# Jay Shah

*Affiliate Professor*

## Education

Ph.D in Mathematics, Massachusetts Institute of Technology, 2017

B.S., in Mathematics, University of Chicago, 2012

## Research Groups

Algebra and Algebraic Geometry, Topology

## Research Area

Algebraic Topology and Higher Category Theory

## Bio

#### Research Interests

I work in the areas of algebraic topology, homotopy theory, and higher category theory. Currently, I am especially interested in the higher categorical and algebraic structures one encounters in equivariant homotopy theory and analogous settings. My thesis established a theory of indexed infinity-categories, focusing on the notion of indexed homotopy limits and colimits; specializing the base infinity-category to the orbit category of a finite group, one obtains equivariant applications. Subsequent to this and in joint work with Barwick, Dotto, Glasman, and Nardin, I am developing infinity-categorical foundations for equivariant higher algebra. For example, we introduce and study a notion of G-symmetric monoidal infinity-category which is a homotopical enhancement of a structure studied by Hopkins and Hill. Geometrically, our viewpoint is informed by stratified homotopy theory; for instance, in a certain precise sense one may view the slice filtration of G-spectra introduced in the seminal work of Hill, Hopkins, and Ravenel as a perversity filtration. Apart from this, I am pursuing some other projects in equivariant homotopy theory and motivic homotopy theory.

#### Selected Publications

- Parameterized higher category theory and higher algebra: Expos´e II - Parameterized homotopy limits and colimits. In preparation.
- Categorifying rationalization (with C. Barwick, S. Glasman, M. Hoyois, and D. Nardin). October 2016. arXiv:1610.07162. Not yet submitted for publication.
- Parameterized higher category theory and higher algebra: Expos´e I - Elements of parameterized higher category theory (with C. Barwick, E. Dotto, S. Glasman, and D. Nardin). August 2016. arXiv:1608.03657. Not yet submitted for publication.
- Parameterized higher category theory and higher algebra: a general introduction (with C. Barwick, E. Dotto, S. Glasman, and D. Nardin). August 2016. arXiv:1608.03654. Not yet submitted for publication.
- Fibrations in 1-category theory (with C. Barwick). August 2016. arXiv:1607.04343. Submitted for publication to the MATRIX conference proceedings.

**Email:** jshah3@nd.edu

**Office:** 255 Hurley Bldg