Periods are numbers that can be expressed as integrals of algebraic differential forms over domains defined by polynomial inequalities with rational coefficients. They form a subring of complex numbers, which contains multiple zeta values and multiple modular values. Although some periods are transcendental, one can work out a Galois theory for them using their defining algebraic data, which is how the classical Galois theory for algebraic numbers were developed. I will discuss Francis Brown's results on multiple zeta values and more recent work on multiple modular values.

TBA

After a brief introduction of modular forms on the upper half plane and vector-space valued differential forms, we will explore a very classical result (independently due to Eichler and Shimura) which relates certain cohomology groups to cusp forms on the upper half plane of corresponding weight. We will then put our algebraic hat on, and recast this result in modern light, using the theory of Automorphic forms developed by (among others) Harish-Chandra and Langlands. I hope to make the talk accessible to most graduate students. Though the topics we will talk about are related to my research, it is not a research talk, more of an exposition. The first part of the talk should be a breeze for anyone with understanding of some fundamental concepts in Complex analysis and Algebraic Topology (holomorphic functions, differential forms, deRham cohomology.) A course in Representation Theory would be helpful in relating to the concepts in the second part of the talk.

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.

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.

Coarsening is the phenomenon where many objects (water drops, molecular islands, particles in a freezing liquid) becoming a smaller number of objects in an orderly way. This talk will examine modeling one such system, tiny liquid drops, through three models: a PDE for the fluid, a coarsening dynamical system for the drops, and an LSW-type ensemble model for the group of drops. We will study self-similarity in the dynamics and extensions of the model to examine very long times when drops grow large enough that gravity distorts their shape.

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.

(Bio)chemical reaction networks are used to model processes that occur in cell biology. A fundamental problem is to characterize chemical reaction networks which admit multiple steady states. The existing literature has focused on identifying necessary conditions for multiple steady states. A natural (but significantly more difficult) problem is to determine sufficient conditions. In joint work with Anne Shiu, we suggest an approach for tackling this problem by defining certain minimal multistationary networks called 'atoms of multistationarity'. These 'atoms' are analogous to prime numbers in the theory of integers, in the sense that every multistationary network is either an atom or contains an atom. The talk will contain many examples from biology and will assume nothing more than calculus.

The instanton equations appear in gauge theory and generalize both the Maxwell equations and the harmonic equation. Their study has been and continues to be a very fertile ground for interactions between physicists and mathematicians. The object of this talk is a description of instanton solutions on S^1xR^3 due to Hurtubise and myself using the Nahm transform, a non-linear transformation that takes a system of PDE and produces a system of ODE or even a system of algebraic equations. This description allows us to answer existence questions for calorons.

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two ends. We study the functor and establish a long exact sequence relating the corresponding Legendrian cohomology categories of the two ends. As applications, we prove that the functor between augmentation categories is injective on objects, and find new obstructions to the existence of exact Lagrangian cobordisms. The main technique is a recent work of Chantraine, Dimitroglou Rizell, Ghiggini and Golovko on Cthulhu homology.

Another Job Audition.

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.

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.

I'm sure it will be great!

TBD

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.

We'll look at a random process on the integer lattice $/mathbb{Z}^2$ known as the voter model. Let's suppose that each point on the lattice represents a single household with one voter who holds one of two possible opinions, 0 or 1 (e.g. Republican or Democrat). Starting with an initial configuration of 0's and 1's on $/mathbb{Z}^2$, a voter changes their opinion at a rate proportional to the number of neighbors holding a different opinion. I'll demonstrate a clever and useful approach to analysing the voter model via the dual process. We'll be able to determine whether or not our process has any nontrivial stationary distributions. I'll also briefly discuss the newest research on variants of the voter model.

We study the dynamics of waves in media with a local periodic structure which varies adiabatically (over many periods of the periodic lattice) across the medium. We focus in particular on the case where symmetries of the periodic structure lead to degeneracies in the Bloch band dispersion surface. An example of such symmetry-induced degeneracies are the `Dirac points’ of media with `honeycomb lattice’ symmetry, such as graphene. Our results are as follows: (1) A systematic and rigorous derivation of the `anomalous velocity’ of wave-packets due to the Bloch band’s Berry curvature. The Berry curvature is large near to degeneracies, where it takes the form of a monopole. We also derive terms which do not appear in the works of Niu et al. which describe a `field-particle’ coupling effect between the evolution of observables associated with the wave-packet and the evolution of the wave packet envelope. These terms are of the same order as the anomalous velocity. (2) Restricting to one spatial dimension, the derivation of the precise dynamics when a wave-packet is incident on a Bloch band degeneracy. In particular we derive the probability of an inter-band transition and show that our result is consistent with an appropriately interpreted Landau-Zener formula. I will present these results for solutions of a model Schr\”{o}dinger equation; extending our results to systems described by Maxwell's equations is the subject of ongoing work. This is joint work with Michael Weinstein and Jianfeng Lu.

Group theory is one of the basic topics of abstract algebra and therefore probably well-known. In this talk, we are going to introduce left-orders on groups and expand the fundamental theorem of homomorphisms from groups to ordered groups. We will see some examples of left-ordered groups that show different levels of orderability and, time permitting, formulate a theorem of A. Rhemtulla which discusses the existence of torsion-free groups without any order.

How does the structure of a network effect how randomness propagates through it? This is a fun mix of probability, differential equations, and graph theory.

Graphics processing units (GPUs) are powerful accelerators that can launch many processes in parallel. Over the past decade, they have been utilized for scientific computation, including molecular dynamics, fluid mechanics, machine learning, and stochastic differential equations. Although dependent on the algorithm, GPUs may execute code faster than CPUs by several orders of magnitude. The mathematics department at Duke hosts 4 older generation GPUs on two nodes that are available for department use. In this seminar I will briefly introduce how GPUs are different than CPUs; the bulk of my time will be a tutorial on how to code CUDA so that attendees may begin to take advantage of these departmental resources for their research. Depending on the attendance, it may be a hands-on tutorial so bring your laptop.

Given a closed, convex curve C in the plane, a billiard path on C is a polygon P inscribed in C such that, at each vertex v of P, the two edges of P incident with v make equal angles with the tangent line to C at v. (Intuitively, this is the path a billiard ball would follow on a frictionless pool table bounded by C.) For most convex curves C, there are only a finite number of triangular billiard paths on C, a finite number of quadrilateral billiard paths, and so on. Obviously, when C is a circle, there are infinitely many closed billiard n-gons inscribed in C, but, surprisingly, the same is true when C is an ellipse. (This is a famous theorem due to Chasles.) The interesting question is whether there are other convex curves, besides ellipses, for which there are infinitely many closed billiard n-gons for some n. In this talk, I'll discuss the above-mentioned phenomenon and show how it is related to the geometry of non-holonomic plane fields (which will be defined and described). This leads to some surprisingly beautiful geometry, which will require nothing beyond multivariable calculus from the audience.

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.

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.

This talk will be geared toward first and second year grad students and/or anyone with limited geometry experience. We will discuss the idea of curvature for curves and surfaces and the notion of "best metrics." The classical Uniformization Theorem will be introduced from a modern angle: Ricci flow. This will motivate studying the Ricci flow in dimension 3 as a tool to understand topology in terms of geometry. Time permitting, we will finish by discussing the Geometrization Theorem.