## Margaret Regan : Using homotopy continuation to solve parametrized polynomial systems in applications

- Graduate/Faculty Seminar,Uploaded Videos ( 1832 Views )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.

## Sarah Schott : Computational Complexity

- Graduate/Faculty Seminar ( 274 Views )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.

## Aaron Pollack : Modular forms on exceptional groups

- Graduate/Faculty Seminar ( 238 Views )Classically, a modular form for a reductive group G is an automorphic form that gives rise to a holomorphic function on the symmetric space G/K, when this symmetric space has complex structure. However, there are very interesting groups G, such as those of type G_2 and E_8, for which G/K does not have complex structure. Nevertheless, there is a theory of modular forms on these exceptional groups, whose study was initiated by Gross-Wallach and Gan-Gross-Savin. I will define these objects and describe what is known about them.

## Joseph Spivey : A How-To Guide to Building Your Very Own Moduli Spaces (they make such great gifts)

- Graduate/Faculty Seminar ( 238 Views )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).

## Dmitry Vagner : Introduction to Diagrammatic Algebra

- Graduate/Faculty Seminar ( 234 Views )We show how algebraic relations can be encoded in suggestive topological diagrams and use this to prove various algebraic equations in a purely pictorial way. We will first go over a few canonical examples: monoids, self-dual objects, Frobenius algebras, and monads. Then we will briefly discuss the underlying theory that makes this miracle rigorous.

## Zhennan Zhou : Semi-classical Schrodinger equation in the electromagnetic field: approximations and numerics

- Graduate/Faculty Seminar ( 228 Views )I will discuss the semi-classical Schrodinger equation with vector potentials, and its challenges in analysis and in numerical simulations. The time splitting spectral method method will be introduced to solve the equation directly, which is believed to have the optimal mesh strategy. Afterwards. a series of wave packet based approximation approaches will be introduced, like the Gaussian beam method, Hagedorn wave packets method and the Gaussian wave packet transformation method.

## Hubert Bray : What do Black Holes and Soap Bubbles have in common?

- Graduate/Faculty Seminar ( 212 Views )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.

## Chung-Ru Lee : Introduction to Trace Formula

- Graduate/Faculty Seminar ( 208 Views )The Trace Formula can be understood roughly as an equation relating spectral data to geometric information. It is obtained via expansion of the trace of certain operators that are associated to the Representation Theory of an affine algebraic group, justifying its name. Therefore, the spectral side of the expansion by nature contains data of arithmetic interests. However, the spectral side is generally less accessible. Meanwhile, the geometric side consists of terms that can be written in a more explicit fashion. The computation of the geometric side, which is now referred to as the Orbital Integrals, thus come on the scene. In this talk, we plan to briefly introduce the general derivation of the (vaguely described) Trace Formula, and demonstrate a few concrete examples of it.

## Pam Miao Gu : Factorization tests and algorithms arising from counting modular forms and automorphic representations

- Graduate/Faculty Seminar ( 203 Views )A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on~$\Gamma_0(N)$ to a simpler function of $k$ and~$N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on~$\Gamma_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found to not be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input. (Joint work with Greg Martin.)

## Yuriy Mileyko : Enter Skeleton: a brief overview of skeletal structures

- Graduate/Faculty Seminar ( 200 Views )Skeletal structures, such as medial axis and curve skeleton, are a particular class of shape descriptors. They have numerous applications in shape recognition, shape retrieval, animation, morphing, registration, and virtual navigation. This talk will give a brief overview of the medial axis and the curve skeleton. The focus will be on the properties of the two objects crucial to applications. We shall show that the rigorous mathematical definition of the medial axis has allowed for an extensive and successful study of such properties. The curve skeleton, on the other hand, is typically defined by the set of properties it has to possess. As a result, numerous methods for computing the curve skeleton have been proposed, each providing mostly experimental verification of the required properties. If time permits, I will mention my work on defining shape skeleta via persistent homology, thus providing a powerful platform for investigating their properties.

## Timothy Lucas : Numerical Solutions of an Immunology Model

- Graduate/Faculty Seminar ( 190 Views )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.

## Bill Allard : The Boundary Finder

- Graduate/Faculty Seminar ( 185 Views )(This abstract is in TeX source code. Sorry!) Fix a small positive number $h$. Let $$G=h\mathbb{Z}^2=\{(ih,jh):i,j\in\mathbb{Z}\};$$ thus $G$ is a rectangular grid of points in $\mathbb{R}^2$. Let $\Omega$ be an bounded open subset of $\mathbb{R}^2$ with $C^1$ boundary and let $E=\{x\in G:x\in\Omega\}$. {\bf Question One.} Given $E$ can one determine the length of $\partial\Omega$ to within $O(h)$? The answer to this question is ``yes'', provided $\Omega$ satisfies a certain natural ``thickness'' condition; without this additional assumption the answer may be ``no''. {\bf Question Two.} Is there a fast algorithm for determining the length of $\partial\Omega$. The answer to this question also ``yes''. In this talk I will describe the proof that the answer to Question One is ``yes'' and I will describe the fast algorithm whose existence is implied in the answer to Question Two. If time permits, I will describe some applications.

## Leslie Saper : Quadratic Reciprocity from Euler to Langlands

- Graduate/Faculty Seminar ( 179 Views )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.

## Michael Reed : The Ear for Mathematicians

- Graduate/Faculty Seminar ( 179 Views )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!

## Abraham Smith : DEs to EDS: How to solve PDEs without being clever

- Graduate/Faculty Seminar ( 174 Views )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.

## Mark Stern : Frommers guide to vector bundles

- Graduate/Faculty Seminar ( 150 Views )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.

## Kash Balachandran : The Kakeya Conjecture

- Graduate/Faculty Seminar ( 148 Views )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.

## Paul Bendich : Topological Data Analysis

- Graduate/Faculty Seminar ( 146 Views )TDA is now about fifteen years old, and is quickly becoming a widely applied tool in data analysis. In this talk, I'll describe how homology groups, a traditional algebraic invariant, can be turned into persistence diagrams, a robust statistical tool for dealing with high-dimensional data or embedded geometric objects. Time permitting, this talk should have some theory, some applications, and some algorithms, and perhaps even a proof.

## Christopher O'Neill : Mesoprimary Decomposition of Binomial Ideals using Monoid Congruences

- Graduate/Faculty Seminar ( 143 Views )Many algebraic objects are notorious for being easy to define, but hard to find explicitly. However, certain algebraic objects, when viewed with the "correct" combinatorial framework, become much easier to actually find. This allows us to compute much larger examples by hand, and often gives us insight into the object's underlying structure. In this talk, we will define irreducible decompositions of ideals, and explore their underlying combinatorial structure in the special case of monomial ideals in polynomial rings. As time permits, we will look at recent results in the case of binomial ideals. This talk will be accessible to anyone who has taken a course in Abstract Algebra.

## Lillian Pierce : Class numbers of quadratic number fields: a few highlights on the timeline from Gauss to today

- Graduate/Faculty Seminar ( 142 Views )Each number field (finite extension of the rational numbers) has an invariant associated to it called the class number (the cardinality of the class group of the field). Class numbers pop up throughout number theory, and over the last two hundred years people have been considering questions about the growth and divisibility properties of class numbers. WeÂ?ll focus on class numbers of quadratic extensions of the rationals, surveying some key results in the two centuries since the pioneering work of Gauss, and then turning to very recent joint work of the speaker with Roger Heath-Brown on averages and moments associated to class numbers of imaginary quadratic fields.

## Sean Lawley : Stochastic Switching with both Mathematical and Biological Surprises

- Graduate/Faculty Seminar ( 134 Views )Motivated by several biological questions, we consider ODEs and PDEs with stochastically switching right-hand sides and PDEs with stochastically switching boundary conditions. In a variety of situations, we prove that the system exhibits surprising behavior. In this talk we will outline this class of problems and highlight some of the most interesting and unexpected results. The talk will be accessible to all graduate students.

## Erin Beckman : A Look at Branching Processes

- Graduate/Faculty Seminar ( 132 Views )In 1873, a man named Francis Galton posed a question in Educational Times, calling for the mathematical study of the extinction of family surnames over time. Within a year, mathematician Henry Watson replied with a solution. But instead of ending there, this question opened up a new direction of mathematics: the study of branching processes. A branching process is a particle system in which the particles undergo splitting or branching events dictated by particular rules. This talk will introduce some examples of these systems (from the basic Galton-Watson model to more general branching-selection models), interesting questions people ask about branching processes, and some recent research done in this area.

## Andrew Goetz : The Einstein-Klein-Gordon Equations, Wave Dark Matter, and the Tully-Fisher Relation

- Graduate/Faculty Seminar ( 131 Views )We describe a geometric theory of dark matter called "wave dark matter" whose underlying equations are the Einstein-Klein-Gordon system of PDEs. In spherical symmetry this system has simple static state solutions which we use to model dark galactic halos. We outline some scaling properties of these states including two new boundary conditions which might account for the existence of an astrophysical scaling relation called the baryonic Tully-Fisher relation.

## Tianyi Mao : Modular Forms and Additive Number Theory

- Graduate/Faculty Seminar ( 131 Views )Abstract: Modular forms, a kind of SL2(Z)-invariant holomorphic functions defined on the upper half plane, are one of the most important objects studied in modern number theory. This talk will start from the basic definitions of modular forms and give some examples and important theorems associated with Eisenstein series. Finally we will use the power of modular forms to solve some classical problems on partitions of integers in additive number theory, including the Ramanujan congruence and sums of squares.