WINGs 2023 Talks
Here are titles and abstracts of the talks that will take place during the conference. We will have seven keynote speakers, three academics in geometry, three in number theory, and one mathematician working outside academia. There will also be short talks given by participants.Keynote talks
Making and breaking post-quantum cryptography from elliptic curves , Chloe Martindale (University of Bristol)
Most of the public-key cryptography in use today relies on the hardness of either factoring or the discrete logarithm problem in a specially chosen abelian group. Here "hard" does not mean mathematically impossible but that the best known algorithm to solve the problem has complexity (sub-)exponential in the size of the input. However, once scalable quantum computers become a reality, both factoring and the discrete logarithm problem will no longer be hard problems, due to Shor's polynomial-time quantum algorithm to solve both problems. Post-quantum cryptography is about designing new cryptographic primitives based on different hard problems in mathematics for which there is no known polynomial-time classical or quantum algorithm. In this talk we will show how to design post-quantum cryptographic primitives from the hard problem of, given two elliptic curves over a large finite field, find and compute and isogeny between them (if it exists). We will then discuss recent work giving an attack on one of these primitives, Supersingular Isogeny Diffie-Hellman (SIDH). This is joint work with Luciano Maino, Lorenz Panny, Giacomo Pope, and Benjamin Wesolowski.
Quivers and higher quivers, Frances Kirwan (University of Oxford)
The theory of quiver representations plays an important role in algebraic geometry and also in geometric representation theory. The aim of this talk is to describe some of this theory and to give a brief glimpse of an extension to 'higher quivers' (which are, roughly speaking, to quivers as higher categories are to categories).
Embedding spaces of links Rachael Boyd (University of Glasgow)
I will introduce embedding spaces – these are fundamental spaces in toppology. In joint work with Corey Bregman we consider the homotopy type of embedding spaces of links (multiple circles), which are especially fun as you can draw pictures! I will introduce you to this topic and sketch our result on the fundamental group of the embedding space of a spit link. I’ll end with an overview of my career to date and some advice.
Careers Talk, Emma Bowley (Mathematician at Defence, Science and Technology Laboratory)
Emma will be speaking about her experiences working as a mathematician in industry.
Tropical Geometry, Diane Maclagan (University of Warwick)
Tropical geometry is a combinatorial shadow of algebraic geometry that replaces varieties with objects from polyhedral geometry and related combinatorics. In this talk I will introduce this area, and describe some of the applications.
Diophantine equations and when to quit trying to solve them, Rachel Newton (King's College London)
The study of integer or rational solutions to polynomial equations with integer coefficients is one of the oldest areas of mathematics and remains a very active field of research. The most basic question we can ask in this setting is whether the set of rational solutions is empty or not. This turns out to be a very hard question! I will discuss some approaches to answering it, and their limitations.
Elliptic curves and modularity, Ana Cariani (Bonn University & Imperial College London)
The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.
Geometry Short talks
Equivariant Minimal Surfaces in Complex Hyperbolic Spaces, Cordelia Webb (University of York)
The moduli space of equivariant minimal surfaces in complex hyperbolic space over a compact surface X, acquires an analytic structure via an embedding into the direct product of the Teichmuller space of X and the character variety of the fundamental group of X. This latter space is homeomorphic to the moduli space of PU(n, 1)-Higgs bundles via the Non-abelian Hodge correspondence. Such minimal surfaces also admit a harmonic sequence of maps. We unpack these definitions and then explore this relation between harmonic sequences and Higgs bundles to better understand these minimal surfaces.
Local stability threshold in del Pezzo surface of degree 2, Erroxe Etxabarri Alberdi (University of Nottingham)
We give a friendly introduction to K-stability, and the motivation behind it. We also introduce del Pezzo surfaces. In particular, we focus on del Pezzo surfaces of degree 2 and we finish with an interesting result about the local stability threshold of it.
Geometric realisation of Connes spectral triples for algebras with central bases., Evelyn Lira Torres (Queen Mary University of London)
In this talk, we will see the construction of the quantum geometrically realised spectral triples or 'Dirac operators' on the noncommutative algebras with central calculus and central spinor bimodule. We will see the outline of the formalism for the quantum Riemannian geometry, a ‘compatible’ version of Connes Spectral Triple, and detail the algebraic features, geometric conditions and the Hilbert space for this algebras. Finally, the fuzzy torus with their standard 2D differential calculus as an example will be given.
Kuga-Satake varieties of a family of K3 surfaces of Picard rank 14, Flora Wing Kei Poon (University of Bath)
The Torelli’s theorem says, a K3 surface with a polarisation lattice is completely determined by the weight two Hodge structure of its second cohomology up to isomorphism. We have a similar statement with polarised abelian variety and the weight one Hodge structure of its first cohomology. In the 60s, Kuga and Satake constructed a weight one Hodge structure from the second cohomology space of a K3 surface, which gives us a polarised abelian variety called the Kuga-Satake variety associated to the K3 surface. Recently in a paper by Clingher and Malmendier, a few families of K3 surfaces of Picard rank 14 with significance in String theory were examined. The aim of the project is to study the ``Kuga-Satake map’’ from one of these families of K3 surfaces to the corresponding family of Kuga-Satake varieties as moduli space of PEL type. Another motivation for studying the Kuga-Satake map for families of K3 surfaces of Picard rank 14 is that the correspondence demonstrates, as a particular case, how the type II locally symmetric domains are embedded into the type IV ones.
Classification of lattice triangles by their first two widths, Girtrude Hamm (University of Nottingham)
The width of a lattice polygon measures the smallest strip of the plane which it is a subset of. Classifying various types of polytope of small width is a recurring problem in polytope classifications. By introducing the second width of a polygon I will suggest a general method for classifying infinitely many polygons of some small width up to affine unimodular equivalence and apply this to classifying lattice triangles. A consequence of this is showing that the sequence counting lattice triangles, up to equivalence, which are a subset of a square has generating function equal to the Hilbert series of a hypersurface in a weighted projective space.
Stability conditions on free abelian quotients, Hannah Dell (University of Edinburgh)
In this talk, I will motivate and introduce Bridgeland stability conditions on triangulated categories. If time permits, I will discuss two approaches to studying stability conditions on derived categories of surfaces that are free quotients by finite abelian groups. One method is via Le Potier functions, which characterise the existence of slope-semistable vector bundles. The second method uses Deligne’s notion of group actions on triangulated categories.
Fukaya-Seidel categories of curve singularities, Ilaria Di Dedda (King's College London)
The idea of this talk is to define the Fukaya-Seidel category of isolated singularities on C^2 (sometimes called a Landau-Ginzburg model) defined by products of real irreducible polynomials, and to provide a nice algorithmic way to compute them (lots of pictures! :)). This fits in the framework of Homological Mirror Symmetry, but I will firmly stay on the A-side.
Deformations of Gorenstein codimension four varieties, Patience Ablett (University of Warwick)
In codimension one, two and three there are explicit structure theorems for the ideals defining projectively Gorenstein varieties. In codimension four and above, less is known. Work of Schenck, Stillman and Yuan exhibits all possible Betti tables for codimension and regularity four Gorenstein varieties. Further work in the area gives explicit constructions for Gorenstein varieties corresponding to each Betti table. In this talk we will discuss recent results of Ablett and Coughlan, which organise Gorenstein varieties with different Betti tables into deformation families.
Derived Geometry Relative to Monoidal Quasi-abelian Categories, Rhiannon Savage (University of Oxford)
In the theory of relative algebraic geometry, we work relative to a symmetric monoidal category C. The affines are now objects in the opposite category of commutative algebra objects in C. The derived setting is obtained by working with a symmetric monoidal model or infinity category C, with derived algebraic geometry corresponding to the case when we take C to be the category of simplicial modules over a simplicial commutative ring k. Kremnizer et al. propose that derived analytic geometry can be recovered when we replace k with a simplicial commutative complete bornological ring. We work more generally with simplicial commutative algebra objects in certain quasi-abelian categories. In this talk, I will briefly introduce these ideas and motivate why this categorical approach is so enlightening.
Fulton-MacPherson configuration spaces and logarithmic geometry, Siao Chi Mok (University of Cambridge)
In 1994, Fulton and MacPherson constructed a compactification of the configuration space of points on a projective manifold. The compactification has excellent properties: it is smooth and projective with normal crossings boundary. As part of their foundational study, Fulton and MacPherson gave an explicit presentation of the cohomology ring of their spaces. In this talk, I will recall the basics of the Fulton-MacPherson spaces, and then explain the basics of a generalization of the construction to pairs. Precisely, given a variety X and a normal crossings divisor D, we explain how one can compactify the configuration space of n distinct points on X that lie away from D.
Machine learning the dimension of a Fano variety, Sara Veneziale (Imperial College London)
The use of machine learning and data analysis techniques in pure mathematics for conjecture formulation has been a growing area of research, with examples in knot theory, representation theory, and string theory. In this talk, we go through a successful example of such an application in algebraic geometry, which positions itself in the context of mirror symmetry for Fano varieties. We study the quantum period, a conjectured invariant of Fanos, for toric Fano varieties and build models that can predict the dimension a variety from it. This inspires and justifies the construction of rigorous asymptotics that prove, in this context, the relationship between quantum period and dimension.
(Chekhov-Eynard-Orantin) Topological Recursion and Enumerative Geometry, Karoline van Gemst (University of Sheffield)
The Topological Recursion formalism is a recursive procedure first introduced in the context of random matrix models by Bertrand Eynard and Nicolas Orantin. The input for the recursion is a so-called spectral curve and the corresponding output is a sequence of multi-differentials. The procedure has been shown to be a universal formalism and is able to recover many results in enumerative geometry such as Mirzakhani’s recursion formula for hyperbolic volumes. In this short talk, I will introduce Topological Recursion and highlight the connection with geometry, enumerative invariants and mirror symmetry.
Number Theory Short talks
q-expansion of Hilbert modular forms, Siqi Yang (Imperial College London)
Hilbert modular forms are generalizations of modular forms from the field of rational numbers to totally real number fields. A Hilbert modular form admits q-expansions at the cusps of the Hilbert modular variety. Based on the Jacquet-Langlands correspondence, Dembélé, Donnelly, Greenberg, and Voight introduced algorithms to compute the Hecke action on Hilbert cusp forms of weight greater than or equal to two by relating them to the quaternionic automorphic forms. In this talk, we will discuss a way to compute the q-expansions of Hilbert modular forms by the duality theorem between Hecke algebras and spaces of cusp forms.
Asymptotics for Integral Points of Bounded Height on a log Fano Variety, Anouk Greven (Universität Göttingen)
While Manin's conjecture predicts the distribution of k-rational points on Fano varieties, such a conjecture does not exist for integral points. By considering integral points on specific examples of Fano varieties, one can build a foundation for a potential conjecture. For my Master’s thesis, I studied a paper by Florian Wilsch in which he determines an asymptotic expression for the number of integral points of bounded height on two open subvarieties of a log Fano threefold. He parametrizes the integral points using the universal torsor method. This method introduces a second variety, the torsor, on which the integral points are lattice points satisfying certain (coprimality) conditions, which simplifies the counting problem. By applying Möbius inversion and changing sums into integrals, he then obtains asymptotic expressions. The goal of my thesis was to see if this method can be extended in a straightforward way to a slightly different log Fano threefold. I showed that it does not, and determined an upper bound for the number of integral points of bounded height on a subvariety of this threefold. In this talk, I focus on the second log Fano threefold and show why the method applied by Florian Wilsch does not extend in a straightforward way.
Quartic points on the Fermat quartic (lying over a quadratic extension of Q(sqrt2)), Maleeha Khawaja (University of Sheffield)
In 1967, Mordell determined all quadratic points on the Fermat quartic using a fairly elementary argument. The proof rests on the fact that the elliptic curves with Cremona label 32a1 and 64a1 have rank 0 over Q. We will see how we can extend this proof to determine all quartic points on the Fermat quartic that lie in a quadratic extension of Q(sqrt 2).
Generalised Jacobians of modular curves and their Q-rational torsion, Mar Curco Iranzo (Utrecht University)
The Jacobian J0(N) of the modular curve X0(N) has received much attention within arithmetic geometry for its relation with cusp forms and elliptic curves. In particular, the group of Q-rational points on X0(N) controls the cyclic N-isogenies of elliptic curves. A conjecture of Ogg predicted that, for N prime, the torsion of this group comes all from the cusps. The statement was proved by Mazur and later generalised to arbitrary level N into what we call generalised Ogg’s conjecture. Consider now the generalised Jacobian J0(N)m with respect to a modulus m. This algebraic group also seems to be related to the arithmetic of X0(N) through the theory of modular forms. In the talk we will present new results that compute the Q-rational torsion of J0(N) for N an odd integer with respect to a cuspidal modulus m. These generalise previous results of Yamazaki, Yang and Wei. In doing so, we will also discuss how our results relate to generalised Ogg’s conjecture.
Additive structure of non-monogenic simplest cubic fields, Magdaléna Tinková (Czech Technical University in Prague)
The simplest cubic fields were introduced by Daniel Shanks in 1974. In this talk, we will focus on non-monogenic subfamilies of these fields. In particular, we will restrict to subfamilies with the integral basis of some concrete forms and discuss which simplest cubic fields (if any) belong to these subfamilies. Then, we will show several results on indecomposable integers, Pythagoras number, or universal quadratic forms over these fields. This is joint work with Daniel Gil-Muñoz.
Distribution of ranks of elliptic curves, Isabel Rendell (University College London)
The aim of this talk is to explain a preprint of Alexander Smith ([1603.08479] The congruent numbers have positive natural density (arxiv.org)). His paper studies quadratic twists of the congruent number curve and we will see how his work has applications both in calculating the rank of these elliptic curves and the congruent number problem. In particular, his paper shows how functions used by Tian, Yuan, and Zhang (1411.4728.pdf (arxiv.org)) can be expressed as determinants of matrices which are constructed using Legendre symbols.
Generalised superalgebras and an application to the lattice isometry problem, Jenny Roberts (University of Bristol)
We construct a superalgebra structure over n-dimensional matrices with elements in some arbitrary field K. This allows us to decompose any matrix in this space with respect to the superalgebra. In the special case taking K to be the field of rational numbers, we can associate the isometry between two lattices with the existence of an integer solution to a quadratic form arising from the superalgebra decomposition.
On the cohomology of congruence subgroups, Anja Meyer (University of Manchester)
The cohomology of matrix groups with entries in fields hs been long studied. However, SL_2(Z/p^nZ) has entries in finite rings, and the methods used previously fail. In this talk I will present a method for finding their mod p cohomology via their Sylow-p-subgroups and present p=3 as worked example.
An explicit Chebotarev Density Theorem on average and applications, Ilaria Viglino (ETH Zurich)
The study of specific families of polynomials and their splitting fields can provide useful examples and evidences for conjectures regarding invariants related to number field extensions and related objects. The main example that can illustrate the relevance of the work, is the family P0 n,N of degree n monic polynomials f with integer coefficients, so that the maximum of the absolute values of the coefficients is less or equal than N, and the splitting field Kf over the rationals Q is the full symmetric group Sn. We let N tend to infinity. With a little work, this can actually be generalized to polynomials with integral coefficients in a fixed number field of degree d over Q. Let π_f,r(x) be the function counting the primes less or equal than x such that f belongs to P0n,N has a fixed square-free splitting type r modulo p. It turns out that the quantity (pi_f,r(x)−delta(r)pi(x)((delta(r)−delta(r)2)pi(x))^(−1/2) is distributed like a normal distribution with mean 0 and variance 1, whenever x is small compared to N, e.g. x = N1/ log logN. Here delta(r) is the coefficient in the asymptotic predicted by the classical Chebotarev theorem. This result leads to interesting applications, as finding upper bounds for the torsion part of the class number in terms of the absolute discriminant, as it was done for other infinite families by Ellenberg, Venkatesh and Heath-Brown, Pierce.
Rational Points on Modular Jacobians , Elvira Lupoian (University of Warwick)
For any prime p which is at least 5, Mazur proved that the rational torsion subgroup of the modular jacobian J_0(p), the Jacobian of the modular curve X_0(p), is a cyclic subgroup of order the numerator of p-1/12, and it is generated by the linear equivalence class of the two cusps 0 and infinity on X_0(p). The Manin and Drinfeld theorem tells us that for any modular curve, the difference of two cusps is a torsion point on the Jacobian, and thus the subgroup of the Jacobian generated by difference of cusps which are fixed by the Galois group is a subgroup of the rational torsion subgroup of its Jacobian. The natural question, following from Mazur's theorem, is whether these groups are equal. In this talk, I will begin with a short introduction to modular curves, their moduli interpretation and their Jacobians. I will then give an overview of variations of the above question, and some interesting results.
Ranks of quadratic twists of elliptic curves with a rational point of order 3 , Mike Vazan (Hebrew University of Jerusalem)
Bhargava and Ho recently showed that the average size of the 2-Selmer group of elliptic curves with a rational point of order three is at most 3. I aim to show that the same upper bound holds for a slightly more general family of elliptic curves, namely elliptic curves admitting a fixed Gal(Qbar/Q)-module of order three as a subgroup. This should also have future applications to the question of bounding the average rank in the family of elliptic curves with a rational subgroup of order 3. In this talk I will present the main ideas behind the proof and describe some of my recent progress.
The role of primes of good reduction in the Brauer-Manin obstruction, Margherita Pagano (Universiteit Leiden)
A way to study rational points on a variety is by looking at their image in the p-adic points. The Brauer-Manin obstruction can be used to study the density of rational points inside the product of the p-adic points. In this talk I will discuss the role that primes of good reduction on a variety might play in the Brauer-Manin obstruction to weak approximation.
