University of Nottingham

Title: Versions of the circle method

Abstract: The circle method is used in combinatorics, subconvex bounds for L-functions, Diophantine equations, and Diophantine inequalities. Increasingly, researchers speak of different versions of the circle method, or different 'delta methods'. I will discuss the four essential features of the method invented by Ramanujan, Hardy and Littlewood and put in its modern form by Vinogradov.* I will then discuss the various innovations and extra ingredients which go into different versions of the circle method as they exist today. *Spoilers: Major arcs/Farey arcs, Hua's lemma/moments/mean values, Weyl differencing/van der Corput differencing, summation formula/trace formula.

Title: Galois groups of low degree points on curves

Abstract: Low degree points on curves have been subject of intense study for several decades, but little attention has been paid to the Galois groups of those points. In this talk we recall primitive group actions, and focus on low degree points whose Galois group is primitive. We shall see that such points are relatively rare, and that they interfere with each other. This talk is based on joint work with Maleeha Khawaja.

Title: Toward criteria for the K-stability of Fano manifolds

Abstract: The Calabi problem for Fano manifolds asks the existence of “good” metrics, so called Kaehler-Einstein metrics.  Around 10 years ago, the above differential geometric problem is shown to be equivalent to an algebraic stability condition called “K-stability”. In this talk, I will present a simplification of the above stability condition using the notion of volume functions. Moreover, I would like to survey recent remarkable progresses in terms of moduli theory of K-stable Fano varieties.

Title: Moments of quadratic Dirichlet $L$-functions over function fields

Abstract: After reviewing recent progress in the moment problem, I will present an asymptotic formula for a "smoothed" fourth moment of quadratic Dirichlet L-functions over function fields, exhibiting infinitely many terms in the asymptotic expansion. The proof involves studying the multiple Dirichlet series associated with the fourth moment, a series in five complex variables satisfying an infinite group of functional equations. We show that this series can be analytically continued to an optimal domain, using a new type of functional equation. This is joint work with Adrian Diaconu and Vicenţiu Paşol.

Title: Abelian number fields with restricted ramification and rational points on stacks

Abstract: A conjecture by Malle gives a prediction for the number of number fields of bounded discriminant. In this talk I will give an asymptotic formula for the number of abelian number fields of bounded height whose ramification type has been restricted to lie in a given subset of the Galois group and provide an explicit formula for the leading constant. I will then describe how counting these number fields can be viewed as a problem of counting rational points on the stack BG and how the existence of such number fields is controlled by a Brauer-Manin obstruction

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Title: Introduction to classification of threefolds of general type

Abstract: In higher dimensional algebraic geometry, the following three types of varieties are considered to be the building blocks: Fano varities, Calabi-Yau varieties, and varities of general type. In the study of varieties of general type, one usually work on "good models" inside birtationally equivalent classes. Minimal models and canonical models are natural choices of good models. 

In the first part of the talk, we will try to introduce some aspects of geography problem for threefolds of general type, which aim to study the distribution of birational invariants of threefolds of general type. In the second part of the talk, we will explore more geometric properties of those threefolds on or near the boundary. Some explicit examples will be described and we will compare various different models explicitly. 

Title: Equivariant birational geometry of Pfaffian cubic threefolds

Abstract: It is known that any smooth cubic threefold Y is given by the vanishing of the Pfaffian of some 6 by 6 skew-symmetric matrix of linear forms. In addition, a Pfaffian representation of Y corresponds to a rank 2 vector bundle on Y. Such a representation also gives rise to a birational map between Y and a smooth Fano threefold X of degree 14. In the first part of the talk, we will review these classical constructions through elementary linear algebra. Then we explore their compatibility with automorphism groups of smooth cubic threefolds. Using this, we give new results on equivariantly stable birationalities between Y and X. This is joint work with Yuri Tschinkel.

Title: Congruences between eigensystems for GL(n)

Abstract: In this talk, I will discuss congruences between modular forms. For example, consider the following question: let p be prime and f be a modular eigenform of level Gamma_0(M), where p divides M. For a given integer m, does there exist another eigenform g congruent to f modulo p^m?

The answer, amazingly, is yes. Even better, such congruences can be captured geometrically via 1-dimensional 'families' of eigenforms (via 'the eigencurve'). This theory, introduced by Hida and Coleman in the 80s/90s, has had (and continues to have) profound consequences in Iwasawa theory and the Langlands program. In this talk, my main aim will be to give a broadly accessible introduction to p-adic families, the eigencurve, and their applications to congruences of modular forms. I'll try to only assume standard knowledge of modular forms, at the level of a typical master's course.

At the end I'll describe joint work with Daniel Barrera and Andy Graham, where we consider some of the problems in generalising these results to automorphic forms of GL(n) (modular forms being the case of GL(2)). Here the picture becomes more subtle -- whilst the original congruences question always has a positive answer for modular forms, in higher dimension often it is conjectured that such systematic congruences don't exist. Daniel, Andy and I study this phenomenon for congruences between symplectic forms.

Title: Some quotients of Cremona groups

Abstract: The group of birational transformations of the projective n-space over a field K, denoted Cr(n,K), is called Cremona group of rank n over K. The structure of these groups depends on the field and on the dimension. For example, the abelianisation of Cr(2,K) is trivial exactly if K is algebraically closed. I will discuss the construction of some quotients of Cremona groups using the Sarkisov program. For example, in a joint work with J. Blanc and E. Yasinsky, we use the birational geometry of non-trivial Severi-Brauer surfaces to show that the free group over the set of complex numbers is a quotient of Cr(n,\mathbb{C}) for n at least 4.

Title: Morava motives and Galois cohomological invariants

Abstract: Galois cohomology is a sequence of abelian groups that is associated to a field K and are non-trivial only if the field K is not algebraically closed. Given an algebraic variety over a field K, one could consider its invariants that take values in Galois cohomology of K. Since these invariants would vanish over the algebraic closure of K, this would be especially meaningful if the variety also 'trivializes' over the algebraic closure, i.e. becomes of some 'simple' or 'canonical' form. There are, however, many other cases which are not captured by this scheme, and there is no known universal definition of Galois cohomological invariant.

In my talk, I will present a framework in which one can work with Galois cohomology (of arbitrary fixed degree)

and smooth projective varieties on the same footing. In this setting the definition of Galois cohomological invariant of an arbitrary smooth projective variety becomes almost a tautology. In order to define this framework one needs to mix the ideas of Grothendieck motives with the algebro-geometric part of the chromatic homotopy theory, namely, Morava K-theories. The talk is based on work in progress, partially joint with A.Lavrenov.

Title: Isotropic world

Abstract: Algebraic Geometry is substantially more complex than Topology. One would like to have local versions of it parametrised roughly by all possible kinds of algebro-geometric points (i.e., by field extensions of the base field), which would allow one to read the information in a simple form. This is achieved with the help of isotropisation, where one annihilates motives of p-anisotropic varieties (i.e., varieties having only closed points of degrees divisible by p). The resulting family of isotropic realisations gives a large number of new points of the Balmer spectrum (a tensor-triangulated generalisation of the Zariski spectrum of a ring) of the category, complementing points coming from the topological realisation. Here “isotropic world” serves as an alternative to topology. A similar (but more general) approach applies to the algebro-geometric stable homotopy category (of Morel-Voevodsky).

Title: Brauer–Manin obstructions on K3 surfaces

Abstract: I will describe some local-global principles used in the study of rational points on algebraic varieties, and the Brauer–Manin obstruction which can explain failures of these local-global principles. I will then discuss some joint work with Martin Bright concerning the primes playing a role in the Brauer–Manin obstruction for varieties with torsion-free geometric Picard group, such as K3 surfaces.

Title: Frieze patterns and Farey complexes

Abstract: Frieze patterns are periodic arrays of integers introduced by Coxeter in the 1970s. Conway and Coxeter discovered an elegant way of classifying frieze patterns of positive integers using triangulated polygons. Recently there has been a resurgence of interest in frieze patterns, motivated by their relationship with cluster algebras. An open problem has been to provide a combinatorial model for frieze patterns over the ring of integers modulo n akin to Conway and Coxeter’s model for positive integer frieze patterns. We describe such a model using the Farey complex of the integers modulo n. Using this model we also solve the problem of when a frieze pattern over the integers modulo n lifts to a frieze pattern over the integers.