About
Department of Mathematics, Northwestern University
301 Lunt Hall
krishna[AT]math.northwestern.edu
rmk4167[AT]northwestern.edu
I am a Boas Assistant Professor at the math department at Northwestern. 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
Last updated: Fall 2018
Research
- A new proof of the Waldspurger formula I (to appear in Algebra and Number Theory) 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.
- The semi-Schrödinger model of an exceptional representation of $\widetilde{\mathrm{GL}}_{2q}$ (In preparation.) I construct a new model space, akin to the Schrödinger model of Weil representation, for the exceptional representation of the double cover of an even rank general linear group.
- 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 examples.
- 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, at least for spherical varieties of rank one.
Teaching
I am not teaching during the Winter 2019 quarter.During the Spring 2019 quarter, I will be teaching two sections of Math 230.
Past classes (at Northwestern):
- Fall 2016, Math 224
- Spring 2017, Math 230
- Spring 2017, Math 482-2, Elliptic curves and modular forms.
- Fall 2017, Math 224
- Spring 2018, Math 224
- 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!
- Fall 2018, Math 230
Past classes (at Columbia):
People
Here are a few friends who share my interests: