An LMS online lecture course in branes, gauge theories and dualities.
An LMS online lecture course in solitons.
The lectures will highlight some recent work on solvable models of topological solitons. The first involves generalisations of the U(1) Abelian-Higgs model whose integrability is intimately related to the geometry of constant curvature Riemann surfaces. The second piece of work is a study of magnetic skyrmions in chiral magnets. Recently a family of soluble models for magnetic skyrmions in chiral magnets was introduced. The energy functional for these models is bounded below by the topological charge, configurations which attain this bound solve first-order equations. The explicit solutions of these first-order equations are given in terms of arbitrary holomorphic functions. Finally I will explain how this model can be interpreted as a gauged non-linear sigma model.
Lecture 1: A primer on solitons. I will introduce the concept of a topological soliton through two prototypical examples, the φ4 and Sine-Gordon models in 1+1 dimensions. Next, we will meet Derrick's theorem and learn why solitons are hard to construct in higher dimensions. Finally, we will meet some examples of higher-dimensional models possessing soliton solutions.
Lecture 2: Solitons in chiral magnets. We will meet a specific model of 2-dimensional chiral magnetic systems which admits soliton solutions. For a special potential term exact, degree 1, skyrmion solutions can be constructed. This leads up to meeting a critically coupled version of the model where there is a whole zoo of analyitic skyrmion solutions.
An LMS online lecture course in moduli spaces.
Moduli spaces of stable maps have been of central interest in algebraic geometry for the last 30 years. In spite of that, the geometry of these spaces in genus bigger than zero is poorly understood, as the Kontsevich compactifications include many components of different dimensions meeting each other in complicated ways, and the closure of the smooth locus is difficult to describe.
In recent years a new perspective on the problem of finding better behaved compactifications, ideally smooth ones, has come from log geometry. This approach has proved successful in a series of examples and log geometry is now becoming a natural setting to study modular resolutions of moduli spaces.
The aim of this series of talks will be to see how log geometry techniques apply to give modular smooth compactifications of moduli spaces of stable maps to projective spaces in genus one and two; we will also explain why the latter are interesting from an enumerative point of view.
In more detail: we begin by studying the deformation theory and the global geometry of moduli spaces of genus one and two stable maps; we then give a brief introduction to log schemes, line bundles on log schemes and log blowups and conclude by exhibiting the log modification resolving the moduli spaces of maps in genus one and two and explaining their modular meaning.
An LMS online lecture course in nonlorentzian geometry.
The topics to be covered are:
• Motivation, definition and properties of homogeneous spaces;
• Examples to build intuition;
• Applications (in physics);
• Introduction to spacetimes;
• Sketch of classification of maximally symmetric spacetimes;
• Discussion of properties and invariants;
• Relevance in physics.
An LMS online lecture course in algebraic geometry.
I will first describe the notion of Bridgeland stability conditions on triangulated categories. Then I will focus on stability conditions on the bounded derived category of coherent sheaves on curves, surfaces and threefolds. In the end, some recent applications of Bridgeland stability conditions in classical algebraic geometry and Donaldson-Thomas Theory will be explained.
An LMS online lecture course in mirror symmetry.
Mirror symmetry conjecturally associates to a Fano orbifold a (very special type of) Laurent polynomial. Laurent inversion is a method for reversing this process, obtaining a Fano variety from a candidate Laurent polynomial. We apply this to construct new Fano 3-folds with terminal Gorenstein quotient singularities.
In this series of talks, I will go through some of the basics of toric geometry to showcase how one can use combinatorial data to systematically build geometric objects. We will restrict our attention to the well-studied Fano case, for which there is concrete evidence that the mirror theorem holds in many cases.
An LMS online lecture course in groups, semigroups and algebras.
An LMS online lecture course in Vinberg theory.
In recent years, Vinberg theory of graded Lie algebras has become relevant in many areas of number theory, from arithmetic statistics (e.g., in the work of Romano-Thorne) to the local Langlands correspondence (e.g., in the work of Reeder-Yu). These lectures will provide the algebraic background for number theory students to engage with research involving graded Lie algebras. We'll start by discussing some of the relevant aspects of the invariant theory of Lie algebras, including the Chevalley restriction theorem and the pioneering work of Kostant on invariant rings. We'll then define graded Lie algebras and look at the graded analogues of these theorems, based on work of Vinberg. Time permitting, we'll look at Slodowy slices and applications to families of algebraic curves. These lectures should give number theory students sufficient background to read, for example, Thorne's paper Vinberg's representations and arithmetic invariant theory and other related papers. But the lectures will also be a useful introduction to some beautiful aspects of Lie theory for students in algebra and representation theory. I'll assume students have some knowledge of Lie algebras, but I will review relevant background and provide examples throughout the lectures.
A series of three lectures on probabilistically checkable proofs.
This is a series of talks on model theory by Jonathan Pila, about point counting, O-minimality and Ax-Schaunel, and the Zilber-Pink conjecture.
The gravitational waves detected recently by LIGO were produced in the final faze of the inward spiraling of two black holes before they collided to produce a more massive black hole. The experiment is entirely consistent with the so called Final State Conjecture of General Relativity according to which, generically, solutions of the initial value problem of the Einstein vacuum equations approach asymptotically, in any compact region, a Kerr black hole. Though the conjecture is so very easy to formulate and happens to be consistent with astrophysical observations as well as numerical experiments, its proof is far beyond our current mathematical understanding, let alone available techniques techniques. In fact even the far simpler and fundamental question of the stability of the Kerr black hole remains wide open.
In my lectures I will address the issue of stability as well as other aspects the mathematical theory of black holes such as rigidity and the problem of collapse. The rigidity conjecture asserts that all stationary solutions the Einstein vacuum equations must be Kerr black holes while the problem of collapse addresses the issue of how black holes form in the first place from regular initial conditions. Recent advances on all these problems were made possible by a remarkable combination of new geometric and analytic techniques which I will try to outline in my lectures.
This series of talks is based on joint works with Oppermann, Grimeland, Labardini and Plamondon. Cluster categories are triangulated categories where quiver mutation appears as a natural operation. A first class of example is given by cluster categories associated with surfaces with marked points. A second class is constructed using the derived category of finite-dimensional algebras of global dimension 2. Mixing both constructions, one may consider surface cut algebras, that are algebras of global dimension 2 constructed from a surface and show how cluster combinatorics permits to deduce information on their derived category.
Let R be a commutative Noetherian ring. Denote by D-(R) the derived category of cochain complexes X of finitely generated R-modules with Hi(X)=0 for i>>0. Then D-(R) has a structure of a tensor triangulated category with tensor product ⊗RL and unit R. In this series of lectures, we study thick tensor ideals of D-(R), i.e., thick subcategories closed under the tensor action by each object in D-(R), and investigate the Balmer spectrum Spc D-(R) of D-(R), i.e., the set of prime thick tensor ideals of D-(R). Here is a plan.
• We give a complete classification of the (co)compactly generated thick tensor ideals of D-(R), establishing a generalized version of the Hopkins--Neeman smash nilpotence theorem.
• We construct a pair of maps between the Balmer spectrum Spc D-(R) and the prime spectrum Spec R, and explore their topological properties.
• We compare several classes of thick tensor ideals of D-(R), relating them to specialization-closed subsets of Spec R and Thomason subsets of Spc D-(R).
If time permits, I would like to talk about the case where R is a discrete valuation ring. My lectures are based on joint work with Hiroki Matsui.
The commutative algebraic groups over a prescribed field k form an abelian category Ck; the finite commutative algebraic groups form a full subcategory Fk, stable under taking subobjects, quotients and extensions. This mini-course will study the categories Ck and Ck/Fk (the isogeny category) from a homological viewpoint, emphasizing the analogies and differences with categories of representations. In particular, we will show that Ck/Fk has homological dimension 1, and we will describe the projective and the injective objects in Ck and Ck/Fk.
Modular representation theory of finite groups seeks to understand, and possibly classify, the algebras - called block algebras of finite groups - which arise as indecomposable direct factors of finite group algebras over a complete local principal ideal domain with residue field of prime characteristic p. The expectation is that 'few' algebras should arise in this way, and that this should in turn lead to significant structural connections between finite groups and their block algebras.
The key feature of block algebras of finite groups is the dichotomy of invariants attached to these algebras.
On the one hand, they have all the typical algebra-theoretic invariants - module categories, their derived categories and stable categories, as well as numerical invariants such as the numbers of isomorphism classes of simple modules, and cohomologivcal invariants such as their Hochschild cohomology.
On the other hand, they have p-local invariants, due to their provenance from group algebras - reminiscent of the local structure of a finite group which includes its Sylow p-subgroups and its associated fusion systems.
Essentially all prominent conjectures which drive modular representation theory revolve around the interplay between these two types of invariants. We describe this interplay with a focus on Hochschild cohomology and analogous cohomology rings which are defined p-locally. This involves a variety of angles - Hochschild cohomology is graded commutative, hence methods and notions from commutative algebra will play a role. Hochschild cohomology in positive degree is also a Lie algebra. We will investigate connections between the algebra structure of block algebras and the Lie algebra structure of its first Hochschild cohomology space.
We'll study the global structure of the stable module category StMod G or, equivalently, the category of singularities of representations of a finite group scheme G over a field of positive characteristic p. The goal of the lectures will be to classify the tensor ideal localizing subcategories in StMod G. The techniques involved in the classification include the theories of support and cosupport in modular representation theory, detection of projectivty for modules, Benson-Iyengar-Krause theory of local cohomology functors, and new methods inspired by commutative algebra which allow to relate local cohomology at closed and arbitrary points. This is based on joint work with Eric Friedlander and Dave Benson, Srikanth Iyengar and Henning Krause.
The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a beautiful structure as a cluster algebra, by a result of J. Scott. Central to this description is a collection of clusters containing only Plücker coordinates, which are described by certain diagrams in a disk, known as Postnikov diagrams or alternating strand diagrams. Recent work of B. Jensen, A. King and X. Su has shown that the Frobenius category of Cohen-Macaulay modules over a certain algebra, B, can be used to categorify this structure.
In joint work with Karin Baur and Alastair King, we associate a dimer algebra A(D) to a Postnikov diagram D, by interpreting D as a dimer model with boundary. We show that A(D) is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay cluster-tilting B-module, i.e. that it is a cluster-tilted algebra in this context. The proof uses the consistency of the dimer model in an essential way.
It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disk. The general surface case can also be considered, and we compute boundary algebras associated to the annulus.
These lectures will be about enumerative K-theory of curves (and more general 1-dimensional sheaves) in algebraic threefolds. In the first lecture, we will set up the enumerative problem and survey what we know and what we conjecture about it. In particular, we will meet the fundamental building blocks of the theory: threefolds fibered in ADE surfaces. In the second lecture, we will learn what geometric representation theory says about these building blocks, and, in particular, meet the present day incarnation of the Weyl group, which is really a fundamental groupoid of a certain periodic hyperplane arrangement, associated to a certain geometrically defined infinite-dimensional Lie algebra. This Weyl group completely determines the curve counts, and so seems like a very fitting topic for Hermann Weyl lectures. In the third lecture, I plan to introduce some of the geometric ideas that go into the actual technical construction of the theory.
These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that allows to prove the existence of phase transitions. Finally, we will discuss certain probabilistic representations and their consequences.
A two-hour course on expanders, thin subgroups of Lie groups, and superstrong approximation.