Academic Year 2023-24
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Abstract: Introduced as recently as 2002 by Fomin and Zelevinsky, cluster algebras are all the craze at the moment. Surely but not exactly slowly, we can all watch them take over the mathematical world. Long gone are the times in which their radius of influence was restricted to algebraists alone. Combinatorialists, algebraic geometers and even physicists have taken a liking to the iteratively defined fellows, to name only a few. If you have somehow been missing out on the fun so far, fear not and instead embark with me on a journey through their history, leading up to the use of Postnikov’s alternating strand diagrams to combinatorially describe the cluster structure of the Grassmannian.
Abstract: Symmetric functions play a fundamental role in various areas of mathematics, such as combinatorics, representation theory, and physics. Three decades ago, Macdonald polynomials were introduced, which generalize many important basis in the theory and have combinatorial properties.
Recently, a generalization of Macdonald polynomials in superspace has been carried out, where these new polynomials depend on anticommutant variables. It has been observed that these polynomials also retain combinatorial properties that extend naturally to the Macdonald polynomials.
In this talk, basic notions of symmetric functions will be presented. We will talk about Macdonald polynomials and their combinatorial properties, to end by talking about the case in superspace.
Abstract: We are all familiar with the classical variational problem of cramming the most amount of material in the shortest amount of time — by the end of this talk, you will be intimately familiar with this problem. The isoperimetric problem is a geometric interpretation of this ancient noumenon, which asks you to find the shape that has the smallest perimeter for fixed area. In this talk I use the isoperimetric problem as a motivation to take you on a whistle-stop tour of as much geometric analysis as I can. Along the way, we will sketch proofs no fewer than THREE WHOLE THEOREMS (time permitting): the isoperimetric inequality for curves, the Sobolev inequality, and the dizzyingly amazing fact of their equivalence. I will mention one of my own results, though briefly: blink, and you’ll miss it!
Abstract: Cohen’s method of forcing was a groundbreaking advance in set theory that allowed one to extend models of ZF(C) with relative ease. The first application showed that the continuum hypothesis may be falsified in ZFC which, alongside Gödel’s constructible universe twenty years prior, answered the millenium question. We shall have a brief look into these techniques to see how it works.
Abstract: The study of optimal stopping problems has a long history, but the seminal work by El Karoui expanded the reach of the analysis. In this talk, I will present the first generalisation of problems of optimal stopping and the solution methods.
Abstract: In this seminar I will introduce lattice statistical mechanics insofar as it is a source of concrete mathematical objects worth computing. The main protagonist in this story is the transfer matrix which provides a computational formalism amenable to a mathematical deconstruction, whereby a deep connection is revealed with objects of interest in the modern field of algebra. Topics may include transfer matrix algebras (including the celebrated Temperley-Lieb algebra), star-triangle identities and commuting transfer matrices, and Baxter’s famous TQ-method.
Abstract: The geodesic approximation models the dynamics of slowly moving solitons in a classical field theory by geodesics on a related moduli space, reducing complex problems in soliton dynamics to more tractable statements in Riemannian geometry.
In this talk I will introduce the geodesic approximation in the particular context of the dynamics of vortices in Abelian Yang-Mills-Higgs theory. We will begin with a brief overview of Abelian YMH theory and the existence of vortex solitons, moving onto the existence and structure of static vortex moduli spaces, and the validity of the geodesic approximation in the low-energy regime. The second half of the talk will focus on finer details about the vortex moduli space including the construction of the metric and some key geometric properties. Time permitting, we will mention some new results on how the vortex metric can itself be approximated in certain parametric limits.
Abstract: The category of posets has partially ordered sets as objects, and monotone maps between them as morphisms. In this talk, we will describe a contravariant endofunctor on this category which takes a poset to its set of upper sets equipped with the inclusion partial order, and takes monotone maps to their inverse image maps, and we will show that it is self-adjoint on the right. This mirrors the contravariant powerset functor on the category of sets, which is similarly self-adjoint on the right - in fact, the powerset functor factors through the upper set functor we are describing. We will also look at some of the combinatorial applications of this adjunction, and, time permitting, we may discuss some other properties of this functor.
Abstract: Orbits of the coadjoint action of a (finite-dimensional) Lie group on the dual of its Lie algebra — called coadjoint orbits — are interesting geometric objects that carry a canonical symplectic structure. This talk will be an introduction to this point of view on a coadjoint orbit as a symplectic manifold, that is, a manifold equipped with a nondegenerate closed 2-form.
In particular, we will discuss a theorem, originally due to S. Lie and rediscovered in the 1960s by A. Kirillov and B. Kostant, that shows that the symplectic leaves of a Lie-Poisson structure coincide with the orbits of coadjoint representation. We will also briefly discuss how these ideas connect to the geometric formulation of classical mechanics where coadjoint orbits appear as natural phase spaces of dynamical systems.
NOTE THE CHANGE OF ROOM: ROGER STEVENS LT 09
Abstract: Large cardinals are an important aspect of modern set theory, allowing us to expand our understanding of infinite cardinals, and providing a rich hierarchy with applications in many areas of mathematics. We will provide the necessary background to understand what large cardinals are, look at the assumptions needed to produce a wordly cardinal, and show that this is independent from ZFC. We will then look at the applications of Large Cardinals within set theory and beyond.
Abstract: One fine morning, as you walk past Henry Price, a geothermal borehole driller excitedly beckons you over. ‘What is it?’ you ask. ‘Did you find geothermal energy?’ ‘What? No, but I can see why you might think that,’ he replies. ‘I found this bounded subset of Euclidean n-space, and I want to break it up into pieces of smaller diameter. How many pieces do you think I would need? Would n+1 suffice?’
This is precisely the content of Borsuk’s Conjecture, which has a surprising resolution using finite combinatorics. We present some notions and results in the theory of set systems, culminating in the beautiful proof of the Frankl-Wilson Theorem on modular intersections, and apply it to help out our driller friend.
Abstract: We will embark on a journey that mirrors my experience some time ago while learning the simplest connection between partial differential equations and probability.
Abstract: In 1984 Grothendieck remarked: ‘General topology kind of sucks. What if there was a new, tamer topology which sucked less?’ In this talk I will be introducing o-minimal structures, one possible realisation of Grothendieck’s vision. The talk will consist of me defining and motivating these structures, then providing some basic and more advanced results relating to them.
Abstract: In the representation theory of associative algebras, we study categories of modules. In this talk, we explain how this is intimately related to the representation theory of quivers. Furthermore, we take a leisurely tour of Auslander-Reiten theory and explain how the “shape” of the finite-dimensional module category over a finite-dimensional algebra is described by its Auslander-Reiten quiver.