Research
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Research Papers

On an invariant bilinear form on the space of automorphic forms via asymptotics, Duke Math. J., to appear. (pdf) (arXiv)
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 functionssheaves dictionary. In the proof, we highlight the connection between the classical nonArchimedean GindikinKarpelevich 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, J. Lie Theory 27 (3), 637655 (2017). (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 quasiaffine 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), Selecta Math. (N.S.) 22 (4), 18251880 (2016). (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 PicardLefschetz 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 FourierDeligne transform, and
the usual Fourier transform on \(\mathcal D\)modules (when the latter are defined).
Undergraduate Research

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), 365372.
(pdf)

The zerodivisor graph associated to a semigroup (with L. DeMeyer, L. Greve, and
A. Sabbaghi), Communications in Algebra 38 (2010), 33703391.
(pdf)