinions, 0 or 1 (e.g. Republican or Democrat). Starting with an initial configuration of 0's and 1's on $/mathbb{Z}^2$, a voter changes their opinion at a rate proportional to the number of neighbors holding a different opinion. I'll demonstrate a clever and useful approach to analysing the voter model via the dual process. We'll be able to determine whether or not our process has any nontrivial stationary distributions. I'll also briefly discuss the newest research on variants of the voter model.

After brief introductions to special relativity and the foundations of differential geometry, we will discuss the big ideas behind Einstein's theory of general relativity. Einstein's theory replaces Newtonian physics not only as the best description of gravity according to experiments, but also as a philosophically pleasing and very geometric idea, which Einstein called his "happiest thought." We will also discuss the predictions made by general relativity, including the big bang and black holes, both of which are strongly supported by observations. We will discuss these ideas from a geometric perspective, and discuss some of the open problems and future directions that are currently being studied.

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.

In the 19th century, people began to study mechanical systems in which motion in a configuration space was constrained by 'no slip' conditions, such as, for example, a wheel or a ball rolling on a plane without slipping. It was immediately noticed that there were many cases in which these 'rolling' constraints did not prevent one from being able to join any two points in a configuration space by an admissible path, and these situations were called 'non-holonomic'. The notion of 'holonomy' arose as a way to quantify and study these 'non-holonomic' systems, and it has turned out to be very fruitful, with many applications in differential geometry and mathematical physics as well as in practical mechanics problems (such as figuring out how to use robot hands to manipulate 3-dimensional objects). In this talk, I'll introduce the ideas that led to the development of the concept of holonomy, show how some simple examples are computed, and describe how even very simple systems, such as a convex surface rolling over another surface without slipping or twisting, can lead to some surprising and exceptional geometry. No expertise in differential geometry will be assumed; if you are comfortable with vector calculus, you can enjoy the talk.

An introductory lecture to the Conley Index theory. We consider the flow case and introduce the key object of study: an index pair of an isolated invariant set. Index pairs are robust under perturbations and their homotopy type is invariant, making them an ideal tool for problems with error terms or even data-generated systems. The relevant tools are algebraic topology and some knowledge of continuous flows.

The ear from the outside in. Eardrum, middle ear, cochlea, 8th nerve, brainstem, cortex. What happens anyway when you listen to Mozart or Van Halen? How do pressure waves become electrical signals? What happens next? Is there deep mathematics in the auditory system? And what are those carteliginous things doing flapping in the breeze on the side of your head? Who says an abstract has to have declarative sentences? Will some of these questions be answered? Come and see!

Optimization problems in material science often require quickly varying composites of materials. In this talk, basic homogenized properties of composites and laminates are discussed. The basic theory is applied to construct and optimize interesting structures, such as field rotators and conducting wheels. The talk will be accessible to everyone.

This talk will be concerned with two problems. The first is the zeros of the derivatives of. Kac random polynomials K_n, which is a random polynomial with i.i.d. coefficients. It has been shown that the empirical measure of zeros of K_n will converge to the uniform measure on the unit circle of complex plane. Same convergence holds true for nay fixed order of derivative of K_n. In a joint work with Renjie Feng, we show if we consider the N_n-th order of derivative of K_n, then asymptotic behavior of empirical measure of this derivative will depend on the limit of \frac{N_n}{n}. In particular, as long as this ratio is greater than 0, the phenomenon of ‘zeros clustering around unit circle’ breaks down. The second talk is about Average Nearest Neighbor Degree (ANND), which is a measure for the degree-degree correlation for complex network. We shall be concerned with the probabilistic properties of ANND in the configuration model. We prove if the variable X generating the network has order of moment larger than 2, then the ANND(k) will converge uniformly to μ2/μ1, where μ2 is the second moment of X, and μ1 is the first moment. For the case that X has infinite variance, we show the pointwise (i.e., for fixed k) scaled convergence of ANND(k) to a stable random variable. This is joint work with Nelly Litvak and Pim van der Hoorn. More recently, Clara Stegehuis showed that when X is sample from the Pareto distribution, then one can obtain a complete spectrum of ANND(k) for the erased configuration model.

THIS JUST IN - An Abstract: I plan to speak about the good old Calabi Conjecture, and its beautiful solution by Yau, that gave gim the Fields Medal. I will start with some background material, and see how far we can get into the proof!

Einstein's General Relativity is a geometric theory of space, time, and gravitation. In some sense, it is the successor of Newton's famous theory of gravitation -- the theory Newton is said to have come up with when an apple fell onto his head. But although Einstein's theory is much better at predicting gravitational effects in our universe, Newton's theory is not at all outdated or even obsolete. In fact, many astrophysical measurements and simulations still heavily rely on Newtonian intuitions, calculations, and concepts. In the talk, I will explain how and to what extent this usage of Newtonian theory in astrophysics and related fields is motivated and mathematically justified. This will lead us to the notion of Newtonian limit. We will also see some examples for the behavior of relativistic quantities like mass and center of mass under this Newtonian limit.

In a chemical reaction system, the concentrations of chemical species evolve in time, governed by the polynomial differential equations of mass-action kinetics. This talk provides an introduction to chemical reaction network theory, and gives an overview of algebraic and combinatorial approaches to determining whether a chemical reaction network admits multiple steady states. In general, establishing the existence of (multiple) steady states is challenging, as it requires the solution of a large system of polynomials with unknown coefficients. However, for networks that have special structure, various easy criteria can be applied. This talk will highlight results from Deficiency Theory (due to Feinberg), criteria for multistationarity for chemical reaction systems whose steady states are defined by binomial equations (in joint work with Carsten Conradi, Mercedes Pérez Millán, and Alicia Dickenstein), and a classification of small multistationary chemical reaction networks (in joint work with Badal Joshi).

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.