The period matrix of a smooth complex projective variety encodes the isomorphism between its singular homology and its algebraic De Rham cohomology. Numerical approximations with high precision of the entries of the period matrix allow to recover some algebraic invariants of the variety, such as the Néron-Severi group in the case of surfaces. In this talk, we will see a method relying on the computation of an effective description of the homology for obtaining such numerical approximations of periods of algebraic varieties, and showcase implementations and applications, in particular to computation of the Picard rank of certain K3 surfaces related to Feynman diagrams.

Let (S,D) be a pair of del Pezzo surface and smooth anticanonical divisor, eg the projective plane and a smooth cubic, and denote by X its mirror. Traditionally (eg in work by N. Takahashi), X is constructed a bit ad hoc and one proves mirror symmetry for (S,D) <-> X by computing two mirror structures on each side, and matching the results of each computation.

In joint work with Ruddat and Siebert, we show that enumerative mirror symmetry - the matching of enumerative structures of (S,D) with period integrals of X - is a consequence of the intrinsic mirror construction of Gross-Siebert. Thus the enumerative mirror theorem becomes a consequence of the intrinsic mirror construction, which itself is based on geometric SYZ duality, a generalisation of lattice duality. All the while, no calculations are needed.

My goal for this talk is to convince you that enumerative mirror symmetry for (S,D) <-> X is a consequence of lattice duality. Time permitting, I will explain consequences thereof.

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The classical Darboux system governing rotation coefficients of three-dimensional metrics of diagonal curvature possesses an equivalent formulation as a sixth-order PDE for a scalar potential (related to the corresponding tau-function). We demonstrate that this PDE is Lagrangian and can be viewed as an explicit scalar form of the 'generating PDE of the KP hierarchy' as discussed recently by Nijhoff in the Lagrangian multiform approach to the Darboux and KP hierarchies. Scalar Lagrangian formulations for differential-difference and fully discrete versions of the Darboux system are also constructed. In the first three cases (continuous and differential-difference with one and two discrete variables), the corresponding Lagrangians are expressible via elementary functions (logarithms), whereas the fully discrete case requires special functions (dilogarithms).

Remarkably, dispersionless limits of the above Lagrangians provide a complete list of 3D second-order integrable Lagrangian densities of the form f(u_{xy}, u_{xt}, u_{yt}).

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Saturated fusion systems (s.f.s) are structures designed to model the local behaviour of finite groups and in many ways behave as if they were themselves finite groups. The analogue of characters of a s.f.s F are F-stable characters. Motivated by applications in topology, there has been a recent interest in studying F-stable characters in a manner similar to the classical character theory of groups. A major obstruction is that F-stable do not admit a unique decomposition into "indecomposable" constituents.

I will give a brief introduction to the theory of fusion systems and their characters before explaining methods we may employ to work around F-stable characters admitting non-unique factorisations. In particular, I will discuss progress made on a conjecture of Semeraro which, if true, allows us to easily find indecomposable bases for the lattice of F-stable characters and reduces the computational complexity required to verify uniqueness of decomposition.

There is an old expectation that birational geometry (in particular, the Minimal Model Program) can be translated into the language of derived categories of coherent sheaves. For example, the blow-up at a smooth centre corresponds to adding several copies of the derived category of the centre to the derived category of the base. However, there is an obstacle for applying derived categories to birational problems: semi-orthogonal decompositions of these categories are essentially non-unique.

I will talk about a possible way to overcome this obstacle by constructing semi-orthogonal decompositions that we call atomic and that have two properties: they are (i) unique up to mutations and (ii) compatible with birational morphisms. Such atomic theory is established only for surfaces; but we can work over arbitrary perfect fields or in the G-equivariant setting. Time permitting, I will explain the role that atoms can play in classification of surfaces over non-closed fields.

This is a joint work with Evgeny Shinder and Julia Schneider. Our work is inspired by the atoms by Katzarkov-Kontsevich-Pantev-Yu, although our approach is different and makes no use of quantum cohomology.

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I will show that a special reduction of period-one closure of the Darboux chain for the Schrödinger operators with matrix potentials coincides with the Brun-Bogoyavlenskij equations of motion of the rigid body about centre of mass in the Newtonian field with a quadratic potential.

The corresponding matrix Schrödinger operators are maximally finite-gap (in some precise sense) with the spectrum explicitly described. The general 2-by-2-matrix case, containing some exotic matrix versions of the harmonic oscillator, will be discussed in more detail.

The talk is based on a recent joint work with V.E. Adler.

We begin with an overview of moduli spaces of stable maps to pairs and their role in enumerative geometry. Specialising to stable maps to toric pairs with prescribed tangency conditions to the toric boundary, we will cover the stratification of the moduli space by combinatorial types of associated tropical maps. Finally, we will use this stratification to study the class of the moduli space in the Grothendieck ring of varieties, describing a wall-and-chamber decomposition of the tangency space such that on open chambers, the class of the moduli space is constant.

Calabi-Yau varieties have taken a prime role in physics and string theory. Their topology and geometry, and that of their moduli spaces, determine to a large extent the physics of a compactification of a 10 dimensional theory on the Calabi-Yau variety. It turns out that the arithmetic structure of these varieties also plays an important role in physics. For example it has an impact in black hole physics and the structure of scattering amplitudes. We will also discuss a number of interesting topics related to the zeta function, the corresponding L-function, and the appearance of modularity for one parameter families of Calabi-Yau manifolds.

We will begin by motivating the study of the arithmetic structure of these varieties. We will explain how counting points over a finite field yields an expression explicitly in terms of the periods of the holomorphic three form of a Calabi-Yau threefold. We will also illustrate this using one parameter families of Calabi-Yau manifolds and show how our expressions also involve p-adic zeta values (work in progress with Eleanora Svanberg)

We will focus on an example for which the quartic numerator of the zeta function of a one parameter family factorises into two quadrics at special values of the parameter. These special values, for which the underlying manifold is smooth, satisfy an algebraic equation with coefficients in Q, so independent of any particular prime. The significance of these factorisations is that they are due to the existence of black hole attractor points in the sense of type II supergravity which predict the splitting of the Hodge structure over Q at these special values of the parameter. Modular groups and modular forms arise in relation to these attractor points, in a way that is familiar to mathematicians as a consequence of the Langland’s Program, but which is a surprise to a physicist. To our knowledge, the rank two attractor points that were found together with Mohamed Elmi and Duco van Straten by the application of number theoretic techniques, provide the first explicit examples of such attractor points for Calabi-Yau manifolds.