## Curtis Porter : CRash CouRse in CR Geometry

- Graduate/Faculty Seminar,Uploaded Videos ( 2252 Views )CR geometry studies real hypersurfaces in complex vector spaces and their generalizations, CR manifolds. In many cases of interest to complex analysis and PDE, CR manifolds can be considered ``curved versions" of homogeneous spaces according to Elie Cartanâ€™s generalization of Kleinâ€™s Erlangen program. Which homogeneous space is the ``flat model" of a CR manifold depends on the Levi form, a tensor named after a mathematician who used it to characterize boundaries of pseudoconvex domains. As in the analytic setting, the Levi form plays a central role in the geometry of CR manifolds, which we explore in relation to their homogeneous models.

## 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.

## Demetre Kazaras:The geometry and topology of positive scalar curvature

- Graduate/Faculty Seminar,Uploaded Videos ( 1663 Views )I will give an informal overview of the history and status of my field. Local invariants of Riemannian metrics are called curvature, the weakest of which is known as "scalar curvature." The study of metrics with positive scalar curvature is very rich with >100 year old connections to General Relativity and smooth topology. Does this geometric condition have topological implications? The answer turns out to be "yes," but mathematicians continue to search for the true heart of the positive scalar curvature conditions.

## Lenhard Ng : Symplectic Techniques in Topology: An Informal Introduction

- Graduate/Faculty Seminar ( 254 Views )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.

## Joseph Spivey! : Mapping Class Groups and Moduli Spaces

- Graduate/Faculty Seminar ( 216 Views )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.

## Andrew Goetz : General Relativity, Wave Dark Matter, and the Tully-Fisher relation

- Graduate/Faculty Seminar ( 211 Views )Abstract: In this talk I will give a quick overview of Einstein's theory of general relativity. I will then move on to discuss the mystery of dark matter: why astrophysicists think it's out there in the universe and what phenomena any successful theory of dark matter will have to explain. One such phenomenon is the Tully-Fisher relation, an intriguing correlation between the visible mass of galaxies and the rotational velocities of their stars. I will wrap up by describing a theory of wave dark matter and how it could possibly explain the Tully-Fisher relation.

## Jianfeng Lu : Surface hopping: Mystery and opportunities for mathematicians

- Graduate/Faculty Seminar ( 201 Views )Surface hopping is a very popular approach in theoretical chemistry for mixed quantum-classical dynamics. Yes, the above sentence looks scary. Let us start over again ... We will examine from a mathematical point of view how stochastic trajectories can be used to approximate solutions to a Schrodinger equation (which is different from what Feynman did). Besides some applications in chemistry, this is a nice topic since it combines ideas from asymptotic analysis, applied probability, and applied harmonic analysis. The only background assumed in this talk is "separation of variables" (and of course some PDEs where separation of variables is applied to).

## Spencer Leslie : Intro to crystal graphs and their connections with number theory

- Graduate/Faculty Seminar ( 199 Views )I will review some basics of crystal bases for highest-weight representations for a semisimple Lie algebra. I will also point to some connections with number theory through Fourier coefficients of Eisenstein series, mostly in type A.

## Kevin Kordek : Geography of Mapping Class Groups and Moduli Spaces

- Graduate/Faculty Seminar ( 192 Views )Mapping class groups are topological objects which can be used to describe the continuous symmetries of a surface. On the other hand, every compact orientable surface has a moduli space, a complex variety whose points parametrize all of its inequivalent complex structures. These concepts turn out to be closely related. In this talk, we'll cover the basics of both mapping class groups and moduli of Riemann surfaces, as well as explore their relationship.

## 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.

## Robert Bryant : The Concept of Holonomy

- Graduate/Faculty Seminar ( 190 Views )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.

## Robert Bryant : Curves, Surfaces, and Webs: An Episode in 19th Century Geometry

- Graduate/Faculty Seminar ( 189 Views )An old question about surfaces in 3-space is: When can a surface be written as a sum of two curves? For example, the elliptic paraboloid z = x^2 + y^2 can be thought of as the sum of the two space curves (x,0,x^2) and (0,y,y^2). However, a little thought shows that most surfaces in space should not be expressible parametrically as X(s) + Y(t) where X and Y are space curves. Surfaces for which this can be done are called `surfaces of translation'. This raises the question of determining whether or not this is possible for a given surface and in how many ways. This simple question leads to some surprisingly deep mathematics, involving complex analysis and overdetermined systems of PDE, and to other questions that are still open today. I will explain some of these developments (and what they have to do with my own work). There will even be a few pictures.

## Nan Wu : Locally Linear Embedding on Manifold with or Without Boundary

- Graduate/Faculty Seminar ( 188 Views )Locally Linear Embedding(LLE), is a well known manifold learning algorithm published in Science by S. T. Roweis and L. K. Saul in 2000. In this talk, we provide an asymptotic analysis of the LLE algorithm under the manifold setup. We establish the kernel function associated with the LLE and show that the asymptotic behavior of the LLE depends on the regularization parameter in the algorithm. We show that on a closed manifold, asymptotically we may not obtain the Laplace-Beltrami operator, and the result may depend on the non-uniform sampling, unless a correct regularization is chosen. Moreover, we study the behavior of the algorithm on a compact manifold with boundary. This talk is based on the joint work with Hau-tieng Wu.

## Hubert Bray : An Overview of General Relativity

- Graduate/Faculty Seminar ( 169 Views )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.

## 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.

## Wenjing Liao : The MUSIC algorithm for line spectral estimation: stability and super-resolution

- Graduate/Faculty Seminar ( 136 Views )The problem of spectral estimation, namely Â? recovering the frequency contents of a signal Â? arises in various fields of science and engineering, including speech recognition, array imaging and remote sensing. In this talk, I will introduce the MUltiple SIgnal Classification (MUSIC) algorithm for line spectral estimation and provide a stability analysis of the MUSIC algorithm. Numerical comparison of MUSIC with other algorithms, such as greedy algorithms and L1 minimization, shows that MUSIC combines the advantages of strong stability and low computational complexity for the detection of well-separated frequencies on a continuum. Moreover, MUSIC truly shines when the separation of frequencies drops to one Rayleigh length and below while all other methods fail. This is a joint work with Albert Fannjiang at UC Davis. The talk involves basic linear algebra and Fourier analysis and it will be accessible to all.

## 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.

## 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.

## Dave Rose : Graphical calculus and quantum knot invariants

- Graduate/Faculty Seminar ( 126 Views )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.

## Shishi Luo : Modelling intrahost influenza dynamics

- Graduate/Faculty Seminar ( 120 Views )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.

## Caitlin Leverson : Legendrian Knots, Augmentations, and Rulings

- Graduate/Faculty Seminar ( 118 Views )Given a contact structure (a plane field) on R^3, one can define a Legendrian knot to be an embedding of the circle such that the embedding is everywhere tangent to the plane field. Surgery along such a knot gives a way to construct new manifolds and so there is interest in classifying Legendrian knots. This turns out to be a finer classification than that of topological knots -- there are many different Legendrian unknots. Given a Legendrian knot, one can associate the Chekanov-Eliashberg differential graded algebra (DGA) generated by the crossings and then find augmentations of this DGA much like those in your standard algebraic topology course. This talk will give an overview of the relationships Joshua Sabloff and Dmitry Fuchs gave between such rulings and augmentations and how it relates to my current work.

## Gabor Szekelyhidi : Extremal Kahler metrics and the Yau-Tian-Donaldson conjecture

- Graduate/Faculty Seminar ( 118 Views )I will first explain with a few simple examples a beautiful picture in geometric invariant theory which relates quotient constructions in symplectic and algebraic geometry. Then we will naively apply this picture in a suitable infinite dimensional setting, leading us to the notion of extremal Kahler metrics and the Yau-Tian-Donaldson conjecture, which is an important problem in Kahler geometry today.

## Mauro Maggioni : A primer on wavelets and their applications

- Graduate/Faculty Seminar ( 117 Views )Wavelets are widely used in signal processing (e.g. analysis of sounds and music) and imaging, for tasks such as denoising and compression (ever wondered how jpeg works?). In harmonic analysis they have been used to understand and solve problems involving integral operators motivated by PDEs. In numerical PDEs they lead to fast algorithms for solving certain types of integral equations and PDEs. I will give a gentle introduction to wavelets and some of their motivating applications, accompanied by live demos. If time allows, I will discuss shortcomings and how they have been addressed in more recent developments and generalizations.

## Chris O'Neill : An Introduction to Ehrhart Theory and Lattice Point Enumeration

- Graduate/Faculty Seminar ( 111 Views )A polytope is a bounded subset of R^d which is the intersection of finitely many half-spaces. Given a polytope P, we can consider integer dilations of P, and ask how many integer points are contained in each dilation, as a function of the dilation factor. Under the right conditions, this counting function is a polynomial with some very interesting and unexpected properties. To demonstrate the usefulness of these results, we will give alternative proofs to some well known results from far outside the realm of geometry.

## Kevin Gonzales : Modeling mutant phenotypes and oscillatory dynamics in the cAMP-PKA pathway in Yeast Cells

- Graduate/Faculty Seminar ( 111 Views )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.

## Dave Rose : Categorification and knot homology

- Graduate/Faculty Seminar ( 110 Views )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!

## Matthew Junge : The inevitable Ewens distribution

- Graduate/Faculty Seminar ( 107 Views )Consider the following three questions: How many uniformly random permutations are needed to invariably generate the symmetric group? How many iterations of a Monte-Carlo algorithm are needed to decide whether a polynomial in Z[x] splits? How many sumsets formed from independent Poisson multistep are needed to have a chance of an empty intersection? And their answer: Four. What is so special about four? Not much. These all are special cases of the ubiquitous Ewens sampling formula. Come and find out why.