Will Merry: Catalogue data in Spring Semester 2015 |
Name | Dr. Will Merry |
Field | Mathematics |
Department | Mathematics |
Relationship | Assistant Professor |
Number | Title | ECTS | Hours | Lecturers | |
---|---|---|---|---|---|
401-3530-15L | Stable Commutator Length and Quasimorphisms Number of participants limited to 12. | 4 credits | 2S | W. Merry | |
Abstract | This seminar is intended for students interested in geometry and topology. The main players in our story are the "stable commutator length" of an element of a group, and "quasimorphisms", which are maps from a group to the real numbers that are "almost" homomorphisms. Our focus will be on using these tools to tackle problems in low-dimensional topology and symplectic geometry. | ||||
Objective | By the end of the seminar, we will have attempted to cover: stable commutator length (scl), basic knot theory, quasimorphisms, Bavard's Duality theorem, the Thurston norm, rationality of scl, the de Rham quasimorphism and rotation numbers, basic symplectic geometry, the Maslov index | ||||
Content | Oriented closed surfaces are classified by their genus (number of holes). The commutator length is the algebraic analogue of the genus in group theory. The first half of this seminar studies the stable commutator length, and discusses applications to low-dimensional topology and knot theory. Quasimorphisms are maps which are "almost" homomorphisms. One of the simplest examples of a quasimorphism is the "rotation number" on the group of orientation-preserving homeomorphisms of the circle. Quasimorphisms crop up in many areas of mathematics. We will focus on applications in symplectic geometry and dynamics. Finally we will discuss the relationship between quasimorphisms and "bounded cohomology", and then establish a link between stable commutator length and quasimorphisms via the "Bavard Duality theorem". | ||||
Literature | An advanced monologue on stable commutator length is: Danny Calegari, "scl", volume 20 of MSJ Memoirs. Mathematical Society of Japan, Tokyo, 2009 (The seminar will only cover a fraction of this book!) Two brief survey articles, both published in the Notices of the AMS "What is..." series, on our two main topics are: Danny Calgari, "What is... stable commutator length?" Notices of the AMS, October 2008, D. Kotschick, "What is... a quasimorphism?" Notices of the AMS, February 2004. | ||||
Prerequisites / Notice | Students should have knowledge of basic algebraic topology (the fundamental group, and (co)homology) as e.g. learned in the course "Algebraic Topology 1 & 2", and some knowledge of basic differential geometry. There are no prerequisites from group theory. | ||||
401-4376-15L | Quantum Mechanics via Symplectic Geometry | 4 credits | 2V | W. Merry | |
Abstract | *This course is intended for mathematicians. It will neither assume nor contain any theoretical physics.* This course is about symplectic geometry and its relations to quantum mechanics and quantum noise. | ||||
Objective | The course begins with the basic notions of symplectic geometry and function theory on symplectic manifolds. We then discuss quantisation in symplectic geometry, and the mathematical formulation of quantum mechanics. The course then goes on to study the relationship between the geometry of covers and quantum noise, using the notion of a symplectic quasi-state. Time permitting, at the end of the course we will discuss elements of Floer theory, and the construction of symplectic quasi-states. | ||||
Content | Here is a tentative syllabus: (1) Hamiltonian dynamics and symplectic geometry (2) Basic principles of quantum mechanics and quantisation (3) Symplectic quasi-states and symplectic approximation theory (4) Quantum noise and the unsharpness principle (5*) Floer homology (*) Time permitting | ||||
Lecture notes | Full lectures notes will be made available on my website. | ||||
Literature | The following two books will be useful for the first half of the course: McDuff and Salamon - Introduction to Symplectic Topology (Oxford University Press, September 2005) Takhtajan - Quantum Mechanics for Mathematicians (AMS Graduate Studies in Mathematics, 2008, Volume 95) The second half of the course will be based on the following book: Polterovich and Rosen - Function Theory on Symplectic Manifolds (AMS, CRM Monograph Series, 2014, Volume 34) | ||||
Prerequisites / Notice | Prerequisites: Mathematics: Essential: Differentiable geometry Desirable but not essential: Basic functional analysis Morse theory Physics: Absolutely nothing |