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.

I will talk about machine learning ideas and techniques used in my art-related projects, in particular the platypus project of cradle-removal in digitized X-ray images of paintings. Using machine learning techniques, we can extract and remove wood-grain patterns from the cradle in X-ray images which cannot be done using any existing Photoshop tools. I will also talk about interesting machine learning problems involved in other projects if time permits.

A very important (and difficult) problem for number theorists is to determine all the rational solutions to a polynomial equation defined over the rational numbers. The oldest nontrivial case, which dates back to Pythagoras, is that of finding all the rational points on a unit circle. In this talk, we will consider the case of elliptic curves, where the rational points have the structure of an Abelian group G under a curiously defined addition law. We will develop some preliminaries and introduce the classical height machinery, a powerful tool which helps us understand the complexity of the points of G. We will look at some important results about the height and about G and see what still remains very elusive.

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.

Simply put, complex Hadamard matrices are scaled unitary matrices with entries drawn from unit complex numbers. They appear as an essential ingredient in quantum information theory and their real members have deep connections to finite geometry and number theory. For us, in this talk, they will be the fixed points of both discrete and continuous dynamical systems. We begin by introducing complex Hadamards and some essential preliminaries. We then discuss a discrete-time dynamical system which can be used to generate complex Hadamards as well as closely related objects known as mutually unbiased bases. Finally, we construct a continuous system whose fixed points are complex Hadamards and exploit classical results from dynamical systems theory to study local structure in spaces of complex Hadamards.

The human heart is a multi chambered organ using valves and compression to create pressure heads, which drive blood flow. Some organisms don't share this type of heart structure, namely, very small invertebrates and embryonic vertebrates. We begin by discussing scaling in fluid dynamics and why merely existing at such a small scales can cause difficulty in effectively transporting fluid. We then move to describing effective pumping mechanisms in valveless tubular hearts and how numerical modeling can help understand how these organisms are able to transport fluid, despite their limitations. Once we have a numerical framework to investigate these organisms we can then begin to add more biological structure in our models and begin to investigate optimal regimes of transport.

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.

Suppose ball-shaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data (i.e. Cech complexes) as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends on more than just the fibrewise homotopy type of the region covered by sensors. In the setting of planar sensors that also measure weak rotation information, we provide necessary and sufficient conditions for the existence of an evasion path, and we pose an open question concerning Cech and alpha complexes. Joint with Gunnar Carlsson.

One of the exciting results of applied algebraic topology for data analysis has been a formulation of the field of "Stochastic topology." This is an intersection of topology and probability/statistics. I will present some research directions in this field: 1) Euler integration for stochastic models of surfaces and shapes: how topological summaries such as persistence homology or Euler characteristics curves can be used to model surfaces and measure distances between bones. 2) Random simplicial complex models: Given m points drawn from a distribution on manifold construct the union of balls of radius r around these points. As m goes to infinity and r goes to zero what can we say about the limiting distribution of Betti numbers or critical points of this random process ? 3) Spectral theory of simplicial complexes: There is a well developed spectral theory for graphs that provides insights on random walks, spectral clustering of graphs, and near linear time algorithms for solving a system of linear equations. How do these ideas extend to simplicial complexes, in particular: a) is there a notion of a Cheeger inequality for clustering to preserve holes ? b) how does one define a random walk on simplicial complexes that have limiting distributions related to the Harmonics of the (higher order) Hodge Laplacian ? c) we conjecture that the question of near linear time algorithms for linear systems is related to a notion of discrete Ricci curvature for graphs. I just expect knowledge of basic math and will focus on motivating concepts rather than details.

In representation theory, you learn about compact Lie groups. In algebraic geometry, you learn about complex algebraic varieties. In this talk, we will see how these two notions come together in the world of Abelian varieties. These are important objects in algebraic geometry, which carry some pretty remarkable and unexpected properties. We will try to cover as many of these as possible. The talk will focus on complex Abelian varieties, so some familiarity with complex analysis would be helpful. Those currently in the Riemann surfaces course may find some interesting parallels.

TBA

A sequence of functions $f = (f_i)$ ($-\infty < i < \infty$) on a surface $S$ is said to be \emph{equi-areal} (or sometimes, \emph{equi-Poisson}) if it satisfies the relations $$ df_{i-1}\wedge df_i = df_i\wedge df_{i+1}\ (\not=0) $$ for all $i$. In other words, the successive pairs $(f_i,f_{i+1})$ are local coordinates on $S$ that induce the same area form on $S$, independent of $i$. One says that $f$ is \emph{$n$-periodic} if $f_i = f_{i+n}$ for all $i$. The $n$-periodic equi-areal sequences for low values of $n$ turn out to have close connections with interesting problems in both dynamical systems and in the theory of cluster algebras. In this talk, I will explain what is known about the classification (up to a natural notion of equivalence) of such sequences and their surprising relationships with differential geometry, cluster algebras, and the theory of overdetermined differential equations. I won’t assume that the audience knows much differential geometry, just basic multi-variable calculus, and the emphasis will be on describing the interesting results rather than on technical details.

Given a metric space and a positive scale parameter, the Vietoris-Rips simplicial complex has a vertex for each point in the metric space, and contains a set of vertices as a simplex if its diameter is less than the scale parameter. I will introduce these complexes and survey how they are used: originally to define a homology theory for metric spaces, later to study hyperbolic groups, and more recently to analyze high-dimensional data. A theorem of Jean-Claude Hausmann states that if the metric space is a Riemannian manifold and if the scale parameter is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. What happens for larger scale parameters? We will see that as the scale parameter increases, the Vietoris-Rips complex of the circle obtains the homotopy type of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible.

TBA

Category theory is a language used to describe mathematical structures. Its focus is on the relationship between mathematical objects rather than the objects themselves. Categorification is then an attempt to find category-theoretic analogues of classical set-theoretic results. This talk will present the basic notions needed to develop a theory of categorification and will show how many familiar mathematical results are, in fact, "shadows" of a hidden categorical structure.

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. We’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.

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.

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.

In this talk, we investigate modular forms and their many connections to number theory. Modular forms are analytic functions on the upper half complex plane that satisfy certain functional equations. They arise in many contexts in number theory: from partitions of integers, to arithmetical divisor functions, to cutting edge research on special values of the Riemann zeta function. We discuss both classical and modern examples, with a view toward illustrating the profound connections between analysis, topology, and number theory.

TBA

One fundamental object in general relativity is the notion of mass. Pointwise energy density and global mass of a spacetime are both well-defined notions of mass. However, the questions of what goes in between the two as the local mass of a given region, and how it relates to the pointwise and global mass are still not well understood. In the case that the spacetime admits a totally geodesic asymptotically flat spacelike slice, the Riemannian Penrose Inequality states that the mass of this slice is lower bounded by the mass of the blackholes. This inequality was proved by Huisken and Ilmanen using inverse mean curvature flow, and by Bray using a different flow. The general Spacetime Penrose Conjecture, which does not assume the existence of such a totally geodesic slice, is still open today. One viable approach is to use the inverse mean curvature vector flow. Such flows do not have a good existence theory. In this talk, we introduce the basic ideas of inverse mean curvature vector flow, and show that there exist many spacetimes in which smooth solutions to such flows exist for all time.

This talk will serve as a brief introduction to Newtonian fluids and focus on flow through channels. It will begin with a brief introduction to the Navier-Stokes and Stokes equations and continue with a derivation of fluid flow through an impermeable channel leading to classical Poiseuille flow. Next, I will present some results for flow through permeable channels. Finally, I will discuss a new result that has been found in collaboration with Jian-Guo Liu, Anita Layton and myself, in which we have determined an analytic solution for Stokes flow in an infinite permeable pipe.

The problem of spectral estimation, namely – recovering the frequency contents of a signal – 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.

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.

The goal of this talk is to show how the search for a proper definition of the second derivative of a function leads us to Yang Mills theory, the mathematics underlying modern particle physics and the source of powerful tools in topology and geometry. Along the way, we will introduce vector bundles, connections, and curvature. We assume a knowledge of multivariable calculus (and the first derivative).

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.

Morse theory relates the topology of a manifold M to the critical point set of a generic real-valued function on M. Most of this talk will be a non-rigorous review of the basic ideas of Morse theory and some simple applications. There will be lots of pictures of tori and other fun manifolds! In the last part of the talk, I’d like to discuss briefly Witten’s novel approach to Morse theory from the 1980’s, which involves studying the asymptotics of a perturbed Laplace operator. I hope this will be interesting and new to most of the audience, even those who have seen some classical Morse theory. It should also illustrate the important role that analysis can play in topological problems.

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.

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.