Many problems that arise in mathematics, science, and engineering can be formulated as solving a parameterized system of polynomial equations which must be solved for given instances of the parameters. One way to solve these systems is to use a common technique within numerical algebraic geometry called homotopy continuation. My talk will start with background on homotopy continuation and parametrized polynomial systems, followed by applications to problems in computer vision and kinematics. Of these, I will first present a new approach which uses locally adaptive methods and sparse matrix calculations to solve parameterized overdetermined systems in projective space. Examples will be provided in 2D image reconstruction to compare the new methods with traditional approaches in numerical algebraic geometry. Second, I will discuss a new definition of monodromy action over the real numbers which encodes tiered characteristics regarding real solutions. Examples will be given to show the benefits of this definition over a naive extension of the monodromy group (over the complex numbers). In addition, an application in kinematics will be discussed to highlight the computational method and impact on calibration.

What does it mean for a problem to be in P, or NP? What is NP completeness? These are questions, among others, that I hope to answer in my talk on computational complexity. Computational complexity is a branch of theoretical computer science dealing with analysis of algorithms. I hope to make it as accessible as possible, with no prior knowledge of algorithms and running times.

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 Â?zeros clustering around unit circleÂ? 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.

In this talk we will explore a mathematical data analysis tool called persistent homology and look specifically into how we can turn topological information into useful data for statistical techniques. The problem is one of translation: persistent homology outputs a module, but statistics is formulated for objects in metric, vector, Banach, and Hilbert spaces. We'll see some of the ways this issue can be dealt with in a special case (single-parameter persistence) and discuss which of those techniques are viable for a more general case (multiparameter persistence).

In a recent paper of Pierce, Turnage-Butterbaugh, and Wood, the authors proved an effective Chebotarev density theorem for families of number fields. Notably D_4-quartic fields have not been treated in their paper. In this talk, I will explain the importance of the case for D_4-quartic fields and will present my proof of a Chebotarev density theorem for certain families of D_4-quartic fields. The key tools are a lower bound for the number of fields in the families and a zero-free region for almost all fields in the families.

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.

Do you ever wanted to know how many lines in 3−space intersect 4 given random lines ? (Answer: 2.) One way to prove this is to do explicit computations in the cohomology of the Grassmannian of lines in the projective space. But interestingly enough, one can also use Representation Theory, or symmetric functions (Schur polynomials), to answer this question. The aim of this talk is to present the basics of Schubert Calculus, as seen from the cohomological point of view. I will define Schubert varieties in Grassmannians, and discuss about how they intersect. The final goal is to show that 2 = 1+ 1 (and I may also use KnutsonÂ?s puzzles for another proof of this).

I will discuss the rich mathematical structures which arise when one asks how to define acceleration in the absence of a preferred coordinate system. I will introduce the Yang-Mills equations, which specialize to give electromagnetism and much of the physics of the standard model. I'll discuss aspects of the geometry, topology, and analysis of the Yang-Mills equations and how too much symmetry can actually make an analysis problem more difficult.

In many problems in fluids and electromagnetics, we may formulate solutions to the Dirichlet and Neumann problems in terms of double and single layer potentials. Such boundary integral representations can result in computational difficulties at points on and near the boundary due to singularities and near singularities. The case of a smooth closed boundary has been well-studied, but I will focus on computational issues that arise from a boundary that is only piecewise smooth, consisting of connected curve segments. I will give an overview of my research in approximating singular and nearly singular integrals, as well as discuss an approach for the computation of double layer potentials at points on and near a curve segment.

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.

Is it true that two polygons in the plane have the same area if and only if they can be decomposed into congruent polygons? What about in three and higher dimensions? And what about the analogous questions for polygons in the hyperbolic plane and polyhedra in higher dimensional hyperbolic spaces? Some aspects of the subject are elementary, while others involve Abel's dilogarithm and arcane subjects, such as algebraic K-theory. This talk will be largely elementary, and fun.

Motivated by the dynamic feedback systems in the kidney, we consider time-delayed transport equations with stochastic left-hand boundary conditions. We first prove the existence and uniqueness of the steady-state solution for the deterministic case with sufficiently small delay (e.g., zero left-hand boundary). Likewise, we prove those for the stochastic case using the similar analytic techniques. In this talk we will show the process of model formulation with biological motivation and address the role of time-delay and stochastic boundary on the solution behaviors. The talk should be accessible to all graduate students who are familiar with basic ODE theory.