Abstract: Describing rational solutions to polynomial equations is one of the foundational goals in number theory, going back to Diophantus of Alexandria. In arithmetic geometry we study these solutions by looking at the variety that the polynomials describe. Questions one might ask are: does this variety contain any rational points? And if so, how many? How are they distributed? In this talk I will give an overview of how mathematicians have aimed to answer these questions for curves and surfaces. I will focus on a special class of surfaces called del Pezzo surfaces and motivate why these are interesting to study. We will go over different ways to talk about 'many' rational points, highlight several results, and show how these relate to some of the major open questions on the arithmetic of surfaces. 

Hoylake Room

Aliaksandra Novik
Title:  Around the DK Conjecture
Abstract: The DK conjecture establishes a link between birational geometry and derived categories, stating that two varieties are K-equivalent if and only if their derived categories are equivalent. Although the general case remains wide open, in this talk we will see key classical results by Kawamata, outline Bridgeland’s proof for the case of Calabi–Yau threefolds, and discuss potential generalizations 

Stefania Vassiliadis
Title: Foliations and MMP: why do we care?
Abstract:  I’ll give a brief introduction to the classical Minimal Model Program (MMP) and the foliated MMP, and I’ll present some applications to the study of foliations.

Birkdale Room

Izzy Rendell
Title: Quadratic Chabauty for modular curves
Abstract: Faltings’ proof of the Mordell Conjecture tells us that for a nice curve of genus at least two, its set of rational points is finite, and therefore we would like to explicitly compute the rational points on these curves. One approach is first to calculate a finite set of p-adic points on the curve, and then extract the rationals from this finite set. Examples of this type of method are the Chabauty-Coleman method and Quadratic Chabauty, where Quadratic Chabauty is an implementation of Kim’s generalisation of the Chabauty-Coleman method. I will give an overview of some of the ideas that go into these methods, some key results and some open problems in this area.

Julie Tavernier
Title: 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.

Hoylake Room

Cat Rust
Title: A Combinatorial Introduction to Expanded Stable Maps
Abstract: We will begin with an introduction to toric varieties and their fans and then move on to discuss the moduli space of expanded stable maps to P2 along with its stratification by combinatorial types. We will then see how these combinatorial types are generated and finish by considering the problem I am currently working on. 

Natasha Diederen
Title: The geometry of minimal surfaces
Abstract: Plateau's problem asks a deceptively simple question: for any given boundary, is there a surface of minimal area that spans it? The search for an answer to this problem lead to the development of geometric measure theory, a field that combines geometry, analysis and measure theory to tackle problems involving non-smooth surfaces. In this talk, we will explore how geometric measure theory expands on the traditional notion of a surface and provides a framework for solving Plateau's problem in a (broadly) satisfactory manner. 

Supriya Weiss
Title: Wading into k-currents!
Abstract: I introduce the notion of a k-current, motivate its development in geometric measure theory as a solution to Plateau's problem, and present some foundational results. Along the way, I will discuss n-rectifiable sets and integral varifolds.

Birkdale Room

Ashleigh Ratcliffe
Title: A systematic approach to solving Diophantine equations
Abstract: This talk reports on the following project: define a suitable notion of "size'' of a polynomial Diophantine equation, order all equations by size, and solve them in this order. On the way, we discuss how to solve Diophantine equations using theory of elliptic curves and the Mordell-Weil sieve to deduce the complete list of integers |n| ≤ 200 which are representable as the difference of two fourth powers.

Kate Thomas
Title: Sums of integers divisible by their base-3 digit sum
Abstract: A base-g Niven number is an integer divisible by the sum of its digits in base-g. We show that any sufficiently large integer can be written as the sum of three base-3 Niven numbers. This follows from an application of the circle method, which we use to count the number of ways an integer can be written as the sum of three integers with fixed, near-average, digit sum.

Mieke Wessel
Title: Solving quadratic forms in restricted variables with the circle method
Abstract: (joint work with Svenja zur Verth) Given a sequence A we study the number of zeroes of bounded height of a quadratic form f with coordinates in A. In particular we give conditions on both A and f such that we can use the circle method to count such zeroes. We build on earlier work of Biggs and Brandes who studied this question in the context of Waring's problem. 

Hoylake Room

María Abad Aldonza
Title: Applications of Grothendieck-Riemann-Roch theorem to Quantum Hall Effect
Abstract: I will introduce the sheaf of Laughlin wavefunctions, explain how the Chern character of such a sheaf encodes important information for quantum physics and give a panoramic view of the subjects of algebraic geometry needed to compute it. This work is a soon-to-be preprint in collaboration with Florent Dupont.

Maartje Wisse
Title: Introduction to knot homologies
Abstract: In this talk I will explain the motivations for the constructions of knot homologies, as well as providing sketches of the constructions. I will then briefly detail some applications to problems in low-dimensional topology. Expect lots of pictures and no previous knowledge of knot theory required!

Birkdale Room

Edwina Aylward
Title: Reduction types of genus 2 curves
Abstract: Tate's algorithm provides a straightforward method for determining the reduction type of an elliptic curve and understanding how it varies in field extensions. In contrast, the reduction types of genus 2 curves are far more intricate, classified into over 100 families by Namikawa and Ueno. In this talk, I will explain how the cluster picture machinery of Dokchitser–Dokchitser–Morgan–Maistret helps describe how the reduction type of a genus 2 curve varies in field extensions. This is joint work with Elvira Lupoian and Vladimir Dokchitser.

Blanca Gil Rosell
Title: Subconvex bounds for SO(n+1) x SO(n)
Abstract:  In 2023 Yueke Hu and Paul Nelson, in their paper "Subconvex bounds for U(n+1) x U(n) in horizontal aspects”, established a subconvex bound valid in certain horizontal aspects for L-functions attached to automorphic representations of unitary groups U(n+1) x U(n).  My current work concerns establishing a subconvex bound for L-functions attached to orthogonal groups SO(n+1) x SO(n).  Since this is still work in progress, I will explain how the unitary and the orthogonal cases are different, what are our approaches to prove the orthogonal case, and focus on the “triple product case”, which relates SO(4) x SO(3) to SL(2) x SL(2) x SL(2) via the exceptional isomorphism.

Hoylake Room

Thais Gomes Ribeiro
Title: A hypergeometric correspondence for a family of K3 surfaces
Abstract: Let k be a field and ψ ∈ k. Consider the family of K3 surfaces in P^3_k:

X_ψ : x^3 y + y^4 + z^3 w + z w^3 − 12 ψ x y z w = 0. We study this family from both arithmetic and complex geometric point of view. On the arithmetic side, that is, when k = F_q, for q a power of a prime, we give explicit formulas for the number of points #X(F_q) in terms of finite field hypergeometric sums. On the complex side, that is, when k = C, we compute some solutions of the Picard-Fuchs equation of the family and show that they are expressed in terms of hypergeometric series whose parameters are closely related to the parameters in the finite field hypergeometric sums. This is a work in collaboration with Adriana Salerno, Eli Orvis, Jessamin Dukes, Leah Sturman, Rachel Davis and Ursula Whitcher.

Sara Sajadi
Title: A unified finiteness theorem for curves
Abstract: In this talk, we present a finiteness theorem for rational points on smooth proper curves over number fields, which unifies several classical results. Our approach encompasses the Birch-Merriman's theorem, Siegel’s theorem, and Faltings’ theorem as special cases. The proof relies on descent techniques and a boundedness argument, leading to a uniform perspective on finiteness phenomena in arithmetic geometry.

Birkdale Room

Mahya Mehrabdollahei
Title: A family of bivariate polynomials and variants of Chinburg’s conjectures
Abstract: In this talk, I present new solutions to Chinburg's conjectures by examining a sequence of multivariate polynomials. These conjectures assert that for every odd quadratic Dirichlet character of conductor f, denoted \chi_{-f}, there exists a bivariate polynomial (or a rational function in the weaker version) whose Mahler measure is a rational multiple of L'(\chi_{-f}, -1). The Mahler measure of a polynomial is the log average of a polynomial over the complex unit torus. To construct such solutions, we investigate a family of polynomials whose Mahler measures can be expressed as linear combinations of Dirichlet L-functions. This approach not only yields explicit solutions for conductors f = 3, 4, 8, 15, 20, and 24 but also provides insights that lead to a generalization of Chinburg’s conjecture. In particular, we extend the conjecture from real primitive odd Dirichlet characters to all primitive odd characters. For this broader setting, we demonstrate that our polynomial family produces solutions for conductors f = 5, 7, and 9. 

Catinca Mujdei
Title: Low-lying zeros of families of L-functions
Abstract: I will define the concept of one-level density of low-lying zeros of a family of L-functions. Through an example I will then explain how one can obtain asymptotics for such densities in the sense of Katz and Sarnak.

Hoylake Room

Silvia Gangeri
Title: Interval exchange transformations and their dynamic
Abstract: One of the first examples of interesting dynamical behaviour is rotations by irrational angles. This problem can be generalised in a multitude and in this talk we'll focus on interval exchange maps (i.e.m.), a problem that hides a lot cool mathematics. To understand the dynamics of an i.e.m. we'll take a short (but hopefully fun) dive in translation surfaces and their moduli spaces.

Ruth Wye
Title: Hilbert schemes as quiver varieties: a case study
Abstract: Quiver varieties are a large class of varieties, and can encode lots of data about the varieties they are isomorphic to. In this talk, we will study one particular quiver, and see how a choice of GIT stability condition can give a large range of varieties, including the Hilbert scheme of points on an ADE singularity, and on crepant partial resolutions of this. The difficulty is knowing what GIT cone to choose, but we will see how certain properties determine the GIT cones exactly. 

Xinrui You
Title: Projective functors on the category O
Abstract: Bernstein and Gelfand classified the projective functors (i.e. direct summands of the tensor by a finite dimensional representation) on the category of Z-finite modules on 1980. We will discuss the restriction of the projective functors on the category O, and the connection to Soergel bimodules.

Birkdale Room

Jessica Alessandrì
Title: The Integral Hilbert Property on some degree-2 del Pezzo surfaces
Abstract: In this talk we will consider a Diophantine equation known also as "Fermat near miss" (for its similarity to Fermat's equation). Studying the existence and abundance of integral solutions of this kind of equations can be translated in studying the integral points on a degree-2 del Pezzo surface.  Rational points on these surfaces have been widely studied, but results on integral points are still missing. We will use a double fibration method to show that on this surface we have "many" (i.e. a dense set of) integral points. This is a work in collaboration with D. Loughran.

Teri Cowen
Title: Modular Forms and Ternary Lattices of prime squared discriminant
Abstract: When considering a ternary quadratic lattice of square-free and non-square discriminant, there has been a lot of research proving that the Hecke-action on the classes of such lattices has eigenvalues in correspondence with the Fourier coefficients of a specific cusp space of weight two modular forms. In this talk, an introduction to relevant background knowledge will be provided to present current research and results on the prime squared discriminant case.

Ana Marija Vego
Title: Asai Euler system in Coleman families
Abstract: The Iwasawa Main Conjecture connects p-adic L-functions with specific Iwasawa modules. In the case of a real quadratic field, if F is a Hilbert modular form, the conjecture states that the characteristic ideal of the Selmer group of F is generated by a p-adic L-function. Using the Asai-Flach Euler system, Lei, Loeffler, and Zerbes establish bounds toward this conjecture for the Asai representation. We will explore possibilities of extending their work by analytically varying the Asai Euler system, as the Hilbert modular forms vary in p-adic Coleman families, potentially yielding further progress on the conjecture.