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 Cartan’s generalization of Klein’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.

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.

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.

Many classes of mathematical objects turn out to be classified in the same way --- two infinite series and 3 "exceptional" objects. These include symmetries of 3-dimensional solids, rigid singularities, certain types of Lie algebras, positive definite even intersection forms, etc. Discovering why such classes should have the same classification has led to many beautiful ideas and observations. I will give a review of some of the basic ideas (assuming very little in the way of prerequisites) and I may have time to say why string theory has been important in this context.

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.

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.

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.

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).

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.

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.

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.

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 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.

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.

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.

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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.

A K3 surface is a 4-dimensional shape that appears in many areas of geometry and mathematical physics. I will define it and explore many of its amazing properties.

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.

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.

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.

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.

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.

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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.

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.

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.

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.

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.

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.