Continuation of MTH 960.

We will cover topics in Diophantine geometry, including some subset of the following: theory of heights, Diophantine approximation (Roth's theorem, Schmidt subspace theorem, Vojta's conjectures), rational and integral points (Mordell-Weil theorem, Siegel's theorem, Falting's theorem). Basic references for the course topics include the book by Hindry and Silverman and the book by Bombieri and Gubler.

The field of topological data analysis involves creating tools for analyzing and encoding the shape and structure of data. These tools include algebraic representations, such as persistent homology and the Euler characteristic transform, as well as graphical representations, such as the Reeb graph, mapper graph, and merge tree. When analyzing data with these tools, it is important to be able to make metrics available, in particular to be able to understand the behavior of the representations in the face of noise. One such important idea arising from this view is the interleaving distance. What started as a metric for comparing persistence modules as a generalization of the bottleneck distance for persistence diagrams has become a much more broad concept. Category theoretic formulations allow for translation of these ideas across different representations of data, including both the algebraic and graph-based signatures of data. In this course, we will both introduce the relevant mathematical constructions from TDA and explore the variations of the definition of interleavings on these frameworks.

This course will cover a variety of techniques, and their applications to questions in low-dimensional topology and geometry. Topics will touch on many aspects of knot theory and 3- and 4-manifolds, and may include: Dehn surgery, unknotting numbers, contact and symplectic geometry, taut foliations and sutured manifold theory, minimal genus problems (e.g. Thurston norm, Milnor and Thom conjectures), Reidemeister-Turaev torsion, the Casson invariant, Floer homology, concordance and homology cobordism groups, mapping class and braid groups. Participants interests will be taken into account when decisions about the emphasis placed on particular subjects are made.

Fundamental group and covering spaces, van Kampen's theorem. Homology theory, Differentiable manifolds, vector bundles, transversality, calculus on manifolds. Differential forms, tensor bundles, deRham theorem, Frobenius theorem.

Riemannian metrics, connections, curvature, geodesics. First and second variation, Jacobi fields, conjugate points. Rauch comparison theorems, Hodge theorem, Bochner technique, spinors. Further topics on curvature or submanifold theory.

Cohomology, products, duality, basic homotopy theory, bundles, obstruction theory, spectral sequences, characteristic classes, and other related topics.

This course intends to introduce students to aspects of Symplectic Geometry (and probably Contact Geometry, its odd dimensional cousin). Symplectic Geometry emerged from Classical Mechanics, and many peculiar phenomena were discovered since the 1980's which sets it apart from Riemannian Geometry. In particular, Rigidity and Flexibility coexist side by side. Versions of Gromov's h-principle which will be introduced in the course reduce geometric classifications to topological results. They yield constructive results and are a manifestation of Flexibility while Rigidity yields obstructive results. The aim of the course is to introduce students to some of these phenomena while keeping prerequisites rather modest (MTH868 should be sufficient).