PhD (York), MPhil (QAU), MSc(PU)
Membership of professional associations
Member of the Australian Mathematical Society and President of Number Theory Down Under.
Area of study
Mathematics and Statistics
I graduated with a PhD (Mathematics) in 2011 from The University of York (England).
I am an active researcher in Analytical Number Theory, Ergodic Theory and Dynamical Systems and the interactions between them. I enjoy working on Number Theoretic Applications to areas such as Electronic Communications, Signal Processing, Partial Differential Equations and Cryptography. I also like to read current research in Mathematical Education and whenever possible contribute to it. See some examples of RESEARCH PROJECTS at the bottom of the page.
Prior to my current position, I have held positions of
- Visiting Faculty Member, Brandeis University (USA), 2016--2017.
- Lecturer, the University of Newcastle (Australia), 2015--2016.
- Associate Lecturer, the University of Newcastle (Australia), 2014--2015.
- Research Associate, the University of Newcastle (Australia), 2013-2014.
- Postdoctoral Research Fellow, Aarhus University (Denmark), 2012—2013.
- Honorary Research Fellow, La Trobe University (Australia), 2010—2012.
- Teaching Assistant, The University of York (England), 2006—2010.
I enjoy teaching and as a testament to my contributions to students learning I received the “2016 Award for Teaching Excellence and Contribution to Student Learning - Individual” from the Faculty of Science and IT, The University of Newcastle. I was also "highly commended for team teaching". In 2016 I was also awarded an Endeavour Fellowship by the Department of Teaching and Training--Australian Government.
Professional Service. I am a regular reviewer for AMS MathSciNet and Zentralblatt Mathematical Reviews and have acted as a referee for many leading international mathematical journals. I have organised a number of international conferences, these include:
- Diophantine approximation and dynamical systems, 6--8 January 2018, La Trobe University.
- Number Theory Down Under (NTDU), annually from 2013 to 2016 at Newcastle and 2017 at La Trobe University Bendigo Campus.
- Capital Number Theory (2016) at The Australian National University.
- Ergodic Theory and its Applications (2016) at The University of Sydney.
The workshops/conferences have been well received and are so far acting as a meeting point harbouring interaction and collaborative research amongst Australian and distinguished overseas researchers.
In 2015, Number Theory Down Under (NTDU) was recognised as the special interest group within the Australian Mathematical Society. I was the first elected General Secretary of this group and as of September 2017, I am the President of this group.
- Analytical Number Theory; metric Diophantine approximation;
- Ergodic Theory and Dynamical Systems;
- MAT2VCA: Vector Calculus
- MAT1ICA: Introductory Calculus
- MAT1DIS: Discrete Mathematics
Here are my latest publications. Most of my publications are available on arXiv and can also be accessed from MathSciNet. A full list is available on my personal website and also on Researchgate and Googlescholar pages. However, if for any reason, you are unable to download them, please contact me to request a copy.
- Layered Interference Alignment: Achieving the Total DoF of MIMO X Channels. Pre-Print:arXiv:1412.7188. or it can be downloaded from here. (with S. H. Mahboubi, A. S. Motahari, A. K. Khandani).
- A problem in non-linear Diophantine approximation. To appear in Nonlinearity. The preprint can be downloaded here. (with Stephen Harrap and Simon Kristensen).
- Hausdorff measure of sets of Dirichlet non-improvable numbers. To appear in Mathematika, preprint: arXiv:1704.03089v1 With (Dmitry Kleinbock, Nick Wadleigh and Bao-Wei Wang).
- The Hausdorff measure version of Gallagher's theorem--closing the gap and beyond. J. Number Theory 186(2018), 211--225. (with David Simmons).
- Two-dimensional shrinking target problem in beta dynamical systems. Bull. Aust. Math. Soc. 97(2018), no.1, 33--42.(With W. Wang).
- A Khintchine--Groshev type theorem in absolute value over complex number systems. New Zealand Journal of Mathematics, Volume 47(2017), 57--67.
- An Inhomogeneous Jarnik type theorem for planar curves. Math. Proc. Camb. Phil. Soc.163 (2017), no. 1, 47--70. (with Dzmitry Badziahin, Stephen Harrap).
- A note on badly approximable sets in projective space. Mathematische Zeitschrift. 285 (2017), no. 1-2, 239--250. (with S. Harrap).
A dichotomy law for the Diophantine properties in $beta$--dynamical systems.Mathematika 62 (2016) 884--897. (with M. Coons and Bao-Wei Wang).
- A Jarnik type theorem for planar curves: everything about the parabola. Math. Proc. Camb. Phil. Soc. (2015), 159, 47--60 or download a preprint pdf.
- A converse to linear independence criteria, valid almost everywhere. Ramanujan J (2015) 38:513–528. (with S. Fischler, S. Kristensen and J. Levesley).
- A note on the weighted Khintchine--Groshev theorem, Journal de Theorie des Nombres de Bordeaux, 26(2014), 385-397. (with T. Yusupova).
Here are some research projects for prospective graduate or undergraduate students. Contact me if you would like to discuss them in detail.
1. Metric Number Theory. The main area of research within metric number theory is metric Diophantine approximation. The fundamental question in this branch is: how well can a real number be approximated by rational numbers quantitatively. To answer this question in various settings requires a deep understanding of the geometry of numbers, measure theory and probability theory. Many field medalists worked in this area of research such as K. Roth, A. Baker, G. Margulis and E. Lindenstrauss. Despite such big names having worked in this area there are multitudes of open problems which needs further investigations. Some examples include the Littlewood conjecture (1930) in multiplicative Diophantine approximation, Duffin-Schaeffer conjecture (1941) on rational approximations to real numbers, open problems on Diophantine approximation on manifolds such as Baker-Schmidt problem (1970). This theory is closely connected with Ergodic theory (for example Shrinking Target problems) and dynamical systems (for example homogeneous dynamics).
2. Ergodic Theory and Dynamical Systems. Within this area of research, I am primarily interested in orbit properties of flows on homogeneous spaces and recurrence properties in measurable dynamical systems and ergodic properties of measure preserving transformations.
3. Number Theoretic Applications. I have solved several open questions concerning the Absolute Value Theory (AVT) of metric Diophantine approximation. My work in AVT is continuously attracting many citations, not only in pure Mathematics (e.g. operator theory, non-linear PDE’s) but also in applied areas such as Signal Processing, see for example my paper on "Layered Interference Alignment" arXiv:1412.7188. There are many open questions on similar applications which can be answered subject to resolution of certain number theoretic problems.
4. Mathematical Education. The mathematical education topics which are of interest to me stem from interesting discussions with colleagues and students and by following the recent trends in teaching techniques and pedagogies. Any interested student is welcome to discuss any topic related to any and all aspects of Mathematics or Statistics.