Andrei Akhvlediani (Oxford University), Quantum mechanics and quantum information
categorically
Henryk Żołądek (University of Warsaw), Quantum mechanics and its
mathematical tools
Quantum mechanics and its mathematical tools
Henryk Żołądek (University of Warsaw)
Abstract: Quantum mechanics was created in the first quarter of the 20th century
as a theory describing the wave properties of particles (like the electrons) on one side and the
particle properties of waves (like the electromagnetic waves) on the other side. It turned out that
this theory can be mathematically well-decribed by means of the linear theory of functional spaces
and differential operators, which is now known as the functional analysis. This apparatus is further
extended by the group representation theory which describes the symmetries of an appropriate
physical system.
I will present some applications of functional analysis and group theory in quantum mechanics. As an
example I will describe the discrete spectrum and the bounded states of an electron in the hydrogen
atom, taking into account the angular momentum and the spin in presence of an external magnetic field.
Categorical Preliminaries
Andrei Akhvlediani (Oxford University)
Abstract: - 2-categories and the idea of bicategories.
- String diagrams for 2-categories.
- The formal theory of adjunctions.
- Monoidal categories and the associated string diagrams.
- Braided-, symmetric-, compact closed- and dagger- monoidal categories.
- Extended graphical calculus and its completeness and soundness.
Quantum Information Categorically
Andrei Akhvlediani (Oxford University)
Abstract: - What is quantum computation?
- The standard mathematical model.
- The quantum teleportation protocol.
- The categorical framework.
- Quantum teleportation = adjunction.
- Frobenius algebras and bases.
- Interacting observables and the zoo of algebraic structures.
Compositional Monoidal Theories
Andrei Akhvlediani (Oxford University)
Abstract: - An outline of Lawvere's functorial semantics.
- Monoidal and symmetric monoidal theories: PROs and PROPs.
- Monads and distributive laws in arbitrary bicategories.
- The bicategories of Spans and of (internal) profunctors.
- Representing categories as monads.
- Composing PROPs; key examples.
- Current work and some open problems.
Additional materials will be distributed by the organizers during the meeting.
Christian Kissig
(University of Leicester),
TQFTs and Invariants of 3-Manifolds
Paweł Traczyk (University of Warsaw), Introduction to Knot Theory
Introduction to Knot Theory
Paweł Traczyk (University of Warsaw)
Abstract: In this talk I will introduce basic notions of Knot Theory, such that are
necessary to understand Khovanov cohomology invariants. Specifically, diagrams of knots,
Reidemeister moves, and the Jones polynomial in Kauffman's version will be described.
TQFTs and Invariants of 3-Manifolds
Christian Kissig (University of Leicester)
Abstract: In this talk we study invariants of 3-manifolds with embedded banded knots.
There are three well-known approaches to these invariants: Chern-Simmons
field theory, 2-dimensional conformal field theory, and quantum groups. We
shall pursue the latter in this talk.
Part I: We review basic notions in monoidal categories with braiding, twists
and dualities, in particular ribbon categories. Distinguishing a dominating
set of objects yields the notion of a modular category. The homsets in modular
categories have the structure of modules for a fixed ground ring K. The
dominating objects generalise the decomposition of modules into direct sum
of irreducible modules.
We give an abstract axiomatic characterisation of modular functors from the
category of 3-manifolds and homeomorphisms into the category of projective
K-modules and K-isomorphisms. 3-dimensional TQFTs extend these modular
functors by maps which assign to each 3-cobordisms a K-homomorphism. We
define non-degeneracy and anomalies at this abstract level. Quantum invariants
arise as maps that assign to 3-cobordisms values in a ground ring K, which -
in this part - we assume only to be unital and commutative.
Part II: We give a concrete construction of a 3-dimensional TQFT for
3-cobordisms with embedded ribbon graphs. The construction uses the decoration
of boundaries of 3-cobordisms with ribbon graphs that are coloured in a
modular category V with ground ring K. An extension of surfaces and gluing
by decoration gives rise to a 3-dimensional TQFT for the ground ring K. The
constructed TQFT is non-degenerate, but has anomalies by definition. We
look into these anomalies in detail in Part III.
Part III: First we introduce anomalies abstractly in terms of certain cocycles
of gluing patterns in abelian groups G. Given such a cocycle, we can tweak the
constructed TQFT by extending the cobordisms and the associated notion of
gluing by weights in G. The so obtained TQFT is anomaly-free. The TQFT
constructed in Part II yields such a 2-cocycle, that we can compute in
terms of the Maslov indices of Lagrangian subspaces of the homologies on the
boundary of decorated 3-cobordisms. The definition will solely depend on the
topological structure of the 3-cobordisms and the decorated notion of gluing.
We talk will come full circle with the theorem that isomorphism clases of
non-degenerate anomaly-free 3-dimensional TQFTs are in bijection with quantum
invariants of closed 3-cobordisms.
The talk will be largely based on the book "Quantum Invariants of Knots and
3-Manifolds" of Turaev, the series of lectures on "Integral TQFT" by Masbaum,
and more remotely on "Algebraic Topology" by Hatcher for the construction
of homologies on surfaces.
Moritz Groth (Universität Bonn),
A short course on ∞-categories
Tomasz Maszczyk (University of Warsaw),
A survey on Gromov-Witten invariants and quantum cohomology
A survey on Gromov-Witten invariants and quantum cohomology.
Tomasz Maszczyk (University of Warsaw)
Abstract: A link between classical enumerative geometry of rational curves in the projective plane
and quantum field theories will be traced. A relation between propagators of the quantum theory
and commutative Frobenius algebras will be described. Gromov-Witten invariants will be defined
and their reconstruction from so called quantum cohomology will be ideologically explained.
Basic notions and the relation with simplicial categories.
Moritz Groth (Universität Bonn)
Abstract: We will give the definition of ∞-categories and construct the associated homotopy
categories. Along the way we will see that we succeeded in finding an axiomatic framework 'for
categories with morphisms of arbitrary dimension' such that compositions exist well-defined up to
a contractible choice. Using the coherent nerve construction due to Cordier, the Joyal model
structure for ∞-categories is defined. The comparison result on the existence of a Quillen
equivalence between the Joyal and the Bergner model structure for simplicial categories is
mentioned.
Categorical constructions and the relation with model categories.
Moritz Groth (Universität Bonn)
Abstract: We transfer some central notions from classical category theory to the world of
∞-categories. Among these are the notions of (co-)cones, join and slice constructions and the
notion of (co-)limits. Then we turn to presentable ∞-categories and describe some main steps in
the argumentation due to Lurie that presentable model categories and simplicial combinatorial
model categories do the same job.
(Symmetric) Monoidal ∞-categories and the notion of fully dualizable objects.
Moritz Groth (Universität Bonn)
Abstract: Motivated by the classical Grothendieck construction, we find a reformulation of the
axioms of a monoidal category. In particular, this reformulation 'hides' the coherence axioms which
is particularly convenient in the ∞ -categorical world since there the pentagon axiom would only be
the first of an infinite list of expected coherence axioms. This reformulation motivates a definition
of monoidal ∞-categories. Then we sketch which modifications are in place in order to obtain a
theory of symmetric monoidal ∞-categories. We finish by introducing the notion of fully
dualizable objects in a symmetric monoidal ∞-category.