My research interests span number theory, operator theory, and geometry. A unifying theme of my research philosophy is the interplay between rigidity and flexibility: specifically, how precise algebraic or geometric constraints interact with analytic or topological invariants.

Examples that motivate this perspective include:

• prime values of linear and polynomial functions via L-functions or exponential sums, as typified in theorems such as Dirichlet's on primes in arithmetical progressions and in conjectures such as Bateman-Horn's on the frequency of primes values of a system of univariate polynomials;

• the interplay between isometric and isomorphic properties, such as in establishing the weak fixed point property in reflexive Banach spaces; and

• geometric configurations on topological spaces, as exemplified by the inscribed square problem. 

These problems are notoriously difficult in their general formulations, often requiring new theoretical tools and unexpected connections. In this regard, since completing my terminal degree, I have been singularly interested in developing novel (or refined) analytic tools and methods to understand where flexible invariants are coupled with rigid constraints, what obstructs their compatibility, and how these tools and methods can be used to resolve problems such as the above. 

In practice, my ongoing research projects focus on the following specific areas of interest. The earliest few of these projects have now been completed, and publications presenting the results are in preparation.

Number Theory:

• Prime Distribution (2011--present)

  This project concerns the distribution of prime values in systems of integer-valued univariate sequences (centrally, polynomial sequences). Principal questions include Dickson’s conjecture, Bunyakovsky’s conjecture (more generally, the Bateman–Horn conjecture) and gaps between primes. My primary motivation is Dirichlet's Theorem on primes in arithmetic progressions, and how the main analytic-number theoretic ideas can be developed via local-global zeta function constructions attached to group objects that encode arithmetic data for a given sequence. To generalise to systems of sequences, I am interested in the development of a "hyperbolic" method (refiningthe classical circle method/exponential sums) to better handle the more difficult minor arcs' estimates that arise under the application of such methods.

• L-functions (2021--present)

  I investigate special values of L-functions at positive integer arguments and the distribution of their zeros, with an interest in connections to the Langlands programme for GL(n) and the Grand Riemann Hypothesis. The ideas being pursued here are intimately connected to my concurrent research project above on prime value distribution in integer-valued sequences.

• Diophantine Approximations & Equations (2023--present)

  I study Diophantine inequalities and homogeneous equations over number fields, with an interest in methods and tools aimed at generalising Dirichlet Approximation Thoerem and Davenport-Schmidt Theorem to tackle the Wirsing–Schmidt conjecture and related problems in transcendental number theory.

Operator Theory:

• Non-linear Operators (2016--present)

  I investigate fixed points of self-maps, focusing on the weak fixed point problem in reflexive Banach spaces and the interaction between geometric and structural properties that govern fixed point behaviour. The research project is aimed a complete geometric characterisation of the weak fixed point property, with affirmative resolution in quasi-very smooth spaces (also known as weakly Hahn-Banach smooth spaces). A related project concerns specialised non-linear operators defined viahomogeneous linear recurrence equations with constant coefficients.

• Operator Algebras (2019--present)

  I study hyperinvariant subspaces of operators, with the invariant subspace problem as a central motivation. The broader aim is to understand how the algebraic and analytic structure associated with an operator constrains the existence and form of its hyperinvariant subspaces; particularly, the how the ideas surrounding Lomonosov's celebrated theorem could be subsumed in a general analytic theory to tackle the problem in reflexive Banach spaces.

Geometry:

• Discrete Geometry (2017--present)

  I study the inscription of polytopes in topological manifolds, with the inscribed square problem as a principal motivation. My approach has been to develop novel methods from geometric function theory, in contrast to the traditional approaches via algebraic geometric/geometric topology; particularly, my interest is in general quadrilateral inscription on quasicircles.

• Convex Geometry (2022--present)

  This project concerns the separation and illumination of convex sets, with Hadwiger’s illumination conjecture as a central problem. I seek to understand the geometric conditions, with a generalisation to infinite-dimensional Banach space setting, that govern illumination and how separation methods can inform the analysis of these conditions. I am also exploring a functional analytic -- Hilbert space -- generalisation of the union-closed sets conjecture and how it elucidates the problem and its resolution.