By means of the use of diffusion semigroups, it is possible to define and study classical operators in harmonic analysis and the fractional Laplacian. We introduce and develop this idea, and show two applications. First, we investigate fractional integrals and Riesz transforms in compact Riemannian manifolds. Secondly, the study of harmonic analysis operators and the fractional powers associated with a discrete Laplacian is carried out.

We consider n-orbifolds modeled on real projective geometry. These include hyperbolic manifolds and orbifolds. There are nontrivial deformations of hyperbolic orbifolds to real projective ones. Among open real projective orbifolds that are topologically tame, we consider ones with radial ends and totally geodesic ends. We will present our work to classify these ends with some natural conditions.

A number of important model linear dispersive equations give rise to interesting curve flows in differential geometry. In this talk we will discuss some of these including the Schrodinger curve flow on the two sphere, the Hodge star mean curvature curve flow in Euclidean and Lorentzian 3-space, and the geometric Airy curve flow on Euclidean space and the affine space. The equations of curvatures of these curve flows turn out to be soliton equations. Hence we can use techniques from soliton theory to study these curve flows. In particular, we can construct infinitely many families of explicit solutions and solve the periodic Cauchy problem.

The theory of solitons has been an active research area since the famous work of Gardner, Greene, Kruskal and Miura on the Korteweg de Vries equation in mid-1960s, and has applications in many areas of mathematics. In this talk, I will explain a joint project with Karen Uhlenbeck, in which we attempt to formulate a uniform geometric structure for soliton equations. Adler, Wilson, and Drinfeld-Sokolov developed a general method for constructing soliton hierarchies from a splitting of a loop group into positive and negative subgroups. We use these splittings to derive in a unified and simple way many remarkable properties of soliton equations including Hamiltonian formulations, commuting conservation laws, formal inverse scattering theory, B\"acklund transformations, tau functions, and Virasoro actions.

In Part I of this talk, I will begin by familiarizing the audience with the notions of the Hasse principle and strong approximation for integral points on varieties. I will then describe a general formalism which explains many counterexamples to these ideas, and in particular I will describe how the Brauer-Manin obstruction fits into this framework. The remainder of this talk will be devoted to providing several known examples of this obstruction from recent literature.

Dynamic, cross-linked, biological fiber networks play major roles in cell and tissue function. We present here a particle-based model for fiber networks inspired from flocking theory, where fibers are modeled as point particles and cross-link interactions are modeled via distance-based potential functions. The model is able to simulate behaviors such as strain hardening, viscoelastic creep, stress relaxation, network rupture, and network reformation, which are common characteristics of biological fiber networks.

I will give a functorial formulation of a classical field theory. Classical mechanics provides a natural example of such a field theory in dimension 1. Extending the definition to higher dimensions, one naturally encounters higher, or derived, symplectic structures. We will see that Poisson manifolds provide natural two-dimensional examples using symplectic groupoids, while Courant algebroids give three-dimensional examples. If I have time, I will explain how a general derived symplectic space gives rise to a classical field theory; for instance, the Chern-Simons and Rozansky-Witten theories arise in this way.

The goal of this talk is to explain how Donaldson's celebrated Theorem A can be used in the study of slice knots. We'll begin by talking about the double branched cover of a knot in the 3-sphere. When such a 3-manifold bounds a negative definite 4-manifold, one obtains an obstruction to the knot being slice by considering the intersection form of the negative definite 4-manifold. After explaining what all these terms mean, I'll focus on a particular construction: the double branched cover of an alternating knot. These techniques have been used in recent years to provide solutions and partial solutions to interesting questions concerning slice knots.

We begin by reviewing the graph partitioning problem and its applications to data clustering. Motivated by a geometric problem, we then introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show converges in a finite number of iterations to a local minimum of the relaxed objective function. We will discuss applications, as well as intriguing connections to other problems such as Nonnegative Matrix Factorization and Reaction Diffusion Equations. This is joint work with Braxton Osting and Edouard Oudet.