## 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

## 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