Part III Essay Collection

The theory of heights is an extremely important tool in the study of the arithmetic of elliptic curves. For $E/K$ an elliptic curve over a number field, the canonical height is a quadratic form $\hat{h}: E(K) \rightarrow \mathbb{R}$ with several desirable properties and far-reaching theoretical applications, playing a role in the celebrated Mordell-Weil Theorem, and constituting a term (the regulator) in the Birch-Swinnerton-Dyer conjecture, one of the most important open problems in mathematics. The utility of heights extends to the computational theory of elliptic curves, where we aim to algorithmically compute the arithmetic invariants of a given elliptic curve $E$. This can be used to aid theory by providing evidence for conjectures (for example, searching for elliptic curves with large rank), with databases such as LMFDB containing millions of elliptic curves over $\mathbb{Q}$ and small number fields.

We consider the properties of cardinals under various versions of the Axiom of Determinacy (AD), a possible set theoretic axiom that conflicts with the full Axiom of Choice. AD asserts the determinacy of certain infinite games which can be used to explore the complexity of subsets of a base set, typically taken to be the real numbers. Under AD some small (infinite) cardinals exhibit properties such as measurability and supercompactness which are normally associated only with very large cardinals. This essay explores some such results.

As first noticed by Achúcarro and Townsend, and subsequently developed by Witten, the topological nature of three-dimensional general relativity can be made explicit by rewriting the theory as a Chern-Simons theory. Chern-Simons theories are a family of topological gauge theories with an extensively studied quantization, so this formulation opens up a promising pathway towards the quantization of gravity. The objective of this essay is to explore the relationship between three-dimensional gravity and Chern-Simons theory, first at the classical level and then in the context of dS3 quantum gravity.

Quantum entanglement between different spacetime regions tells us how strongly correlated the regions are. However, the basic definition of entanglement does not work in gauge theories, as the underlying Hilbert space is not a tensor product space. The Standard Model is a gauge theory (and so are other important theories), so it would be nice to have a suitable redefinition. This essay is about redefining entanglement entropy to mitigate this problem (and what works and what does not).

We define the quantum cohomology ring $QH^*(X)$ of a symplectic manifold, which is a deformation of the ordinary cohomology ring $H^*(X)$ by "higher-order terms", or more concretely, using Gromov-Witten invariants.

The Seidel representation is a map $\pi_0(\widetilde G) \to QH^*(X)$, where $\widetilde G$ is a covering space of the free loop space on $Ham(X)$. To define this, we will also define Hamiltonian Floer (co)homology, and study $S^1$-actions on symplectic manifolds.

To conclude, we present two applications of the theory. The first is using the Seidel representation to find elements in $\pi_1(Ham(X))$ of infinite order. The second is to use $S^1$-actions and the Seidel representation to compute the quantum cohomology ring of toric manifolds.

QCD is the theory of quarks. Yet, because of confinement we don’t see individual quarks. Instead we see protons and neutrons. QCD also predicts other particles, known as pions. These are quark-anti-quark composites and are the lightest particles in the spectrum. They decay through the weak force, which is why we don’t see them every day, but if we restrict attention to just the strong force then it’s important to understand the physics of the pion and its relatives. This physics is largely constrained by symmetry. The pions should rightly be viewed as massless Goldstone bosons for spontaneously broken chiral symmetry (even though pions have a mass of 100 million electron volts!). This perspective allows us to write down an effective theory for the low-energy dynamics of pions, known as the chiral Lagrangian, from which we can recover the proton and neutron as a soliton, known as a Skyrmion.

We present a solution, due to Krstić, to a classical problem in universal algebra—finding necessary and sufficient conditions for a monoid to embed into a group—using the methods of geometric group theory. Following this, we extend the technique to prove a new topological characterization of monoids that embed into abelian groups, and Boolean groups.

We introduce the local Langlands correspondence for GL_n and define all of the objects involved in the statement of the theorem. Emphasis is placed on thorough exposition and careful explanation of results encountered.

Algebraic theories can be interpreted ‘over’ a topological space. For example, an interpretation of the theory of groups over a topological space $X$ assigns a group $G(U)$ to each open set $U$ of $X$, in a suitably compatible way: for instance, if $V \subseteq U$, then there is a canonical homomorphism $G(U) \to G(V)$. Such a collection of groups, called a sheaf, can be thought of as encapsulating different states of knowledge about a single object $G$, which we can analyse in terms of the sentences that it satisfies. Any given sentence in the language of groups is classically either true or false for each $G(U)$. Analogously to Heyting semantics, the truth value of a sentence $\varphi$ interpreted in $G$ will be an open set of $X$, namely, the union of the open sets $U$ for which $\varphi$ holds in $G(U)$. For example, the group axioms will take the maximal truth value $X$, so in this sense $G$ ‘is’ a group. We will show that this method of reasoning is logically sound, provided that we only use a constructive form of logic, and that the sentences in question are suitably geometric. In this essay, we will study the idea of interpreting theories over spaces in detail.

QCD is the theory of quarks. Yet, because of confinement we don’t see individual quarks. Instead we see protons and neutrons. QCD also predicts other particles, known as pions. These are quark-anti-quark composites and are the lightest particles in the spectrum. They decay through the weak force, which is why we don’t see them every day, but if we restrict attention to just the strong force then it’s important to understand the physics of the pion and its relatives. This physics is largely constrained by symmetry. The pions should rightly be viewed as massless Goldstone bosons for spontaneously broken chiral symmetry (even though pions have a mass of 100 million electron volts!). This perspective allows us to write down an effective theory for the low-energy dynamics of pions, known as the chiral Lagrangian, from which we can recover the proton and neutron as a soliton, known as a Skyrmion. The purpose of this essay to is explain this framework.

Suppose we have a function f : R^p → R, the sampling problem is to output samples X ∼ π, where π(x) = (1/Z) exp(−f(x)), where Z is the normalization constant. In most cases, the normalization constant is computationally intractable, and so we would only have access to π up to proportionality.

This essay is an introduction to the theory of topoi, a certain kind of structured category introduced by Grothéndieck for describing his new idea of the étale cohomology. We will be using it to analyse logic in a novel, categorical way for fun and profit!

The term “KPZ” stands for the initials of three physicists, namely Kardar, Parisi and Zhang, which, in 1986 conjectured the existence of universal scaling behaviours for many random growth processes in the plane.

A process is said to belong to the KPZ universality class if one can associate to it an appropriate “height function” and show that its 3:2:1 (time : space: fluc- tuation) scaling limit, see 1.2, converges to a universal random process, the KPZ fixed point. Alternatively, membership is loosely characterised by having: 1. Local dynamics; 2. A smoothing mechanism; 3. Slope-dependent growth rate (lateral growth); 4. Space-time random forcing with the rapid decay of correlations.

The central object that we will study is the so-called KPZ fixed point, which belongs to the KPZ universality class. Many strides have been made in the last couple of decades in this field, with constructions of the KPZ fixed point from certain processes such as the totally asymmetric simple exclusion process (with arbitrary initial condition) and Brownian last passage percolation.

In this essay, we: 1. delineate the origins of KPZ universality; 2. describe and motivate canonical models; 3. give an overview of recent developments, especially those in the 2018 Dauvergne, Ortmann and Virag (DOV) paper; 4. present the strategy of and key points in the proof of the absolute continuity result of the KPZ fixed point by Sarkar and Virag; 5. conclude with remarks for future directions. The presentation is such that the content is displayed in a way that is as self-contained as possible and aimed at a motivated audience that has mastered the fundamentals of the theory of probability.

In string theory, our universe is hypothesized to be a product of Minkowski spacetime and a Calabi-Yau 3-fold. Curves in a CY 3-fold will be locations where endpoints of strings can attach. Studying the enumerative geometry of curves on CY 3-fold has thus aroused interest among both mathematicians and physicists. My essay will explain the notion of virtual counting, focussing on the Donaldson-Thomas resp. Pandharipande-Thomas invariants. We will dive into the deformation theory, derived category and the motivic Hall algebra to better understand the corresponding two compactifications of the moduli spaces of curves on CY 3-folds.

Let $G$ be an abelian group. By a (non-trivial) corner in a set $A ⊆ G × G$ we mean a choice of $x, y$, and $d \ne 0$ in $G$ such that $(x, y)$, $(x + d, y)$, and $(x, y + d)$ all lie in $A$. The existence of corners in subsets $A$ of $[N]×[N]$ of positive density was first proved by Ajtai and Szemerédi in 1974, and also follows–at least qualitatively–straight from the multidimensional Szemerédi theorem.

This essay surveys the existing literature on the quantitative aspects of the problem and gives a detailed exposition of Shkredov’s upper bound on the density of corner-free sets of $\mathbb F_p^n × \mathbb F_p^n$ as set out by Green in his expository survey, as well as recent lower-bound constructions in that setting motivated by links between this problem and questions in communication complexity. In addition, the essay explains how to adapt Shkredov’s argument to prove a quantitative version of the Ajtai-Szemerédi result.

In this work we aim to construct alternatives, namely toposes, to the category of sets as basics objects of mathematics, and we will see how we can construct models and reason about them in this setting.

The problem of determining which finite groups can act freely on a sphere was first addressed by P. A. Smith in 1944, and resolved by I. Madsen, C. B. Thomas and C. T. C. Wall thirty years later. In this essay we show the negative part of the answer, namely that groups $\mathbb{Z}/p \oplus \mathbb{Z}/p$ and dihedral groups $D_p$ cannot act freely on a sphere for any prime $p$. We also summarise T. Petrie’s construction of a free action on $S^{2q−1}$ of a non-abelian group of order $pq$, with $p, q$ odd primes.

The notion of a "topos" has its roots in two separate mathematical disciplines: On the one hand, an Elementary Topos can be viewed as a generalised mathematical universe; that is, as a foundation for higher-order intuitionistic logic rather than the classical set-theoretic setting. From this perspective, a topos is simply a category sufficiently rich in categorical structure that it admits an "internal language" which can be reasoned about according to the deduction rules of intuitionistic logic, and we have a natural notion of models of theories inside such categories.

The second viewpoint - Grothendieck Toposes - arose from Grothendieck's work in the 1960s reshaping algebraic geometry with the invention of scheme theory. This class of toposes is a generalisation of the category of sheaves on a topological space, and thus can be analysed via arguments of a geometric nature. Placing these "geometric" constraints on toposes imposes structure on their internal logic; in particular, we can show that the theories whose models are preserved by geometric morphisms between Grothendieck toposes (that is, functors which preserve the geometric structure) are those which admit axiomatisations whose axioms are all of a specific form. We call such theories geometric theories.

In this essay, we will introduce the basic theory of elementary and Grothendieck toposes, and show how logic can be interpreted in these categories. We will use the machinery we have built to show that for any geometric theory $\mathbb{T}$, there is a Grothendieck topos $\mathbf{Set}[\mathbb{T}]$ which contains a universal model of $\mathbb{T}$ inside, in the sense that any model of $\mathbb{T}$ in a Grothendieck topos can be viewed as the inverse image of this model along some geometric morphism. We call $\mathbf{Set}[\mathbb{T}]$ the classifying topos of $\mathbb{T}$.