Research

My research lies in the areas of algebraic geometry and number theory, in particular arithmetic geometry, algebraic number theory and analytic number theory. My research interests include rational points, harmonic analysis, additive combinatorics, multiplicative number theory, sieve theory, Brauer groups, L-functions, class field theory, etale cohomology, Hodge theory, algebraic stacks, Hilbert schemes, abelian varieties, computer algebra, automorphic forms, algebraic groups, and birational and equivariant geometry. Geometers think that I'm a number theorist, and number theorists think that I'm a geometer.

My work so far can be roughly sorted into the following themes.

Manin's conjecture: This conjecture concerns the distribution of rational points of bounded height on algebraic varieties. My papers [2] and [3] prove some cases of this conjecture for certain del Pezzo surfaces. My works [4] and [7] also concern this conjecture, where we construct new counter-examples to Manin's conjecture, and also use the Weil restriction to obtain new cases of this conjecture. In my paper [18], we obtain new lower bounds for Manin's conjecture for conic bundle surfaces.

Equivariant geometry: This topic concerns the study of algebraic varieties equipped with group actions. In my papers [1] and [5], we classify those (possibly singular) del Pezzo surfaces which admit a freely transitive action of some linear algebraic group. Such surfaces are of interest as Manin's conjecture is already known to hold in such cases.

Rational points in families of varieties: In [6] and [12], I study the problem of counting the number of varieties in a family which contain a rational point. In [6] I formulate some conjectures on this problem and prove some special cases via harmonic analysis on toric varieties, together with answering some cases of a question of Serre on specialisation of Brauer group elements. In [12], we also consider the problem of studying weak approximation for certain families of varieties. In my joint work [14], we find geometric conditions which force very few of the varieties in a family to have a rational point, in a precise quantitative sense. In my recent paper [36], we put forward a new conjecture concerning counting problems of this type.

Hasse principles: Suppose that some object X over a number field k satisfies some property P over every completion of k. Then does X necessarily satisfy P over k? If the answer is yes, then one says that a Hasse principle holds. We study problems of this type for the existence of lines on surfaces and intersections of two quadrics in [8] and [10], which includes both positive and negative results. In [13] we study problems of this type for the existence of norms for abelian extensions. I enjoy studying the Hasse principle via the Brauer-Manin obstruction for explicit equations [25, 31, 32].

Good reduction of algebraic varieties and the arithmetic of moduli stacks: A famous theorem of Faltings (formally the Shafarevich conjecture), states there are only finitely many abelian varieties of fixed dimension over a given number field k with good reduction outside of a given set of places of k. Such problems are closely related to the study of integral points on moduli stacks. We study problems of this type for flag varieties and complete intersections in projective space in [9] and [11], respectively. We also obtain a version of the Shafarevich conjecture for sextic surfaces in [16]. My later papers in this area have further explored the applications of stacks to such problems [26, 34].

Counting using harmonic analysis: I view harmonic analysis in the broad sense as the representation theory of topological groups. This is very useful for counting due to the following: When summing a function over a topological group G, the resulting sum is G invariant. This allows one to perform a spectral decomposition and rewrite the sum as a sum over irreducible representations and eigenfunctions. In most cases I consider G is abelian, where this process is simply Poisson summation. Cases I've considered is when G is the set of rational points on algebraic torus [6], or G is the absolute Galois group of Q, where I have developed new techniques with my co-authors for counting abelian number fields of bounded discriminant [13, 27, 36].

Del Pezzo surfaces: I have a soft spot for del Pezzo surfaces. Such surfaces were intensively studied by classical algebraic geometers over the complex numbers, with the most famous examples being cubic surfaces. Over non-algebraically closed fields, however, the arithmetic of del Pezzo surfaces is a very rich topic and an active area of research. Many of my papers concern these surfaces [1, 2, 3, 5, 8, 17, 18] and they also make fleeting appearences in my other works, as useful examples. In my work [17] we give a complete answer to the following question of Serre: over a given finite field k, what are the possible number of rational points that a smooth cubic surface over k can have? We expanded upon this in [28], and completely determined the possibilities for the Galois action on the lines of a cubic surface over every finite field.