Overview

Starting from my PhD thesis Towards a Bezout-type Theory of Affine Varieties written at University of Toronto under the supervision of Pierre Milman, my research in math falls under two broad themes:

• Compactification of affine varieties
• Affine Bézout problem

I am in particular interested in one of the simplest cases of the first problem:

• Compactification of $\mathbb{C}^2$

Book

How many zeroes?

Papers

Compactification of affine varieties

• Projective completions of affine varieties via degree-like functions, Asian Journal of Mathematics, 18 (4), 2014; arXiv:1012.0835.

Compactification of $\mathbb{C}^2$

Preprints

• Normal equivariant compactifications of $\mathbb{G}^2_a$ of Picard rank one, arXiv:1610.03563.
• Mori dream surfaces associated with curves with one place at infinity, arXiv:1312.2168.
• Analytic compactifications of $\mathbb{C}^2$ part II – one irreducible curve at infinity, arXiv:1307.5577.

Publications

• Is the intersection of two finitely generated subalgebras of a polynomial ring also finitely generated? Arnold Mathematical Journal, 3 (3), 2017; arXiv:1301.2730.
• How to determine the sign of a valuation on $\mathbb{C}[x,y]$, Michigan Mathematics Journal 66 (4), 2017; arXiv:1301:3172.
• Algebraicity of normal analytic compactifications of $\mathbb{C}^2$ with one irreducible curve at infinity, Algebra & Number Theory, 10 (8), 2016; arXiv:1510.00998.
• Analytic Compactifications of $\mathbb{C}^2$ part I – curvettes at infinity, C. R. Math. Acad. Sci. Soc. R. Canada, 38 (2), 2016; arXiv:1110.6905.
• How fast do polynomials grow on semialgebraic sets?, with Tim Netzer, Journal of Algebra, 413, 2014; arXiv:1305.1215.
  Note: I had signed the pledge to boycott Elsevier. I came to know of the submission of this article to the Journal of Algebra only after the fact (Tim made this submission, and he was not aware of my pledge). I was too shy to ask him to withdraw the submission – this I regret now.
• Normal analytic compactifications of $\mathbb{C}^2$, Automorphisms in birational and affine geometry, Springer Proc. Math. Stat. 79, 2014; arXiv:1308.3286.
• Compactifications of $\mathbb{C}^2$ via pencils of jets of curves, C. R. Math. Acad. Sci. Soc. R. Canada, 34 (3), 2012.

Affine Bézout problem

Preprints

• Intersection multiplicity, Milnor number and Bernstein’s theorem, arXiv:1607.04860

Publications (primarily announcements)

• Bezout-type Theorems for the Affine Plane, C. R. Math. Acad. Sci. Soc. R. Canada, 34 (4), 2012.
• An affine Bezout type theorem and projective completions of affine varieties, C. R. Math. Acad. Sci. Soc. R. Canada, 30 (4), 2008.