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Dr Mumtaz Hussain Senior Lecturer-Mathematics or Statistic, Mathematics & 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.
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. 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, La Trobe University 2018.
- Number Theory Down Under (NTDU), annually from 2013 to 2016 at Newcastle and 2017 at La Trobe University Bendigo.
- 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.

Here is a list of available PhD or Masters or Undergraduate projects for interested students.

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, homogeneous dynamics of group actions, conformal iterated function systems and its generalisations, measure rigidity theory, 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.



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