# Research

My research lies in Complex and non-Archimedean Dynamics, Complex and Algebraic Geometry, Pluripotential Theory, and Several Complex Variables. The main objects of my research are complex varieties or projective varieties over a field other than complex numbers (e.g. the field of $p$-adic numbers), their geometric properties, and (meromorphic or rational) maps between them. I am also interested in applications of these fields to other fields, such as Mathematical Physics, Partial Differential Equations, and Statistics.

A) SOME RESEARCH HIGHLIGHTS.

In joint works with Fabrizio Catanese and Keiji Oguiso (see papers 1 and 3 below), I show the rationality and unirationality of some finite quotients of complex 3-tori. This solves a challenging question in Algebraic Geometry posed by Kenji Ueno in 1975. This also gives, for the first time, examples of unirational and rational 3-folds having primitive automorphisms of positive entropy.

In joint works with Tien-Cuong Dinh and Viet-Anh Nguyen (see papers 6 and 8 below), I give criteria to check that a meromorphic selfmap does not preserve any non-trivial meromorphic fibration, and prove equidistribution and density results for isolated periodic points of a quite general class of meromorphic maps.

I solve a question posed by Vincent Guedj in 2004 on the first dynamical degree of a meromorphic map. This result has applications to dynamics of complex 3-tori and more generally pseudo-automorphisms in dimension 3. See paper 7 below.

I give an algebraic approach to define dynamical degrees for rational maps over an arbitrary algebraic closed field of characteristic zero. This has applications to dynamics over Berkovich analytic spaces. See paper 7 below.

B) SELECTED WORKS IN PROGRESS.

With Charles Favre, Topological entropy of endomorphisms over non-Archimedean fields. Preliminary report, July 2013.

With Tien-Cuong Dinh and Viet-Anh Nguyen, Number of isolated periodic points of meromorphic maps. Preliminary report, June 2013.

With Muhammed Alan, Monge-Ampere equations with singular right hand side.

C) SELECTED PUBLICATIONS.

Automorphisms of positive entropy + (Uni)rational varieties.

1) With Fabrizio Catanese and Keiji Oguiso, Unirationality of the Ueno-Campana’s threefold. Preprint: arXiv:  1310.3569. [pdf]

2) With Keiji Oguiso, Salem numbers in dynamics of K\”ahler $3$-folds and complex tori. Submitted for publication, preprint arXiv: 1309.4851. [pdf]

3) With Keiji Oguiso, Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy. Submitted for publication, preprint arXiv: 1306.1590. [pdf]

4) On automorphisms of blowups of projective manifolds. Submitted for publication, preprint arXiv: 1301.4957. [pdf]

5) On automorphisms of blowups of $\mathbb{P}^3$. Submitted for publication, preprint arXiv: 1202.4224. [pdf]

Equidistribution of isolated periodic points.

6) With Tien-Cuong Dinh and Viet-Anh Nguyen, Equidistribution for meromorphic maps with dominant topological degree. Submitted for publication, preprint arXiv: 1303.5992. [pdf]

Dynamical degrees.

7) The simplicity of the first spectral radius of a meromorphic map. This paper was divided into two parts. Part 1: accepted in Michigan Journal of Mathematics (pending minor revisions) [pdf]; Part 2: submitted for publication. Preprint arXiv: 1212.1019. [pdf]

8) With Tien-Cuong Dinh and Viet-Anh Nguyen, On the dynamical degrees of meromorphic maps preserving a fibration, Communications in Contemporary Mathematics 14, 1250042, 2012. [pdf]

9) Degree complexities of birational maps related to matrix inversions: Symmetric case, Mathematische Zeitschrift 270 (2012), no 3–4, 725–738. [pdf]

10) With Eric Bedford, Degree complexities of birational maps related to matrix inversion, Communications in Mathematical Physics 298 (2010), no. 2, 357–368. [pdf]

11) Degree complexity of a family of birational maps: II. Exceptional cases, Mathematical Physics, Analysis and Geometry 12 (2009), no 2, 157–180. [pdf]

12) With Eric Bedford, Kyounghee Kim, Nina Abarenkova, and Jean-Marie Maillard, Degree complexity of a family of birational maps, Mathematical Physics, Analysis and Geometry 11 (2008), no 1, 53–71. [pdf]

Positive closed currents and Pullback by meromorphic maps.

13) With Dan Coman, Geometric properties of upper level sets of Lelong numbers on projective spaces. Submitted for publication, preprint arXiv: 1305.0864. [pdf]

14) Some dynamical properties of pseudo-automorphisms in dimension $3$. Transactions of the American Mathematical Society, accepted. Preprint arXiv: 1304.4100. [pdf]

15) Pullback of currents by meromorphic maps, Bulletin de la Soci\’et\’e Math\’ematique de France, accepted. Preprint arXiv: 1107.1743. [pdf]

Analytic functions, Plurisubharmonic functions, and Applications.

16) With Dang Duc Trong, Cao Xuan Phuong, and Dinh Ngoc Thanh,  Tikhonov’s regularization to deconvolution problem, Communications in Statistics: Theory and Methods, accepted. [pdf]

17) With Dang Duc Trong, Alain Pham Ngoc Dinh, and Phan Thanh Nam,  Determination of the body force of a two dimensional isotropic elastic bodyJournal of Computational and Applied Mathematics 229 (2009), 192–207. [pdf]

18) Sets non-thin at $\infty$ in $\Bbb C ^m$, Journal of Mathematical Analysis and Applications 356 (2009), no 2, 517–524. [pdf]

19) With Dang Duc Trong, The growth at infinity of sequences of entire functions of bounded orders, Complex Variables and Elliptic Equations, Vol 53 (2008), No 8, 717–743. [pdf]