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We give examples of smooth quasi-projective varieties over complex numbers, in the context of connected Shimura varieties, which are not homeomorphic to a conjugate of itself by an automorphism of the complex numbers.

We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in…

The absolute sets of local systems on a smooth complex algebraic variety are the subject of a conjecture of N. Budur and B. Wang based on an analogy with special subvarieties of Shimura varieties. An absolute set should be the…

We study contact structures on smooth complex projective varieties with a simple normal crossing divisor, generalizing some well-known results concerning the non-logarithmic case. In particular, we describe the structure of elementary log…

We prove the Kawamata-Morrison cone conjecture for Q-factorial terminal projective primitive symplectic varieties with second Betti number greater than five defined over a field of characteristic zero. As an application, we prove that the…

We prove estimates for the Betti numbers of constructible sheaves in characteristic p>0 depending only on their rank, stratification and wild ramification. In particular, given a smooth proper variety of dimension n over an algebraically…

This is an expository note explaining how the geometric notions of local connectedness and properness are related to the $\Sigma$-type and $\Pi$-type constructors of dependent type theory.

This is my dissertation about digraphs ordered by pp-constructability. We study in particular smooth digraphs, i.e., digraphs without sources or sinks, tournaments and semicomplete digraphs, orientations of paths and cycles, digraphs with…

We give new examples of algebraic integral cohomology classes on smooth projective complex varieties that are not integral linear combinations of classes of smooth subvarieties. Some of our examples have dimension 6, the lowest possible.…

We introduce a theory of motivic cohomology for quasi-compact quasi-separated schemes, which generalises the construction of Elmanto--Morrow in the case of schemes over a field. Our construction is non-$\mathbb{A}^1$-invariant in general,…

We prove that the monodromy of an irreducible cohomologically complex rigid local system with finite determinant and quasi-unipotent local monodromies at infinity on a smooth quasiprojective complex variety $X$ is integral. This answers…

This small text was written for the AMS Notices. It is a survey of integrality properties of complex local systems, where I tried to single out one example which is not entirely explicit in the literature. The focus is on the obstruction it…

We study the local cohomology modules for the secant variety of lines of a smooth projective variety $Y$ and for higher secant varieties of smooth projective curves. We show that the local cohomological defect in the first case is related…

I consider the class of surfaces $X$ over algebraically closed fields with numerical invariants given in the title. In characteristic zero, this class contains fake projective planes which were introduced by David Mumford. I prove that in…

We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of…

In the several contexts such as combinatorial number theory, families of sets of positive integers closed under taking subsets have been investigated. Then it is sometimes useful to give bijections between the set of the one-sided infinite…

We construct a projective variety with discrete, non-finitely generated automorphism group. As an application, we show that there exists a complex projective variety with infinitely many non-isomorphic real forms.

Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…

We classify the smooth projective symmetric G-varieties with Picard number one (and G semisimple). Moreover we prove a criterion for the smoothness of the simple (normal) symmetric varieties whose closed orbit is complete. In particular we…

This paper describes several classical constructions of thin bases of finite order in additive number theory, and, in particular, gives a complete presentation of a beautiful construction of J. W. S. Cassels of a class of polynomially…