# Research in geometry and topology

In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss-Bonnet theorem and Chern–Weil theory. *Source: Wikipedia*

## Professors with interests in geometry and topology

The group consists of professors and graduate students engaged in original research in geometry and topology and its relationships to other fields. Several professors are supported by grants from the National Science Foundation. Currently, several graduate students are working towards their Ph.D.s, but the group is always looking for more students. The group's professors are constantly discovering topics that are suitable for Ph.D. projects.

## Activities

The group meets most weeks of the semester for a research seminar. We invite many speakers from universities throughout the nation, as well as occasionally from international institutions. Students and professors from Wayne State also contribute talks to the seminar.

The group is responsible for teaching undergraduate and graduate courses in topology and geometry, including:

- MAT 5520 (undergraduate topology)
- MAT 5530 (undergraduate differential geometry)
- MAT 6500 (point-set topology)
- MAT 7500 (differentiable manifolds)
- MAT 7510 (algebraic topology)
- MAT 7520 (more algebraic topology)
- MAT 7570 (topics course, often homological algebra)

## Research areas in geometry and topology

Several professors are engaged in research in algebraic topology, a discipline that uses the tools of modern algebra to attack geometric problems. Many projects draw connections between algebraic topology and other fields, such as differential geometry, algebraic geometry, analysis, and mathematical physics.

The following is an incomplete list of specific recent projects.

- Vector fields, immersions, and embeddings for smooth manifolds
- Stable homotopy groups of spheres
- Connections between algebraic topology and parts of analysis (specifically operator algebras, approximation theory and measure theory on manifolds)
- Group actions on manifolds
- Group cohomology and extraordinary cohomology theories of classifying spaces of groups
- Multiplicative structures in homotopy theory
- Computer calculation of algebraic structures used in topology
- Localization and periodicity
- Abstract homotopy theory of pro-categories
- Sums-of-squares formulas in positive characteristic
- Computational motivic homotopy theory
- Homotopical sheaf theory