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 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.
Duality and vanishing theorems for topologically trivial families of smooth h-cobordisms (arXiv:2111.03188)
While in graduate school I served as a teaching assistant for the following courses.
336-1: Introduction to the Theory of Numbers
344-1: Introduction to Topology
214-0: Single Variable Calculus III
300-0: Foundations of Higher Mathematics
285-3: Accelerated Mathematics for Mathematical Methods in Social Sciences
320-2: Real Analysis
344-1: Intro to Topology
230-0: Differential Calculus of Multivariable Functions (two sections)
224-0: Integral Calculus of One Variable Functions
331-3: Menu Abstract Algebra
224-0: Integral Calculus of One Variable Functions (two sections)
220-0: Differential Calculus of One Variable Functions (two sections)
Here is a list of previous and upcoming travel.
2022 Higher algebraic structures in algebra, topology and geometry, Institut Mittag-Leffler
2020 Talbot Workshop: Ambidexterity in Chromatic Homotopy Theory, Plymouth Massachusetts (postponed to 2021)
2021 Viva Talbot!, Online
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