Hi, my name is Yajit Jain. I am a Tamarkin Assistant Professor of Mathematics and NSF Postdoctoral Fellow at Brown University. Previously I was a graduate student at Northwestern University working with Prof. John Francis. Before that I was an undergraduate student at MIT.

My current research is in manifold topology and homotopy theory.

I am also an organizer of the MIT Talbot Workshop.

Research

I am a manifold topologist studying exotic smoothings of manifold bundles. These are natural extensions of exotic smooth structures on individual manifolds, e.g. Milnor spheres, to the parametrized setting. The key invariant appearing in this area is the higher smooth torsion, an invariant studied independently by Igusa and Klein, Dwyer, Weiss, and Williams, and Bismut and Lott. 

Work on these invariants was pioneered by Igusa and Waldhausen using intricate models for moduli space functors including the Whitehead space, the stable parameterized h-cobordism space, and algebraic K-theory of spaces. In my work I use excisive approximations of these functors, which I model using the n-Disk categories and zero-pointed spaces appearing in work of Ayala and Francis on factorization (co)homology. Coming up with refinements of the higher smooth torsion invariants in this setting gives me the footing I need to execute generalizations of theorems from classical differential topology, such as the Poincaré–Hopf theorem, to excisive functors that generalize the Euler characteristic.

In my thesis I prove a conjecture of Goette and Igusa which states that after rationalizing and stabilizing the dimension there are no exotic smoothings of bundles with even dimensional closed fibers. In practice this means proving a vanishing theorem for a refinement of the higher smooth torsion invariant. This statement is in stark contrast to the odd dimensional setting in which the same authors construct all possible exotic bundles on a given smooth bundle.  

Teaching

While in graduate school I have served as a teaching assistant for the following courses. This entails leading weekly discussion sections, holding office hours, and grading homework.

Winter 2020
336-1: Introduction to the Theory of Numbers
344-1: Introduction to Topology

Fall 2019
214-0: Single Variable Calculus III
300-0: Foundations of Higher Mathematics

Spring 2019
285-3: Accelerated Mathematics for Mathematical Methods in Social Sciences

Winter 2019
320-2: Real Analysis
344-1: Intro to Topology

Fall 2017
230-0: Differential Calculus of Multivariable Functions (two sections)

Spring 2017
224-0: Integral Calculus of One Variable Functions
331-3: Menu Abstract Algebra

Winter 2017
224-0: Integral Calculus of One Variable Functions (two sections)

Fall 2016
220-0: Differential Calculus of One Variable Functions (two sections)


Mentorship

I have mentored several advanced undergraduate students serving as TA’s in the math department.

I founded the Directed Reading Program at Northwestern with Prof. Ezra Getzler and currently serve as an organizer.

Talbot Workshop

I am currently an organizer of the MIT Talbot Workshop alongside Calista Bernard, Morgan Opie, and Lucy Yang.

In 2019 Talbot was held in Brady, TX. The topic was moduli spaces of manifolds, mentored by Prof. Soren Galatius and Prof. Oscar Randal-Williams. We made it into the local newspaper!

Here are the notes for the workshop, originally live TeX-ed by Lucy Yang, and edited by several members of the workshop including myself.

The 2020 workshop, Ambidexterity in Chromatic Homotopy Theory, has been postponed to 2021 and will be held in Nagodoches, TX. It will be mentored by Prof. Jacob Lurie and Prof. Tomer Schlank.

Travel

Here is a list of previous and upcoming travel.


2020 Young Topologists Meeting - University of Copenhagen

2020 Talbot Workshop: Ambidexterity in Chromatic Homotopy Theory, Nagodoches, TX

2020 (∞, n)-categories, factorization homology, and algebraic K-theory, MSRI

2020 Tensor categories and topological quantum field theories, MSRI

2020 Introductory Workshop: Higher Categories and Categorification, MSRI

2019 HCM Workshop: Automorphisms of Manifolds - Hausdorff Center for Mathematics, University of Bonn

2019 Four Manifolds: Confluence of High and Low Dimensions - Fields Institute, University of Toronto

2019 Talbot Workshop: Moduli Spaces of Manifolds - Brady, TX

2018 Manifolds Workshop, Homotopy Harnessing Higher Structures Program - Isaac Newton Insti- tute, University of Cambridge

2018 Young Topologists Meeting - University of Copenhagen

2018 Seminaire de mathematiques superieures: Derived Geometry and Higher Categorical Struc- tures in Geometry and Physics - Fields Institute, University of Toronto

2017 Topological and Geometric Methods in QFT - Montana State University

2017 Floer Homology and Homotopy Theory Summer School + Conference - UCLA

2016 Talbot Workshop: Equivariant Stable Homotopy Theory and the Kervaire Invariant - Salt Lake City, Utah

2016 Young Topologists Meeting - University of Copenhagen