# Jonathan Wang

I am a fifth year graduate student in mathematics at the University of Chicago. My advisor is Vladimir Drinfeld.

In 2011-2012, I completed the Master of Advanced Study in Mathematics at the University of Cambridge with funding from the Churchill Scholarship. I graduated in 2011 with an A.B.-A.M. degree in Mathematics from Harvard University.

My CV may be found here (updated November 2016).
Email: jpwang [at] math [dot] uchicago [dot] edu

## Writings

### Research

• On an invariant bilinear form on the space of automorphic forms via asymptotics. (pdf)
We generalize the definition of the bilinear form $$\mathcal B$$ to an arbitrary split reductive group over a function field. The definition of $$\mathcal B$$ relies on the asymptotics maps defined using the geometry of the wonderful compactification of $$G$$. We show that this bilinear form is naturally related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary. In the proof, we highlight the connection between the classical non-Archimedean Gindikin-Karpelevich formula and certain factorization algebras acting on geometric Eisenstein series. We also give an alternate definition of $$\mathcal B$$ using the constant term operator and the standard intertwining operator.

• On the reductive monoid associated to a parabolic subgroup, submitted. (pdf) (arXiv)
Let $$G$$ be a connected reductive group over a perfect field $$k$$. We study a certain normal reductive monoid $$\overline M$$ associated to a parabolic $$k$$-subgroup $$P$$ of $$G$$. The group of units of $$\overline M$$ is the Levi factor $$M$$ of $$P$$. We show that $$\overline M$$ is a retract of the affine closure of the quasi-affine variety $$G/U(P)$$. Fixing a parabolic $$P^−$$ opposite to $$P$$, we prove that the affine closure of $$G/U(P)$$ is a retract of the affine closure of the boundary degeneration $$(G \times G)/(P \times_M P^−)$$. Using idempotents, we relate $$\overline M$$ to the Vinberg semigroup of $$G$$. The monoid $$\overline M$$ is used implicitly in the study of stratifications of Drinfeld's compactifications of the moduli stacks $$\mathrm{Bun}_P$$ and $$\mathrm{Bun}_G$$.

• On a strange invariant bilinear form on the space of automorphic forms (with V. Drinfeld), accepted by Selecta Mathematica. (arXiv)
Let $$F$$ be a global field and $$G:=SL(2)$$. We study the bilinear form $$\mathcal B$$ on the space of $$K$$-finite smooth compactly supported functions on $$G(\mathbb A)/G(F)$$ defined by $$\mathcal B (f_1,f_2):=\mathcal B_{naive}(f_1,f_2)-\langle M^{-1}\text{CT} (f_1)\, ,\text{CT} (f_2)\rangle$$, where $$\mathcal B_{naive}$$ is the usual scalar product, CT is the constant term operator, and $$M$$ is the standard intertwiner. This form is natural from the viewpoint of the geometric Langlands program. To justify this claim, we provide a dictionary between the classical and 'geometric' theory of automorphic forms. We also show that the form $$\mathcal B$$ is related to S. Schieder's Picard-Lefschetz oscillators.

• Radon inversion formulas over local fields, Math. Res. Lett. 23(2), 535–561 (2016). (pdf) (arXiv)
Let $$F$$ be a local field and $$n \ge 2$$ an integer. We study the Radon transform as an operator $$M : \mathcal C_+ \to \mathcal C_-$$ from the space of smooth $$K$$-finite functions on $$F^n \setminus \{0\}$$ with bounded support to the space of smooth $$K$$-finite functions on $$F^n \setminus \{0\}$$ supported away from a neighborhood of $$0$$. These spaces naturally arise in the theory of automorphic forms. We prove that $$M$$ is an isomorphism and provide formulas for $$M^{-1}$$. In the real case, we show that when $$K$$-finiteness is dropped from the definitions, the analog of $$M$$ is not surjective.

• A new Fourier transform, Math. Res. Lett. 22(5), 1541–1562 (2015). (pdf) (arXiv)
In order to define a geometric Fourier transform, one usually works with either $$\ell$$-adic sheaves in characteristic $$p>0$$ or with $$\mathcal D$$-modules in characteristic $$0$$. If one considers $$\ell$$-adic sheaves on the stack quotient of a vector bundle $$V$$ by the homothety action of $$\mathbb G_m$$, however, Laumon provides a uniform geometric construction of the Fourier transform in any characteristic. The category of sheaves on $$[V/\mathbb G_m]$$ is closely related to the category of (unipotently) monodromic sheaves on $$V$$. In this article, we introduce a new functor, which is defined on all sheaves on $$V$$ in any characteristic, and we show that it restricts to an equivalence on monodromic sheaves. We also discuss the relation between this new functor and Laumon's homogeneous transform, the Fourier-Deligne transform, and the usual Fourier transform on $$\mathcal D$$-modules (when the latter are defined).

#### Papers from Undergraduate Research Projects

• Thin Lehman matrices and their graphs, Electronic Journal of Combinatorics 17 (2010), R165. (pdf)
• A new infinite family of minimally nonideal matrices, Journal of Combinatorial Theory, Series A 118 (2011), 365-372. (pdf)
• The zero-divisor graph associated to a semigroup (with L. DeMeyer, L. Greve, and A. Sabbaghi), Communications in Algebra 38 (2010), 3370-3391. (pdf)

### Exposition

• Introduction to $$D$$-modules and representation theory (pdf)
Cambridge Part III Essay. This document attempts to provide a succinct yet thorough introduction to some basic properties of algebraic $$D$$-modules. I hope to modify/update this essay as my knowledge of the subject improves.
• The moduli stack of $$G$$-bundles (pdf) (arXiv)
Harvard University Senior Thesis. This paper provides an expository account of the geometric properties of the moduli stack of $$G$$-bundles.
• Functor of points description of the flag variety (pdf)
Personal note on the connection between $$G$$-equivariant line bundles on the flag variety $$G/B$$ and the functor of points of $$G/B$$ using Plücker relations.
• Existence of the quotient scheme $$G/H$$ (pdf)
Proof of the representability of the fppf sheaf $$G/H$$ by a scheme for (not necessarily reduced) algebraic groups $$H \subset G$$ over an arbitrary field. Also proves that the quotient is a geometric quotient in the case $$H$$ and $$G$$ are smooth.
• Local algebra in algebraic geometry (pdf)
An overview of some facts from local algebra and how they relate to algebraic geometry. Based on the course Math 233B. Theory of Schemes, taught by Dennis Gaitsgory at Harvard, Spring 2010.
• Theorem on formal functions, Stein factorization, and Zariski's Main Theorem (pdf)
Discussion and proofs of the theorem on formal functions, Stein factorization, and the various forms and applications of Zariski's Main Theorem. Everything is proved for proper morphisms (as opposed to only projective morphisms).
• Higher direct images of coherent sheaves under a proper morphism (pdf)
A proof that for a proper map of noetherian schemes, higher direct images of a coherent sheaf remain coherent. Includes a short introduction to derived functors.

### Notes

The following are less formal notes/thoughts of mine from various courses/talks.
• Topics in calculus and algebra (html) Taught by Ian Grojnowski at University of Cambridge, Lent 2012.
• Moduli stacks of vector bundles (pdf) Notes from my talk for the Part III Algebraic Geometry Seminar, Lent 2012.
• Abstract algebra (pdf) Math 55A lecture notes, taught by Dennis Gaitsgory at Harvard, Fall 2007.
• Cryptography (pdf) Computer Science 220R lecture notes, taught by Michael O. Rabin at Harvard, Fall 2009.

### Useful websites

• Sloane
On-line encyclopedia of integer sequences
• MathSciNet
Database of math publications
• Detexify
LaTeX handwriting recognition

### Summer programs

• Duluth REU
Summer research program run by Joe Gallian at the University of Minnesota Duluth. I was a participant in the summer of 2009. I highly recommend this program!