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.

The cellular cytoskeleton is made up of protein polymers (filaments) that are essential in proper cell and neuronal function as well as in development. These filaments represent the roads along which most protein transport occurs inside cells. I will discuss several examples where questions about filament-cargo interactions require the development of novel mathematical modeling, analysis, and simulation. Protein cargoes such as neurofilaments and RNA molecules bind to and unbind from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since these transport models can be analytically intractable, we have proposed asymptotic methods in the framework of partial differential equations and stochastic processes which are useful in understanding large-time transport properties. I will discuss a recent project where we use stochastic modeling to understand how filament orientations may influence sorting of cargo in dendrites during neural development and axonal injury.

The problem of recovering a sparse Fourier signal from samples comes up in signal processing, imaging, NMR spectroscopy, and machine learning. Two major challenges involve dealing with off-grid frequencies, and dealing with signals lacking separation between frequencies. Without a minimum separation condition, the problem of frequency recovery is exponentially ill-conditioned, but the signal can still be efficiently recovered in an "improper" manner using an appropriate filter. I will explain such an algorithm for sparse Fourier recovery, and the theory behind why it works - involving some clever analytic inequalities for Fourier-sparse signals. Finally, I will discuss recent work with Xue Chen on applying these ideas to system identification. Identification of a linear dynamical system from partial observations is a fundamental problem in control theory. A natural question is how to do so with statistical rates depending on the inherent dimensionality (or order) of the system, akin to the sparsity of a signal. We solve this question by casting system identification as a "multi-scale" sparse Fourier recovery problem.

This talk consists of two aspects about solving the radiative transport through the integral formulation. The radiative transport equation has been numerically studied for many years, the equation is difficult to solve due to its high dimensionality and its hyperbolic nature, in recent decades, the computers are equipped with larger memories so it is possible to deal with the full-discretization in phase space, however, the numerical efficiency is quite limited because of many issues, such as iterative scheme, preconditioning, discretization, etc. In this talk, we first discuss about the special case of isotropic scattering and its integral formulation, then walk through the corresponding fast algorithm for it. In the second part, we try to trivially extend the method to anisotropic case, and talk about the method’s limitation and some perspectives in both theory and numerics.

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.

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.

In his letter, "Pursuing Stacks," Grothendieck advocated to Quillen for the use of "higher" categories to encode the higher homotopy of spaces. In particular, Grothendieck dreamt of realizing homotopy n-types as n-groupoids. This powerful idea both opened the field of higher dimensional algebra but also informed a paradigm in which the distinction between topology and algebra is blurred. Since then, work by Baez and Dolan among others further surveyed the landscape of higher categories and their relationship to topology. In this talk, we will explore this story, beginning with some definitions and examples of higher categories. We will then proceed to explain "the periodic table of higher categories" and the four central hypotheses of higher category theory. In particular, these give purely algebraic characterizations of homotopy types, manifolds, and generalized knots; and account for the general phenomena of stabilization in topology. No prerequisites beyond basic ideas in algebraic topology will be expected.

The Positive Mass Theorem in general relativity states that a spacelike hypersurface of a spacetime satisfying the dominant energy condition must have nonnegative total mass. In the special case in which the hypersurface is totally geodesic, local energy density coincides with scalar curvature, and the above theorem becomes a purely geometric statement about complete, asymptotically flat Riemannian manifolds. I will try to present the necessary background for one to understand the statement of the theorem. I will also discuss attempts to better understand the relationship between scalar curvature and total mass. Note that this talk is especially geared towards early graduate students and people specializing in other fields, and thus I will assume no previous knowledge of smooth manifolds, Riemannian geometry or general relativity.

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.

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).

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.

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.

Many simple card tricks rely on mathematical principles and logic. I will be talking about some of these tricks, and the interesting ideas behind them. Hopefully I will have time to show you two or three tricks, and the proof to how they work. I will be using work from Mathematical Magic by Diaconis and Graham. The exciting thing about these tricks is they do not rely upon sleight of hand, and come out looking stunning nonetheless.

We will discuss recent work on computing angle preserving mappings (a.k.a. conformal mappings) using linear methods. We will begin with an intro/reminder on what these mappings are, and why would one to compute them. Then we will discuss the results themselves which show that when choosing a good target domain, computation of angle preserving mappings can be made linear in the sense that (i) They are a solution of a linear PDE (ii) They can be approximated by solving a finite dimensional linear system and (iii) the approximates are themselves homeomorphisms and "discrete conformal".

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.

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.

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.

What do eigenvalues have to do with geometry and topology? The first part of the talk will provide a few answers to that very broad question, including a discussion of the Euler characteristic from a spectral theory perspective. The second part of the talk will be a brief introduction to my research in analytic torsion, a topological invariant defined in terms of eigenvalues. In particular I'll explain some similarities and differences between analytic torsion and Euler characteristic.

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.

Many systems employ sensors to interpret the environment. The target-tracking task is to gather sensor data from the environment and then to partition these data into tracks that are produced by the same target. The goal of sensor fusion is to gather data from a heterogeneous collection of sensors (e.g, audio and video) and fuse them together in a way that enriches the performance of the sensor network at some task of interest. This talk summarizes two recent efforts that incorporate mildly sophisticated mathematics into the general sensor arena, and also comments on the joys and pitfalls of trying to apply math for customers who care much more about the results than the math. First, a key problem in tracking is to 'connect the dots:' more precisely, to take a piece of sensor data at a given time and associate it with a previously-existing track (or to declare that this is a new object). We use topological data analysis (TDA) to form data-association likelihood scores, and integrate these scores into a well-respected algorithm called Multiple Hypothesis Tracking. Tests on simulated data show that the TDA adds significant value over baseline, especially in the context of noisy sensor data. Second, we propose a very general and entirely unsupervised sensor fusion pipeline that uses recent techniques from diffusion geometry and wavelet theory to compress and then fuse time series of arbitrary dimension arising from disparate sensor modalities. The goal of the pipeline is to differentiate classes of time-ordered behavior sequences, and we demonstrate its performance on a well-studied digit sequence database. This talk represents joint work with many people. including Chris Tralie, Nathan Borggren, Sang Chin, Jesse Clarke, Jonathan deSena, John Harer, Jay Hineman, Elizabeth Munch, Andrew Newman, Alex Pieloch, David Porter, David Rouse, Nate Strawn, Adam Watkins, Michael Williams, Lihan Yao, and Peter Zulch.

We'll undertake a gentle introduction to class field theory by investigating cyclotomic fields, including a proof of quadratic reciprocity. The results we'll discuss complement Les Saper's Grad Faculty seminar talk, though by no means is the latter a prerequisite. As a special treat, I will reveal a completely new, elementary proof of Fermat's Last Theorem.

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.)

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).

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.

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.