Postgraduate level Mini-Courses
The Department of Mathematics of the University of Manchester will be offering four graduate level mini-courses during the academic year 2022-2023. The courses have been designed for and are aimed at beginning PhD students at a level comparable to that of taught course networks operating in the UK, such as MAGIC and TCC. Each mini-course will run for five weeks, followed by an assessment week. There will be of two hours of lectures per week plus a support session hour per week. First year PhD students can take these for credit, as they do with MAGIC courses; other members of the Department (senior PhD students, postdoctoral researchers, faculty etc.) are also welcome to attend.
Please see below for more details; anyone wishing to attend should email the respective lecturer.
Introduction to stable groups (first half of semester 1):
Lecturer: Jan Dobrowolski, jan.dobrowolski@manchester.ac.uk
Course Website: Please click this link
Course Description: The aim of the course is to survey the main results on groups satisfying the model-theoretic property of stability and its strengthenings such as ω-stability and finiteness of Morley rank. The main question about groups of finite Morley rank, so-called Cherlin-Zilber conjecture, asks if every simple group of finite Morley rank is an algebraic group over an algebraically closed field. Despite this main question being still open in rank higher than 3, the study of groups of finite Morley rank has brought about a number of deep structural results, e.g. a striking theorem of Zilber saying that a connected solvable non-nilpotent group of finite Morley rank interprets an infinite field. We will discuss some of these fundamental results on groups of finite Morley rank along with some results on stable groups in broader generality. In particular, we will discuss the concepts of a connected component and a generic element in a stable group, which may be viewed as generalisations of the familiar concepts from the theory of algebraic groups to much wider contexts.
Prerequisites: Basic knowledge of model theory; basic knowledge of group theory (recommended).
Syllabus:
1. Preliminaries: Stable theories, ω-stable theories, Morley rank, central series of a group, solvability, nilpotence, group actions. Basic properties of stable groups: chain conditions, connected components. Examples of stable groups.
2. Zilber’s indecomposability theorem. Binding groups.
3. ω-stable fields.
4. Groups of finite Morley rank.
5. Generics, connected components, and local ranks in stable groups.
Related MAGIC courses: MAGIC004: Applications of model theory to algebra and geometry
Schedule-Location: Tuesdays 10-11 & Wednesdays 10-12 @ Frank Adams 2.
Geometry of algebraic curves (second half of semester 1):
Lecturer: Kostas Karagiannis, konstantinos.karagiannis@manchester.ac.uk
Course Description: The aim of this course is to develop and explore the theory of algebraic curves from the viewpoint of modern algebraic geometry. The subject matter combines tools from commutative and homological algebra, Galois theory and algebraic number theory, among others. One of the main objectives will be to justify the use of "geometry" in the course's title as every effort will be made to accompany the algebraic formalism with geometric intuition.
Prerequisites: Familiarity with the basic concepts of commutative algebra and Galois theory is assumed. Some exposure to differential geometry, homological algebra and category theory would be helpful but is not deemed necessary to succeed in the course.
Syllabus:
1. Affine and projective plane algebraic curves.
2. Morphisms, rational maps and function fields.
3. Meromorphic functions, divisors and differentials.
4. The Riemann-Roch theorem, genus and applications.
5. (Time permitting) Introduction to sheaf cohomology.
Related MAGIC courses: MAGIC073: Commutative algebra, MAGIC074: Algebraic geometry, MAGIC112: Computational algebra and geometry.
Schedule-Location: Tuesdays 10-11 & Wednesdays 10-12 @ Frank Adams 1.
Introduction to homological algebra (first half of semester 2):
Lecturer: Simon Crawford, simon.crawford@manchester.ac.uk
Course Description: Homological algebra is ubiquitous in modern representation theory, algebraic geometry, and topology. The aim of this course is to develop some of the fundamental tools of homological algebra in the context of modules over rings. The course will culminate with a proof of Hilbert's Syzygy Theorem, which gives an application of many of the concepts introduced in the course.
Prerequisites: Some familiarity with basic commutative algebra is assumed. Knowledge of what a category would also be helpful.
Syllabus: Basic category theory, hom, tensor, adjoint functors, projective/injective/flat modules, homology, derived functors, Tor and Ext, Hilbert's syzygy theorem.
Related MAGIC courses: MAGIC073: Commutative algebra, MAGIC074: Algebraic geometry, MAGIC009: Category theory/
Schedule-Location: TBA.
Groups and formal languages (second half of semester 2):
Lecturer: Alex Evetts, alex.evetts@manchester.ac.uk
Course Description: This course will explore some of the rich connections between formal language theory and combinatorial and geometric group theory. We will start by seeing how finitely generated infinite groups can be given the structure of a metric space, and exploring some of the fundamental questions in this field, such as Dehn's word problem. We will then introduce regular and context-free languages and see some of the ways that they can be applied to gain insight into these fundamental questions, culminating in the Muller-Schupp Theorem, characterising virtually free groups as those with context-free word problem.
Prerequisites: Familiarity with the basics of group theory and graph theory. Some exposure to decision problems or automata theory would be advantageous but will not be assumed.
Syllabus: Large-scale geometry of finitely generated groups, Dehn's word problem, growth of groups, formal languages and automata, hyperbolic groups, the Muller-Schupp Theorem.
Related MAGIC courses: MAGIC096: Spectra and geometry of graphs and networks, MAGIC107: Computability Theory and Reverse Mathematics
Schedule-Location: TBA.