Geometry & TOPOLOGY

The pairing of geometry & topology is indispensable in modern mathematics & all its applications. The origins of geometry involve quantitative notions of space. For example, length, angle, area, volume, etc.. While the origins of topology involve qualitative notions of space; in particular, properties which allow for a less rigid description of how points can be organized in a space than mere number alone allows. For example, the qualities of nearness, connectedness, compactness, etc.. These foundational ideas are formalized and elevated into a robust (widely applicable, useful) and effective (algorithmically computable, ideally) framework for investigating the shape of spaces and the rich interrelationships between different structures on spaces. 

A concrete and illustrative example we suggest investigating as a case study is the idea of a topological manifold. --- The Earth is a topological manifold: locally it is flat, globally it is not, and an "atlas" organizes all the data of the local "coordinate patches" in a globally appropriate (can be made mathematically precise) manner! 

Examples of thematic researh questions in geometry & topology are: 

• Given a topological manifold, which geometric structures does it admit?  — At the risk of being pedantic we emphasize the interdependence of all mathematical disciplines. In algebraic geometry one might first ask how to concretely build manifolds. While in number theory one might want to know how to concretely build manifolds over the integers. Our research interests align more with the theory of deformation spaces of geometric structures on manifolds, for which the initial quesiton is a natural departure point.

• How do you tell manifolds apart? — As a means of providing proofs that answer this question in various contexts, the following disciplines have been created, or at least have nontrivial overlap, with geometry & topology: dynamics, knot theory, combinatorial group theory, obstruction theory, surgery theory, gauge theory, geometric group theory. Manifolds can be assigned a wide variety of invariants in order to tell them apart. For example, numerical invariants, groups, polynomials, compactifications, bordifications, or dynamical systems can be used as invariants of manifolds. Numerical invariants include: "dimension" [differential topology], or multiple dimensional "Betti numbers" [algebraic topology], or "scalar curvature" [Riemannian geometry]. Dimension is perhaps the most intuitive invariant. Yet all invariants, even dimension, are surprisingly subtle. The (incomplete) technical list above hints at the layers of sophistication.

• What can we do with it? — Happily, applications of this theoretical framework abound! Everything from Physics (old and exciting story!) to Machine Learning (new and exciting story!) have foundations in geometry & topology. It is certainly more challenging to think of something that has nothing to do whatsoever with geometry & topology, than things that do. So, pick your pleasure(s) and lean into your training!

We recommend that you see some of the Explorations created by our lab members and shared on their member profiles! We can also suggest reading the following surveys:

[J. Milnor, Topology Through The Centuries: Low Dimensional Manifolds] 

[W. Thurston, How To See 3-Manifolds]

[F. Bonahon, The Hyperbolic Revolution: From Topology To Geometry And Back] 

Our board: In the realm of gauge theory one desires to minimize the Hermitian Yang-Mills flow along a complex gauge orbit of fixed stable G-Higgs bundle on a compact Riemann surface. A theorem of Karen Uhlenbeck guarantees nice enough regularity of such a process for the existence of the limiting object to be a critical point of the flow. More work to argue the same results holds for polystable bundles leads to the Hitchin-Simpson theorem which guarantees the existence of a so-called Hermitian-Einstein metric (on the limiting stable Higgs bundle that lives in the same isomorphism class of the original bundle) which satisfies Hitchin's equations. This theorem is dual to the Corlette-Donaldson-Labourie theorem which guarantees the existence of a harmonic metric on flat G-bundles associated to a reductive representation of the fundamental group of the compact Riemann surface. Kevin Corlette and Karen Uhlenbeck are mathematical giants and also are members of historically underrepresented groups of people in mathematics. Each of them are mathematical champions of ours. We are reviewing the details of the [Nonabelian Hodge Correspondnece]. For more you can also see the [Hitchin-Kobayashi correspondence].