Results 11 to 20 of 28
Abstract: Gödel’s incompleteness theorems state that any recursively axiomatized theory which can interpret sufficient arithmetic is incomplete and cannot prove its own consistency. I will give an overview of their proofs, how they relate to recursion theory, and go over some examples of statements which are undecidable in the theory of Peano arithmetic.
Abstract: A semigroup is a set equipped with an associative binary operation. The definition is simple, but semigroups exhibit rich internal structure — one that is surprisingly different to groups. In this talk I will introduce the notion of a semigroups along with central notions to semigroup theory including Green's relations, key tools to understanding the structure of semigroups which partition elements according to how they generate ideals. This talk will give an overview of the area aimed at building intuition rather than deep technical knowledge.
Abstract: Set theory is often accepted as a foundation of mathematics because of our ability to embed mathematical objects as sets, and prove classical theorems from the axioms of set theory. However, this answer does not examine what ‘job’ we want a foundation of mathematics to do, what it is that makes a theory foundational. We will discuss the response of the philosopher Penelope Maddy to this question, and time permitting also discuss category theory and homotopy type theory through this lens.
Abstract: In order to scandalise the growing number of Italians in this department, I will present an unhinged rant rethinking this well-loved dish. In contrast to more standard recipes, the main ingredient will be a functorial link homology theory in the thickened 3-ball. In fact, my talk will be primarily concerned with unpacking, as much as time allows, the following sentence: “skein lasagna modules are four manifold invariants defined using functoriality of Khovanov-Rosansky homology, which is a categorification of the Jones polynomial”.
Abstract: Coxeter groups are groups which can be generated by involutions and certain simple relations. We will discuss how they relate to geometric reflection groups and we'll go over as many remarkable properties as possible. For example, the word problem is solvable: there is an algorithm to decide if two words in the group represent the same element (in general this is not possible!).
Abstract: Solitons are solutions to field equations arising from classical field theory that are localised in some sense, topological solitons are those which are stabilised by topological features of the target space. They are used to model a wide variety of physical phenomena, and often have a rich geometric and dynamical structure. In this talk we will take a tour through soliton theory, describing some important topological solitons and their key features. Time permitting, we will discuss some connections between the existence of solitons and the geometry of the underlying space.
Abstract: Skew braces are an algebraic structure with close historical and practical ties to the set-theoretic Yang-Baxter equation. We will discuss existing results about skew braces, covering gamma functions, the connection between skew braces and regular subgroups of the holomorph of a group, results about bi-skew braces, and ideals. We will use this to completely classify the finite bicyclic skew braces, and, among them, which are bi-skew.
Abstract: A D-set is a relational structure on the leaves of a tree, consisting of a quaternary relation defined such that we have D(x,y,z,w) if and only if the path from x to y is disjoint from the path from z to w. In this talk I will introduce several model-theoretic notions, namely ultrahomogeneity, indiscernible sequences, dp-minimality and distality, and I will discuss how they arise during the study of D-sets and what they tell us about structures in general.
Abstract: Elliptic PDEs are a widely studied class of partial differential equations. I will discuss a collection of interesting results in regularity and existence, some of them rather remarkable! In doing this I will discuss the H\"older space approach and the Sobolev space approach and hopefully give a couple of the key ideas of the proofs.
Cluster algebras were introduced in 2002 by Fomin and Zelevinsky in the context of Lie theory as commutative algebras defined combinatorially through an iterated mutation process. Since then, they have appeared in many areas of mathematics, including mathematical physics, algebraic combinatorics, and algebraic geometry. Buan, Marsh, Reineke, Reiten, and Todorov later introduced the cluster category, providing a natural categorical model for the combinatorics of cluster algebras. This talk will be a gentle, pictorial introduction to cluster algebras and their categorification via cluster categories, illustrated with plenty of examples.