Here are titles and abstracts of the talks that took place during the conference.

Keynote talks

Low dimensional interactions, Sarah Dean Rasmussen (University of Cambridge)

Low dimensional topology mostly studies objects that occur in 4 and fewer dimensions. Since a lot can happen in 4 and fewer dimensions, this provides many opportunities to explore interactions between tools and structures from different areas of mathematics, such as between algebraic geometry and knots, between geometric group theory and real dynamics, and many other examples. I’ll try to discuss how I became interested in some of these questions and where different explorations have taken me.

Institutional support for women in STEM: don’t miss out!, Ana Lecuona (University of Glasgow)

A career in academia is a dream come true for many of us. While it is difficult for everyone, there is overwhelming evidence that women and other minorities have it more difficult within STEM subjects at least. There are several associations, grants and networks trying to address this issue. In this short talk, I will try to signpost some which might be able to support your own careers. We will focus mainly in the work of the LMS and opportunities in the UK.

Elliptic curves and the Birch and Swinnerton-Dyer conjecture, Céline Maistret (Univeristy of Bristol)

Number theory is the branch of mathematics concerned with studying numbers and solving equations. This talk will address the latter by introducing a particular set of equations which define objects called elliptic curves. Solving these equations has proven extremely difficult due to their complex mathematical structure. The quest for their solutions started over a century ago and reached a milestone in the 1960’s when Birch and Swinnerton-Dyer proposed a formula to find all their solutions. In this talk, I will present the Birch and Swinnerton-Dyer conjecture and explain how it allows to find all solutions. Following the talk, I’ll discuss my career path and experience as a woman in Mathematics.

Enumerative geometry, Cristina Manolache (University of Sheffield)

Enumerative geometry is concerned with classical questions such as: "How many lines pass through two points?", "How many ellipses pass through five points in a plane?" or more modern questions such as: "How many lines are there on a quintic threefold?". In this talk I will give a short overview of the mathematical challenges in this field and a few personal challenges along my career.

The Accidental Number Theorist, Vandita Patel (Univeristy of Manchester)

In this talk, I will describe my career path into academia. Along the way, I will discuss key results obtained in the realm of Diophantine equations. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, explicit bounds for linear forms in logarithms, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves.

Short talks

The homological minimal model program, Charlotte Bartram (University of Glasgow)

The minimal model program is a procedure that takes a variety and simplifies it through a series of birational transformations. In the 1990's Bondal and Orlov proposed a homological interpretation of the minimal model program, wherein the birational transformations are carried out at the level of derived categories. In this talk I aim to give a brief overview of this theory and its development.

Formalizing the ring of adèles in Lean, Marı́a Inés de Frutos-Fernández (Imperial College London)

We will present a formalization of the ring of adèles and group of idèles of a global field in Lean 3, an interactive theorem prover with an ever-growing mathematics library. We will give a quick introduction to Lean and explain how these definitions were formalized, with a focus on the kind of decisions one has to make during the formalization process. We will also discuss some applications, including the statement of the main theorem of global class field theory and a proof that the ideal class group of a number field is isomorphic to an explicit quotient of its idèle class group.

Identifying topological signals and matching cycles in image persistent homology, Inés García-Redondo (LSGNT)

Persistent homology is a technique that applies classical homology to data settings and has been implemented with great success over the past 20 years in a wide range of applications. It provides a topological summary of the data by conveying a dynamical understanding of homology by studying the progression of homology groups over a filtration, i.e., a nested sequence of simplicial complexes. From this process, we are able to determine that some topological features of data are more prominent and “have higher persistence” which is a main indicator of signal in topological data analysis, however, it misses some important subtleties. In this talk, I will present ongoing work that adapts notions of topological “prevalence” and signal matching via cycles as proposed by Reani and Bobrowski (2021) to the more computationally efficient cohomological setting underlying Ripser (2021), which is the state-of-the-art for persistent homology computations. This work takes place in the setting of image persistent homology, developed by Cohen-Steiner et al. (2009). I will also present results of numerical experiments and applications to real-world data. No prior knowledge of computational topology or persistent homology will be assumed. This is joint work with Anna Song and Anthea Monod (Imperial College) and funded by the EPSRC “London School of Geometry and Number Theory” Centre for Doctoral Training.

Mirror symmetry as an isomorphism of Frobenius manifolds, Karoline van Gemst (University of Sheffield)

In this short talk I will introduce Frobenius manifolds and discuss what they have to do with mirror symmetry. I will focus particularly on the B-side of mirror symmetry which in this context builds a Frobenius manifold from a function called a Landau-Ginzburg superpotential. I will say a little bit about the joint work with Andrea Brini (Arxiv:2103.12673) where we found B-models of a special class of Frobenius manifolds related to the quantum cohomology of certain orbifolds, and, if time permits, briefly mention a topological application of these results.

Being a mathematician in industry: a personal perspective, Laura Guthrie (FITZ Partners)

In this relatively ‘untechnical’ talk, I will discuss my own personal experience within both academia and ‘the industry’. After completing my university undergraduate degree, I tried many different careers in non-academic fields but returned to university to begin a PhD in Geometry and Number Theory at the LSGNT. However, mathematical research was not the path for me and so I chose to leave (before completion). I hope to be able to share the lessons I have learned whilst transitioning between what can often feel like vastly different worlds.

Geometric Dirac operator on the Fuzzy Sphere, Evelyn Lira-Torres (Queen Mary University of London)

In this talk we will briefly introduce talk about how we construct a Connes spectral triple or `Dirac operator' on our chosen algebra the non-reduced fuzzy sphere \(\mathbb{C}_\lambda[S^2]\) as realised using quantum Riemannian geometry with a central quantum metric \(g\) of Euclidean signature and its associated quantum Levi-Civita connection. It was found that the Dirac operator is characterised uniquely up to unitary equivalence within our quantum Riemannian geometric setting and an assumption that the spinor bundle is trivial and rank 2 with a central basis.

The Modular Approach to Diophantine Equations over totally real fields, Diana Mocanu (University of Warwick)

In this talk I will give a brief overview of the modular approach for solving Diophantine equations over the rationals pioneered by Wiles, Ribet, and Mazur in solving Fermat’s Last Theorem. I will show how it generalizes over totally real fields to give asymptotic results. If time permits, I will present a few examples of recent results involving this method.

Formalization of \(p\)-adic \(L\)-functions in Lean, Ashvni Narayanan (LSGNT, Imperial College London)

The \(p\)-adic \(L\)-functions connect algebraic and analytic number theory. In this talk, we shall see how the \(p\)-adic \(L\)-functions associated to Dirichlet characters have been defined (non-canonically) in the automated theorem prover Lean in terms of \(p\)-adic measures, and that this is equivalent to the standard definition. If time permits, we shall also look into making a “functor” out of the definition.

The convergence of integer continued fractions, Margaret Stanier (The Open University)

Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive coefficients converges. This is not the case, however, if the coefficients are not necessarily positive. We use a geometric representation of an integer continued fraction to find a simple test which determines whether such a fraction converges or diverges. This is joint work with Ian Short.

The Defect is bad, deal with it!, Vaidehee Thatte (King's College London)

The 'defect' (or ramification deficiency) is one of the main obstacles to nice results, such as obtaining resolution of singularities in positive residue characteristic. In this talk, I will discuss why this is a serious obstruction, and how generalized ramification theory helps us understand and treat it. We will take this journey through some examples rather than theory, in keeping with the spirit of the retreat.

Dehn surgery on knots in \(S^3\), Laura Wakelin (LSGNT, Imperial College London)

Dehn surgery is the process of drilling out a tubular neighbourhood of a knot \(K\) in a 3-manifold \(M\) and then filling it back in to obtain a new 3-manifold \(M_K(p/q)\), where the “slope” \(p/q\) describes the choice of gluing map used in the construction. We will consider the case of knots in \(S^3\) and look at the geometry and topology of \(S^3_K(p/q)\) for different choices of \(K\) and \(p/q\).

Derived categories and semiorthogonal decompositions, Fei Xie (University of Edinburgh)

I will introduce bounded derived categories of coherent sheaves on algebraic schemes and study their semiorthogonal decompositions. I will give examples to discuss the relation between derived categories and rationality, and explain how to measure singularities via derived categories.

Congruences between Modular forms of integer and half-integer weight, Sadiah Zahoor (University of Sheffield)

The theory of half-integral weight modular forms may be adopted to prove 'congruences' between Selmer type groups. William McGraw and Ken Ono used this approach by passing from 'congruences' between modular forms of integer weight to congruences between modular forms of half-integer weight. For example, recall the famous 'congruence modulo 11' between the normalised Discriminant function ‘Delta’ of weight 12 and the newform f of weight 2 attached to Elliptic curve of conductor 11. Using Shimura's correspondence and Kohnen’s isomorphism, which connects modular forms of weight 2k for a positive integer k with half-integer modular forms of weight k+(1/2), our congruence descends to a congruence modulo 11 between half integer modular forms of weight 3/2 and 13/2. The talk shall begin with a brief introduction to modular forms of integer and half-integer weight leading to statement and overview of the main result I have been working on. I will also give an overview of current progress and generalisation of Theorem of McGraw and Ono to Hilbert modular forms of integer and half-integer weight.