Talks

Elliptic curves over finite fields are central to modern number theory and cryptography, yet they are rather difficult to visualize — their points form finite sets without obvious geometric structure, and the group law that makes them so useful is obscured in most pictures. This is quite different from the more familiar story over the complex numbers, where geometry runs the show: every elliptic curve is a torus, and the group law is simply addition. In this talk, we explore a way to bridge the gap between these worlds. Using ideas from lattices with complex multiplication, we construct a way to lift any elliptic curve over a finite field to a (subset of a) complex torus, in a way that makes the group structure, the action of Frobenius, and the points over all field extensions simultaneously visible in a single picture. We will begin with an introduction to elliptic curves aimed at a general mathematical audience, building from curves over the reals and complex numbers to the finite field setting, before describing the lifting construction and the pictures it produces. This is joint work with Nadir Hajouji.

Accurately ray-tracing a scene containing multiple black holes requires following light along null geodesics of a curved spacetime — in general, a computationally demanding problem involving nonlinear ODEs on a manifold known only numerically. This talk describes a beautiful exception: the Majumdar-Papapetrou exact solutions to the Einstein-Maxwell equations, describing multiple extremally charged black holes in static equilibrium, where gravitational attraction is precisely canceled by electrostatic repulsion. By exploiting conformal invariance of null geodesics, we reduce the relativistic raytracing problem to a classical one: tracing light through flat space with a spatially varying refractive index encoding the black hole configuration. Any renderer supporting variable indices of refraction can then produce exact images of these spacetimes — no relativistic simulator required.

An introduction to knot theory at the undergraduate level: we introduce the idea of knots and prove that knots exist (by constructing the tricolorability invariant). Then we prove all knotted loops come undone in 4 dimensions, but begin an exploration of the linking of surfaces with loops and knotted spheres.

We describe work on applying the geometry of symmetric spaces to graph embedding problems in machine learning. Using the rich structure of the space of symmetric positive definite matrices, we develop tools from Riemannian geometry to more faithfully represent complex graph data. Joint work with Fede Lopez, Anna Wienhard, Bea Pozzetti and Michel Strube at the University of Heidelberg.

Surprising effects in classical physics in a curved background space.

Building a path tracer with vectors and calculus. This talk highlights how to make wonderful images using lower division undergraduate mathematics, following the process of building a naive path tracer using only math from a multivariable calculus course.

Rendering 3-manifolds in light of the Geometrization Theorem. An updated version of a talk on joint work with Remi Coulon, Henry Segerman and Sabetta Matsumoto visualizing the eight Thurston geometries, with new work on putting these pieces back together with connect sums and JSJ decompositions.

While the 19th century witnessed a near complete understanding of 1 and 2 dimensional manifolds, much progress during the 20th century in 3 dimensions was guided by a conjecture of Poincare, first formulated in 1904. Poincare's conjecture — essentially that simple 3 dimensional spaces can be probed effectively using 1-dimensional loops — proved much more difficult than originally hoped, remaining unsolved for nearly 100 years. Following a century of work, its eventual resolution by Perelman in 2002 provided a new and powerful tool — called Geometrization — to the study of all 3-dimensional spaces. My goal in this talk is to give an overview of this exciting story lying at the heart of modern topology, from what was asked to what the mathematical community has learned.

The resolution of Painleve's conjecture.

A public talk broadly discussing modern geometry and its surprising intersections with machine learning, touching on geometric transitions and the use of curved spaces in data science.

A visual introduction to curved geometry in math and physics, with one goal being to understand how curvature leads to Einstein rings and images of black holes like in the movie Interstellar.

An attempt to 'discover' the idea of neural networks from a mathematical perspective, aimed at introductory undergraduates familiar with basic probability, vectors, and matrices.

Is my experience of the color red the same as yours? Do movies look realistic to dogs? While these questions may at first seem similarly unanswerable, they are in fact quite different! While philosophers still debate on the coherence of the former, the latter is resolved mainly through the use of Linear Algebra. As an exciting interdisciplinary adventure involving linear algebra, real analysis, cell biology, and evolutionary history, we work together in this talk to precisely express the question (is the response of a dog's retina to the stimuli of a natural vista, and a TV displaying the same image equal?), and lay out a path towards its solution. Along the way we discover the mathematical underpinnings of why humans believe in three primary colors, how newspapers, phones, and televisions exploit the imprecision of our eyes, and the vast difference between sight and hearing. Perhaps, the star of the show relating all these things is the Rank-Nullity theorem of elementary Linear Algebra!

The Geometrization Theorem of Thurston and Perelman provides a roadmap to understanding topology in dimension 3 via geometric means, by studying 3-manifolds which admit each of the eight Thurston Geometries. This talk concerns a recent project to produce provably correct intrinsic images and computer simulations of these geometric manifolds, by generalizing various algorithms from Euclidean geometry to locally homogeneous spaces. Doing so involves a careful analysis of the Riemannian-geometric features of each Thurston geometry, and we work out some new elementary properties of Nil, SL(2,R) and Sol. Joint work with Remi Coulon, Sabetta Matsumoto and Henry Segerman.

Manifolds are ubiquitous in modern mathematics, from the familiar low dimensional examples of curves and surfaces in calculus, to higher dimensional abstract examples in geometry, physics, data science and beyond. They come in a bewildering variety — and even the basic question of 'what kinds of manifolds are possible in each dimension' remains an active area of research. Over the past almost two centuries, incredible progress has been made on this classification. This has resulted in a rather clear picture in low dimensions, and a clear demarcation of the obstacles we must confront as the dimension increases. And while the arguments involved can get quite technical, the big-picture story is a beautiful interplay of shape, symmetry and geometry which deserves to be more widely known.

Hyperbolic Dehn Surgery is a procedure for producing closed hyperbolic 3-manifolds from hyperbolic knot complements, by 'filling in the cusp' with a solid torus. This procedure produces infinitely many examples of closed hyperbolic manifolds, as Thurston (its discoverer) proved that all but finitely many ways of performing the gluing result in a manifold which admits a hyperbolic structure. This talk provides a graduate-student-friendly introduction to this area of mathematics, leading up to the statement (but not proof) of Thurston's theorem, and placing it in the wider context of geometric topology and geometric group theory.

A bonus lecture exploring the geometry and algebra of SU(2), given at the end of an undergraduate group theory course.

To a first approximation, the behavior of light appears quite simple: it always travels in a straight line from the source to the viewer. But upon deeper inspection, various surprising effects — from mirrors to mirages — show the need for a more sophisticated theory. These seemingly disparate phenomena are consequences of a single underlying principle: in every circumstance light endeavors to take the most efficient path between two points, or the path of least flight time. From this observation springs forth a deep connection of the theory of ray optics to differential geometry, modeling the trajectory of light in a varying medium as the shortest path — or geodesic — in an abstract curved space. In this talk we demonstrate the power of this viewpoint, producing computer simulations of lenses and mirages by solving for geodesics in the appropriate metric. Finally, we investigate telltale signs of curvature in the real world — including ring-like mirages appearing in images from the Hubble space telescope, and consider Einstein's great insight: that gravity is not a force, but just a consequence of living in a curved world.

The Geometrization Theorem of Thurston and Perelman provides a roadmap to understanding topology in dimension 3 via geometric means. Specifically, it states that every closed 3-manifold has a decomposition into geometric pieces, and the zoo of these geometric pieces is quite constrained: each is built from one of eight homogeneous 3-dimensional Riemannian model spaces (called the Thurston geometries). In this talk, we approach the question of 'what does a 3-manifold look like' from the perspective of geometrization. Through animations of simple examples in dimensions 2 and 3 we review what it means to put a geometric structure on a manifold, and construct an example admitting each of the Thurston geometries. Using software written in collaboration with Remi Coulon, Sabetta Matsumoto and Henry Segerman, we explore these manifolds 'from the inside' by raytracing along geodesics, then re-assemble these geometric pieces to understand an inside view of general 3-manifolds.

Functional analysis is a powerful, modern branch of mathematical analysis where topics that are familiar from the theory of real numbers (sequences, series, convergence, etc) are generalized to the infinite dimensional world of functions. This talk provides an introduction to this field for students who have mastered elementary real analysis, tying together both historical motivations and the modern viewpoint. The first half of the talk focuses on the pre-history of the subject, looking at the mathematics of the geocentric model of the heavens developed by the Ptolemies and Ibn al-Shatir, among others. While the notion of epicycles very naturally leads to precise mathematical questions about sequences of functions and convergence, it took us many further centuries to fully understand what lies just beneath the surface of this idea. In the second half we visit Fourier and his study of heat, formalizing the questions these historical figures may have asked, and laying out the roadmap leading from here to the modern theory of Functional Analysis.

Created over 150 years ago, the periodic table of elements reveals many interesting similarities, and periodic behaviors in the elementary constituents of nature. Elements with similar chemical properties are placed on top of one another into columns, and the number of columns populated increases as one moves down the table. But inside this organizational tool one can find many mathematical patterns, which historically provided welcome hints to the nature of quantum theory itself. One such pattern is simply noting that the number of elements in any given row of the table is always twice a perfect square. Why is that? Amazingly, this feature of nature is forced upon us by symmetry and mathematics. In this talk, we use the periodic table as a lighthouse, illuminating our way on a journey through infinite dimensional linear algebra, symmetries, eigenvalues, eigenfunctions, and separable PDEs.

Be it in math classes or donut shops, all of us have seen many different shaped tori throughout our lives. Mathematically, we know that the conformal structures on the torus are parameterized by a dimensional space, and so it's natural to ask which tori have we really seen? What do the conformal structures look like, which don't arise from identifying opposing sides of a Euclidean rectangle? In this talk I will illustrate a beautiful result of Ulrich Pinkall, who showed that all possible conformal structures on the torus are realizable as embedded surfaces in the three sphere using the geometry of the Hopf fibration. We will interact with these tori in three ways: by conformally projecting into R^3, by viewing them projectively from a vantage point in R^4, and finally intrinsically as submanifolds of S^3, rendered geometrically correct (so light follows great circles).