These seminars (unless otherwise stated) will take place on Tuesdays at 1pm via Zoom (https://ucl.zoom.us/j/97636994687) on an almost weekly basis.

**Zoom link: https://ucl.zoom.us/j/97636994687**

## 18 Jan 2022

**Speaker: Alp Muyesser**

Supervisor: Dr A Pokrovskiy

##### Title: Partitioning groups into zero-sum sets

**Abstract:**

Let G be a finite abelian group. Can we partition G into sets of size 3 such that each set sums to the identity? Although simple to state, this turns out to be hard problem with interesting applications in finite geometry and experiment design. We will talk about a combinatorial attack on problems of this type that has recently lead to the resolution of several longstanding conjectures. This is joint work with my advisor, Alexey Pokrovskiy.

## 25 Jan 2022

**Speaker: Sam Harris**

Supervisor: Prof NR McDonald

##### Title: The Formation of Wildfire Fingers

**Abstract:**

Wildfires are among the deadliest natural disasters the planet faces, and as the effects of climate change worsen, they are becoming increasingly more common across the globe. In the UK, recent, notable examples include: the 2018 Saddleworth Moor Fire (England) and the 2019 Moray wildfire (Scotland). Predicting how wildfires will spread is vital in protecting homes, preserving wildlife and saving lives. In this talk, three key mechanisms of wildfire spread - constant rate of spread, curvature and oxygen effects - are introduced and a resulting two-dimensional model is constructed. The oxygen effect, which is not seen explicitly in previous wildfire models, is dynamical and requires the solution of the steady advection-diffusion equation. This adds a destabilising effect to the fire line – the free boundary separating burnt and unburnt regions – causing the development of so-called “fire fingers”, analogous to the problem of viscous fingering in a Hele-Shaw cell. To finish, numerical solutions are computed using a conformal mapping method.

## 08 Feb 2022

**Speaker: Marta Benozzo (LSGNT)**

Supervisor: Prof P Cascini (Imperial College London)

##### Title: Iitaka conjecture

**Abstract:**

Classifying varieties is one of the main problems in geometry. In dimension 1, it is completely solved: curves are classified by their genus. What is the invariant we need to use in higher dimension? The genus of a curve is strictly related to the sections of its canonical bundle (dual of the tangent). Thus, the idea is to look at sections of the canonical bundle also in higher dimension. Using them, it is possible to define a new invariant: the Kodaira dimension. This invariant allows us to identify three main “building blocks” of varieties. What we would like to do next is splitting each variety into its building blocks. This is done by means of fibrations and this is where the Iitaka conjecture comes into play. It predicts a relation between the Kodaira dimension of the source, the fibres and the base of every fibration.

## 15 Feb 2022

**Speaker: Ignacia Fierro Piccardo**

Supervisor: Prof T Betcke

##### Title: An OSRC Preconditioner for the

**Abstract:**

The Electric Field Integral Equation (EFIE) is commonly used to solve high-frequency electromagnetic scattering problems. However, the EFIE being a First Kind Fredholm operator, needs a regulariser in order to use iterative solvers. A regulariser alternative is the exact Magnetic-to-Electric (MtE) operator, which has the disadvantage of being as expensive as solving the EFIE. However, Bouajaji et al. have developed a local surface approximation of the MtE for time-harmonic Maxwell’s equations that can be efficiently evaluated through the solution of sparse linear systems. In this research we demonstrate the preconditioning properties of the approximate MtE operator for the EFIE using a Bempp implementation and show a number of numerical comparisons against other preconditioning techniques like the Calderón Preconditioner.

## 22 Feb 2022

**NO SEMINAR**

## 01 Mar 2022

**Speaker: Luke Debono**

Supervisor: Prof HJ Wilson

##### THIS SEMINAR HAS BEEN CANCELLED.

## 08 Mar 2022

**Speaker: Pascale Voegtli (LSGNT)**

Supervisor: Prof P Cascini (Imperial College London)

##### Title:A Gentle Introduction To K-Stability

**Abstract:**

Initially introduced as a criterion to characterize the existence of a special type of metrics on Fano varieties in differential geometry, K-stability has recently transpired to be a notion that can be reformulated in purely algebraic terms using well established tools from higher dimensional algebraic geometry. In the talk we will introduce some of them and try to hint at how they can beneficently be applied in the construction of a moduli space for Fano varieties, the so called K-moduli space.

## 15 Mar 2022

**Speaker: Hannah Tillmann-Morris (LSGNT)**

Supervisor: Prof T Coates (Imperial College London)

##### Title: Can a computer detect blowups of Fano manifolds?

**Abstract:**

Mirror symmetry gives us a correspondence between Fano manifolds and certain types of Laurent polynomials. This could potentially be used to reduce the classification of Fano manifolds to combinatorial problem. But many aspects of this conjectured correspondence are still unknown. For example, given two Fano manifolds and two corresponding Laurent polynomials, can the information of whether one Fano is a blowup of the other be extracted just from the data of the Laurent polynomials? I will explain how this Fano/Laurent polynomial correspondence arises and how Laurent polynomials can reveal the blowup relationship in some simple examples.

## 22 Mar 2022

**Speaker: Alessio Di Lorenzo**

Supervisor: Dr L Foscolo

##### Title: Asymptotically conical Calabi-Yau conifolds

**Abstract:**

In the 50’s Calabi asked whether we could find a metric with some a priori given Ricci curvature in the Kahler class of a Kahler metric, on a compact complex manifold. The answer was affirmative and has been proven to be so in the 70’s by Yau. In the particular case in which the real first Chern class of the manifold is zero, this amounts to say that the manifold admits a Ricci-flat metric in any class, and these manifolds are called Calabi-Yau. Several approaches have been tried and have been successful in extending this result to non-compact setting. The non-compact setting can be thought as twofold: either the manifold is complete (e.g. C^n), or it is not (e.g. singular projective varieties). We will give an idea of what is needed in the case the manifold has singularities and it is moreover similar to a cone at infinity.