• 1.01 Sciences

Accepting PhD Students

Personal profile

Key Research Interests

My research is primarily in Model Theory and its connections with number theory and analytic functions.

A major theme is the complex exponential function, which is used to describe both exponential growth and decay and sinusoidal wave patterns. Despite it being one of the most central of mathematical tools, there are still basic questions about the theory of the exponential function which are completely open. 

Complex numbers which can be described as solutions to a polynomial equation, like the square root of 2, are called algebraic. If they are irrational, this description does not pin them down exactly, for example 2 has both a positive and negative square root. The paper The algebraic numbers definable in various exponential fields with Macintyre and Onshuus, explains when we can use exponentiation to pick out exactly one solution.

In A geometric approach to some systems of exponential equations, Aslanyan, Mantova and I gave new situations where one can find complex solutions to equations. The general conjecture, called Exponential-Algebraic Closedness, is still open.

This Exponential-Algebraic Closedness is built into Zilber's exponential field, and I have done much work on the theory of that structure, such as in Pseudo-exponential-maps-variants-and-quasiminimality with Bays, and its connections with the complex exponential such as in Blurred-complex-exponentiation.

Connections with number theory include A Schanuel-property-for-exponentially-transcendental-powers with Bays and Wilkie on (functional) transcendence and with diophantine geometry such as Exponentially-closed-fields-and-the-conjecture-on-intersections-with tori with Zilber.

I am also interested in other analytic functions, for example the work Differential-existential-closedness-for-the-j-function with Aslanyan and Eterovic, and via o-minimality with Jones and Servi in Local interdefinability of Weierstrass elliptic functions.

Model Theory is an approach to doing mathematics which stems from the idea of formalising, or "mathematising" the language we use when doing mathematics. It has its own collection of techniques which are related to ideas from other areas of mathematics including combinatorics (often infinite, set theoretic combinatorics), geometry, algebra, and topology.

Within Model Theory, I am particularly interested in extending the techniques to applications slightly outside the usual scope of "classical first order" model theory, such as to positive logic as in Existentially-closed-exponential-fields with Haykazyan and to the very general "formalism-free" context of abstract elementary categories as in Quasiminimal-structures-and-excellence with Bays, Hart, Hyttinen and Kesälä, and the work of my former PhD student Mark Kamsma.

 

 

Follow this link for details of current PhD opportunities in Mathematics. But feel free to email me to discuss projects outside these areas and alternative sources of funding.

 

See my personal webpage for more information about my research and teaching.

Career

Reader and Associate Professor in Mathematics, UEA, 2019 -

Senior Lecturer in Mathematics, UEA, 2014 - 2019

Lecturer in Pure Mathematics, UEA, 2009 - 2014
EPSRC Postdoctoral Fellow, University of Oxford, 2007 - 2009

Research Assistant Professor, University of Illinois at Chicago, 2006 - 2007

DPhil, University of Oxford, 2002 - 2006

MSc in Mathematical Logic, University of Manchester, 2001 - 2002

MA and Certificate of Advanced Studies in Mathematics, University of Cambridge, 1997 - 2001

Teaching Interests

I have taught a variety of subjects for undergraduates including Model Theory, Mathematical Logic, Real and Complex Analysis, Calculus, Combinatorics, and Computability. I have also taught Category Theory to PhD students via the MAGIC consortium of universities. 

My book An Invitation to Model Theory is suitable for students at the beginning Masters or PhD level, and can also be read by researchers in other fields as an overview of the basic concepts of Model Theory. 

Key Responsibilities

  • Deputy Director of Teaching in MTH 2022 - 2023
  • LMS Departmental Representative for UEA  2013 -
  • Senior Advisor in MTH 2014 - 2022
  • Disability Liaison Officer for MTH 2014 - 2022
  • Chairman of MTH/ENG Extenuating Circumstances Panel 2014 - 2021

I have also been the Student Partnership Officer for MTH, the Director of Equality and Diversity in MTH, and the MAGIC coordinator.

Collaborations and top research areas from the last five years

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