Department of Mathematics, Brandeis University
204 Goldsmith
rahulkrishna[AT]brandeis.edu

I am a postdoctoral instructor at the math department at Brandeis. I study number theory in general, and automorphic forms in particular. My research is mainly focused on the relative trace formula and the local problems that arise in its applications.

## Curriculum Vitae

PDF

Last updated: Fall 2018

## Research

• On the global Gross-Prasad conjecture for orthogonal groups (In preparation.)
• Building on "A new proof of the Waldspurger formula I," I construct an RTF comparison strategy aimed at the global Gross-Prasad conjecture for orthogonal groups. This relies on conjectures of "smooth transfer" and "fundamental lemma" type, which I verify by hand in some low rank examples.
• A new proof of the Waldspurger formula I (Algebra and Number Theory, 2019)
• I construct a new comparison between two very different relative trace formulas (RTFs). These RTFs are designed so that their spectral sides "encode" Waldspurger's formula for toric periods; they are also designed keeping the interpretation of Waldspurger's formula as the low rank version of the global Gross-Prasad in mind. I set up an orbit-by-orbit matching, and conjecture and prove statements of "smooth transfer" and "fundamental lemma" type in this setting. This approach should, with further work on regularization of these RTFs, lead to a completely new and independent proof of Waldspurger's formula.
Here are a few projects that have gotten temporarily put on the back burner.
• Beyond endoscopy for spherical varieties of rank one: the fundamental lemma (Joint with D. Johnstone, and in preparation.)
• We verify the fundamental lemma of Y. Sakellaridis in his "relative beyond endoscopy" program for spherical varieties of rank one.
• A new proof of the Waldspurger formaula II (In preparation.)
• I explain how to regularize the relative trace formulas in "A new proof of the Waldspurger formula I," and deduce a global character identity equivalent to Waldspurger's formula.

## Teaching

I am currently teaching MATH 15A, "Applied linear algebra" and MATH 23B, "Introduction to proofs". The course pages are available on LATTE.

Past classes (at Brandeis):
• Fall 2019, MATH 15A, Applied linear algebra
Past classes (at Northwestern):
• Spring 2019, Math 230, Differential multivariable calculus (Calc III)
• Fall 2018, Math 230, Differential multivariable calculus (Calc III)
• Spring 2018, Math 482-2, An introduction to the trace formula. A very incomplete set of lecture notes are available here, and a partial collection of homework problems is available here. By clicking on the links above, you agree to inform me of any errors (mathematical or otherwise) you find. A quick remark: the notes cover much of the classical theory of the trace formula for finitely generated Fuchsian groups acting on the upper half plane. I also discussed the adelic perspective for $\mathrm{GL}_2$ in class; however, these notes have yet to be written. Be warned!
• Spring 2018, Math 224, Integral single variable calculus (Calc II)
• Fall 2017, Math 224, Integral single variable calculus (Calc II)
• Spring 2017, Math 482-2, Elliptic curves and modular forms.
• Spring 2017, Math 230, Differential multivariable calculus (Calc III)
• Fall 2016, Math 224, Integral single variable calculus (Calc II)

Past classes (at Columbia):
• Autumn 2015: Precalculus
• Summer 2011: Integral single variable calculus (Calc II)
• Summer 2014: Differential multivariable calculus (Calc III)

## People

Here are a few friends (in no particular order) who share my interests:

## Everytopic seminar

Corey Bregman and I are co-organizing the Everytopic seminar at Brandeis. Talks are colloquium style and are suitable for a broad audience of mathematicians, including graduate students. Each seminar is 50 minutes and is followed by pizza. We usually meet Mondays at 5PM in Goldsmith 300 . The schedule for Spring 2020 can be found below.

### Title and abstract

Jan. 27 Elden Elmanto (Harvard) Cobordisms and moduli spaces in algebraic geometry:
Naively, one might guess that a cobordism between two smooth varieties is defined by another smooth variety (of one dimension higher) admitting a projective morphism to P^1. This is almost correct, save for some serious transversality issues. In this talk, we will give a survey of the theory of algebraic cobordism (due to Voevodsky Levine-Morel, and Levine-Pandharipande), give some examples and say what the the theory is good for. We will then give some recent progress on how to solve transversality issues using derived algebraic geometry. Part of this is joint work with Bachmann, Hoyois, Sosnilo and Yakerson.

Feb. 3 No speaker
Feb. 10 No speaker
Feb. 24 Andreas Mihatsch (MIT) Moduli of p-divisible groups:
The aim of this talk is to provide some background for two recent breakthroughs in arithmetic intersection theory, namely the proof of the Arithmetic Fundamental Lemma (AFL) by Wei Zhang and the proof of the Kudla-Rapoport (KR) Conjecture by Chao Li and Wei Zhang. Both results concern intersection numbers on moduli spaces of p-divisible groups.
We will begin by defining p-divisible groups and by explaining how they come up in arithmetic geometry. We will then define their moduli spaces and illustrate some of their geometric properties. We end with a brief discussion of the two aforementioned results.

Mar. 2 Sebastien Picard (Harvard) Non-Kahler Calabi-Yau manifolds:
We will discuss a certain class of manifolds introduced by string theorists C. Hull and A. Strominger. These spaces are non-Kahler Calabi-Yau threefolds. We propose to study this geometry by using the Anomaly flow, which is a nonlinear flow of non-Kahler metrics. This talk will contain joint work with T. Fei, D.H. Phong, and X.-W. Zhang.

Mar. 9
Mar. 16
Mar. 23 Anthony Conway (Max Planck) TBD
Mar. 27
(note the unusual time!)
Matt Kerr (WUSTL) TBD
Apr. 20
Apr. 27 Rachael Norton (Fitchburg State) TBD
May 4

Past semesters: