## Postgraduate Forum

### 2018-19

### Abstracts

Evolution algebras were introduced in 2006 in order to study non-Mendelian genetics. A pioneer in this research was Tian who in 2008 showed connections between the evolution algebras and several different fields.

We will see the genetical motivation of these algebras, as well as defining them algebraically. During the talk, we will see also the connection of the evolution algebras and the graph theory, and we will establish some basic definitions that will allow us to work out if an evolution algebra is decomposable or not. Time permitting, we will speak about the classification for these algebras when their dimension is 2 and 3.

Deformation theory can be thought of as the study of infinitesimal thickenings of geometric or algebraic objects. Equivalently, it is the study of infinitesimal neighbourhoods in moduli spaces. In this talk, I'll provide an introduction to some deformation-theoretic concepts. I'll explain Deligne's philosophy that in characteristic zero, deformation problems are "controlled" by differential graded Lie algebras (dglas). I'll talk about derived deformation theory, and how it clarifies some concepts within the classical theory, in particular the proofs of Lurie and Pridham that dglas are equivalent to derived moduli problems via Koszul duality. Time permitting, I'll talk about an application of noncommutative deformation theory to birational geometry in the form of the Donovan-Wemyss contraction algebra, and I will mention how derived noncommutative deformations also fit into the picture.

A tiling is a decomposition of Euclidean space into pieces that fit together without gaps or overlaps. The most basic examples consist of periodic structures such as a tiling by unit cubes, which can be used to model the structure of a crystal of table salt. More interesting examples can be obtained by requiring that our tiles are unable to tessellate the space in a periodic manner. Physical realizations of these aperiodic tilings are called quasicrystals and were first observed in a man-made material in 1984, with the first naturally occurring specimen discovered in 2009. In this talk I will provide an introduction to tiling theory and explain some ways that we can associate dynamical systems to a tiling.

We will cover the basics of finite and infinite matroids and give examples.

On an infinite, connected, locally finite graph, we can define "internal diffusion-limited aggregation" (IDLA), in which a large number of explorers starting from the same point perform random walks until they all find a free spot to settle. For the integer lattice, the scaling limit of the explored region is known to be a Euclidean ball, and we will see some variations of the model for which this is either known and conjectured to hold.

We will also look at some of the more interesting properties enjoyed by IDLA and other settlement processes, and the link between IDLA and chip-firing games.

For every prime \(p\) the \(p\)-completion functor of Bousfield and Kan relates a space to its nilpotent completion. A space is \(p\)-good if the induced map is a homology isomorphism. We give examples of spaces which are good and bad at different primes in the sense of Bousfield and Kan in any arbitrary combination. Moreover we investigate which impact the existence of a Sylow \(p\)-subgroup has on the homotopy type on the classifying space and under which conditions the homotopy type of wedges of classifying spaces is good or bad for a solid ring \(R\). We give results relating to various other \(R\)-homological structures and a collection of examples.

The classical definition of an amenable group relies on the notion of an invariant mean, which turns out to be equivalent to the Følner condition. This condition depends on a Cayley graph of the group and so we can talk about amenable graphs. In the talk I will discuss graphs with an additional notion of weight function \(w\) on their vertices and give motivation for considering such structures.

Random growth occurs in many real world settings, for example, we see it exhibited in the growth of tumours, bacterial growth and lightning patterns. As such we would like to be able to model such processes to determine their behaviour in their scaling limits. Well studied models to describe these different processes include the Eden model and DLA. In a 1998 paper Hastings and Levitov introduced a one parameter family of conformal maps \(\mathrm{HL}(\alpha)\) which can be used to model all such Laplacian growth processes and allows us to vary between the previous models by varying alpha. We will consider a regularised version of this model and show that at certain values of \(\alpha\) a phase transition on the scaling limits occurs.

The Random Walk in a Random Environment (RWRE) is a generalization of the well-known random walk. In the RWRE, the transition probabilities itself are random. A natural application of this is the propagation of heat, or the diffusion of matter through an irregular environment. As we all know, the independent increments of the Simple Random Walk (SRW) make it an easy class of processes to study. However, since the walker of an RWRE gets more information about its environment as time progresses, this process is no longer Markovian. This makes it a lot more challenging to obtain results on the limiting behaviour of the walk, and indeed many basic questions in dimension \(d\geq 2\) remain open. I will first give an introduction of the field of RWRE, and will introduce trapping effects. Informally, this is the occurrence of pockets in the random environment where the walk spends a long time with relatively high probability.

These trapping effects will be further explored in a specific model of a RWRE, namely of a biased random walk on the 2-dimensional lattice after performing supercritical percolation. In this model, a walker with a small bias in the positive \(x\)-direction will have a positive velocity. However, as the bias increases, the velocity vanishes. This is due to the walker spending a superlinear amount of time in dead ends in the cluster. These results were obtained by Berger, Gantert and Peres in 2003 and by Snitzman in 2003. I will give an overview of the main ideas in their proofs.

Constructing a profinite group through inverse limits of finite groups, we can characterise a topology of subgroups of this topological group.

Using this we establish notions of \(G\)-spaces and what it means to be essentially finite or almost finite. In particular, we consider Hall's \(p\)-construction and its ideals and the infinite pro-\(p\) partition group.

Evolution algebras were introduced in 2006 in order to study non-Mendelian genetics. A pioneer in this research was Tian who in 2008 showed connections between the evolution algebras and several different fields.

We will see the genetical motivation of these algebras, as well as defining them algebraically. During the talk, we will see also the connection of the evolution algebras and the graph theory, and we will establish some basic definitions that will allow us to work out if an evolution algebra is decomposable or not. Time permitting, we will speak about the classification for these algebras when their dimension is 2 and 3.

Deformation theory can be thought of as the study of infinitesimal thickenings of geometric or algebraic objects. Equivalently, it is the study of infinitesimal neighbourhoods in moduli spaces. In this talk, I'll provide an introduction to some deformation-theoretic concepts. I'll explain Deligne's philosophy that in characteristic zero, deformation problems are "controlled" by differential graded Lie algebras (dglas). I'll talk about derived deformation theory, and how it clarifies some concepts within the classical theory, in particular the proofs of Lurie and Pridham that dglas are equivalent to derived moduli problems via Koszul duality. Time permitting, I'll talk about an application of noncommutative deformation theory to birational geometry in the form of the Donovan-Wemyss contraction algebra, and I will mention how derived noncommutative deformations also fit into the picture.

A tiling is a decomposition of Euclidean space into pieces that fit together without gaps or overlaps. The most basic examples consist of periodic structures such as a tiling by unit cubes, which can be used to model the structure of a crystal of table salt. More interesting examples can be obtained by requiring that our tiles are unable to tessellate the space in a periodic manner. Physical realizations of these aperiodic tilings are called quasicrystals and were first observed in a man-made material in 1984, with the first naturally occurring specimen discovered in 2009. In this talk I will provide an introduction to tiling theory and explain some ways that we can associate dynamical systems to a tiling.

We will cover the basics of finite and infinite matroids and give examples.

On an infinite, connected, locally finite graph, we can define "internal diffusion-limited aggregation" (IDLA), in which a large number of explorers starting from the same point perform random walks until they all find a free spot to settle. For the integer lattice, the scaling limit of the explored region is known to be a Euclidean ball, and we will see some variations of the model for which this is either known and conjectured to hold.

We will also look at some of the more interesting properties enjoyed by IDLA and other settlement processes, and the link between IDLA and chip-firing games.

For every prime \(p\) the \(p\)-completion functor of Bousfield and Kan relates a space to its nilpotent completion. A space is \(p\)-good if the induced map is a homology isomorphism. We give examples of spaces which are good and bad at different primes in the sense of Bousfield and Kan in any arbitrary combination. Moreover we investigate which impact the existence of a Sylow \(p\)-subgroup has on the homotopy type on the classifying space and under which conditions the homotopy type of wedges of classifying spaces is good or bad for a solid ring \(R\). We give results relating to various other \(R\)-homological structures and a collection of examples.

The classical definition of an amenable group relies on the notion of an invariant mean, which turns out to be equivalent to the Følner condition. This condition depends on a Cayley graph of the group and so we can talk about amenable graphs. In the talk I will discuss graphs with an additional notion of weight function \(w\) on their vertices and give motivation for considering such structures.

Random growth occurs in many real world settings, for example, we see it exhibited in the growth of tumours, bacterial growth and lightning patterns. As such we would like to be able to model such processes to determine their behaviour in their scaling limits. Well studied models to describe these different processes include the Eden model and DLA. In a 1998 paper Hastings and Levitov introduced a one parameter family of conformal maps \(\mathrm{HL}(\alpha)\) which can be used to model all such Laplacian growth processes and allows us to vary between the previous models by varying alpha. We will consider a regularised version of this model and show that at certain values of \(\alpha\) a phase transition on the scaling limits occurs.

The Random Walk in a Random Environment (RWRE) is a generalization of the well-known random walk. In the RWRE, the transition probabilities itself are random. A natural application of this is the propagation of heat, or the diffusion of matter through an irregular environment. As we all know, the independent increments of the Simple Random Walk (SRW) make it an easy class of processes to study. However, since the walker of an RWRE gets more information about its environment as time progresses, this process is no longer Markovian. This makes it a lot more challenging to obtain results on the limiting behaviour of the walk, and indeed many basic questions in dimension \(d\geq 2\) remain open. I will first give an introduction of the field of RWRE, and will introduce trapping effects. Informally, this is the occurrence of pockets in the random environment where the walk spends a long time with relatively high probability.

These trapping effects will be further explored in a specific model of a RWRE, namely of a biased random walk on the 2-dimensional lattice after performing supercritical percolation. In this model, a walker with a small bias in the positive \(x\)-direction will have a positive velocity. However, as the bias increases, the velocity vanishes. This is due to the walker spending a superlinear amount of time in dead ends in the cluster. These results were obtained by Berger, Gantert and Peres in 2003 and by Snitzman in 2003. I will give an overview of the main ideas in their proofs.

Constructing a profinite group through inverse limits of finite groups, we can characterise a topology of subgroups of this topological group.

Using this we establish notions of \(G\)-spaces and what it means to be essentially finite or almost finite. In particular, we consider Hall's \(p\)-construction and its ideals and the infinite pro-\(p\) partition group.