# Research in geometry and topology

## Faculty with interests in geometry and topology

- Robert Bruner
- Luca Candelori
- Vladimir Chernyak
- Bruce Corrigan-Salter
- Daniel Drucker
- Po Hu
- Daniel Isaksen
- John Klein
- Andrew Salch

The group consists of faculty, graduate students, and undergraduate students engaged in original research in geometry and topology and its relationships to other fields. Some of this work is supported by external funding from the National Science Foundation, the Simons Foundation, and other sources. Currently, several graduate students are working towards their Ph.D.s. The group is always looking for more students.

## 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.

We also hold a weekly modular forms seminar, which brings together topologists and number theorists to discuss projects relating algebraic fopology and algebraic number theory, such as the construction of topological Maass forms and topological overconvergent *p*-adic modular forms.

Most semesters, an undergraduate research group conducts research on an original topic related to geometry and topology. Recent projects have included:

- Applications of topological data analysis to MRI data in neuroscience.
- Jordan decompositions of tensor products.
- The Nottingham group of power series.
- Lie algebra cohomology.

We offer a variety of 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 I).
- MAT 7520 (algebraic topology II).
- MAT 7570 (topics course on varying subjects of current interest).

## Research interests

Some of our faculty 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 algebraic number theory, arithmetic geometry, differential geometry, algebraic geometry, physics (statistical mechanics), and stochastic processes.

Some typical recent projects include:

- Topological data analysis.
- Relationships between stable homotopy theory and L-functions.
- Cohomology of profinite groups.
- The algebraic topology of moduli spaces of stochastic processes.
- Intersection theory via the Wiener measure.
- Algebraic K-theory.
- Poincaré duality complexes.
- Vector fields, immersions, and embeddings for smooth manifolds.
- Stable homotopy groups of spheres.
- 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.
- Computational motivic homotopy theory.
- Equivariant stable homotopy theory.