December 2009, January 2010, February 2010, March 2010, November 2010, December 2010, March 2011.

Contact seminar organizers at: qft 'at' mimuw.edu.pl.

Joint meeting of the Seminar on Quantum Field Theories &

Polish Seminar on Category Theory dedicated to**Saturday, March 12th, 2011**

at the Department of Mathematics, Computer Science, and Mechanics, room 4070.

## Speakers

## Talks

Polish Seminar on Category Theory dedicated to

Quantum Information and Quantum Computing

took place on at the Department of Mathematics, Computer Science, and Mechanics, room 4070.

Aleks Kissinger
(Computing Laboratory, Oxford University),
*Frobenius States and a Graphical Language for Multipartite Entanglement*

Marek Zawadowski
(Department of Mathematics, University of Warsaw),
*Monoidal categories, an introduction*

*Frobenius States and a Graphical Language for Multipartite Entanglement*

Aleks Kissinger (Oxford University)

**Abstract:**
While multipartite quantum states constitute a key resource for quantum computations and protocols,
obtaining a high-level, structural understanding of entanglement involving arbitrarily many qubits
is a long-standing open problem in quantum computer science. We approach this problem by identifying
the close connection between a special class of highly entangled tripartite states, which we call
Frobenius states, and the standard notion of a commutative Frobenius algebra. We use these states
(and their induced algebras) as a primitive in a graphical language that is both universal for quantum
computation and capable of highlighting the behavioural differences in types of Frobenius states. This
graphical language then suggests methods for state preparation, classiﬁcation, and efﬁcient classical
modeling of certain kinds of quantum systems.

Some backround material can be found here and
here.

These papers can serve as lecture notes: paper 1 and
paper 2.

And here are some slides from a similar talk (by the same speaker).

*Monoidal categories, an introduction*

Marek Zawadowski (MIM UW)

**Abstract:**
This talk is meant to be an introduction to the theory of monoidal
categories and some of its extensions. After reviewing the basic
notions concerning monoidal categories I will discuss the coherence,
the graphical language, and some 2-dimensional aspect of the theory.
Then I will present some extension of the notion of a monoidal
category: braided, autonomous, and traced categories among others.
These notions will be illustrated with some typical examples.

The sixth meeting dedicated to
**Saturday, December 18th, 2010**

at the Department of Mathematics, Computer Science, and Mechanics, room 4070.

## Speakers

## Talks

Perspectives in Quantum Field Theory II

took place on at the Department of Mathematics, Computer Science, and Mechanics, room 4070.

Tomasz Maszczyk
(Department of Mathematics, University of Warsaw),
*Quantum sets*

Jerzy Lewandowski
(Department of Physics, University of Warsaw),
*Gravity Quantized. Mathematically exact models of quantum gravity*

*Quantum sets*

Tomasz Maszczyk (MIM UW)

**Abstract:**
As we know from the lecture of prof. Woronowicz, quantum geometry has
no underlying set of points. The aim of the present lecture is to show that quantum geometry has
something "quantum" instead. I will explain categorical foundations necessary to extend the classical
adjunction between sets and commutative algebras to an adjunction between "quantum sets" (some
generalization of sets) and associative algebras. The extension consists of two adjoint functors F
and P extending the functor of functions on a given set and the functor of points of a given
commutative algebra. The main motivation for this extension is to give a noncommutative geometric
interpretation of bialgebras as coordinate algebras of "quantum algebraic monoids", i.e. monoids in
the monoidal category of "quantum algebraic sets". This lecture can be regarded as an introduction
to a lecture I will give at the Seminar on Noncommutative Geometry at IMPAN on December 20th, 2010.

*Gravity Quantized. Mathematically exact models of quantum gravity*

Jerzy Lewandowski (FUW)

**Abstract:**
There are known models of quantum Einstein's gravity that from the
mathematical point of view are exact and complete theories. They are formulated within the framework
called Loop Quantum Gravity. The question of whether any of them describes the reality is an open
issue. But given a shortage of candidates, they deserve attention. The aim of this lecture is to
present the mathematical structure of these models.

The first 60 minute lecture will be a pedagogic introduction to the mathematical part of the theory:
spaces of connections, diffeomorphism invariant measures, diffeomorphism invariant representations
of certain *algebras called the quantum holonomy-flux algebras, spin-networks. With those tools the
Hilbert spaces of the quantum states of geometry will be defined.

The second 60 minute lecture will be an introduction to the models. In general relativity a
spacetime point is not a well defined concept. Space and time emerge from relations between the
fields. That feature has important consequences in the quantum theory.

The fifth meeting dedicated to
**Saturday, November 20th, 2010**

at the Department of Mathematics, Computer Science, and Mechanics, room 4070.

## Speakers

## Talks

Perspectives in Quantum Field Theory

took place on at the Department of Mathematics, Computer Science, and Mechanics, room 4070.

Stanisław Lech Woronowicz
(Department of Physics, University of Warsaw),
*Is there a geometry without points.*

Wojciech Lubawski
(Department of Mathematics, Polish Academy of Sciences),
*Cobordisms with chronology and different versions of Khovanov homology for knots and links.*

*Cobordisms with chronology and different versions of Khovanov homology for knots and links*

Wojciech Lubawski (IM PAN)

**Abstract:**
We will define chronological Khovanov homology for links and show its
basic properties and relations to the standard Khovanov homology and so called odd-Khovanov homology. We
will discuss how to generalize the construction to the case of virtual links and knots/Dynkin diagrams.

*Is there a geometry without points*

Stanisław Lech Woronowicz (FUW)

**Abstract:**
A concept of point seems to be very fundamental
in any geometrical system. In fact geometrical spaces considered so far
are collections of points organised in a certain way. In the first part
of the talk we shall discuss the concept of point as it appears
in physics. In particular we shall show that under the natural
assumptions (axioms of relativistic quantum field theory) no physical
information can be stored in one point of the space-time (Wightman
theorem). The second part will be devoted to non-commutative (quantum)
spaces. These are objects that are not built of points. Nevertheless
they may carry geometrical structures similar to that of classical
spaces. One of the first example of a non-commutative space is the
quantum phase space that was discovered (introduced) by W. Heisenberg.

The fourth meeting dedicated to
**Sunday, March 28th, 2010**

at the Department of Mathematics, Computer Science, and Mechanics, room 5070.

## Speakers

## Talks

Categorical methods in Quantum Mechanics

took place on at the Department of Mathematics, Computer Science, and Mechanics, room 5070.

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.

The third meeting dedicated to
**Saturday, February 27th, 2010**

at the Department of Mathematics, Computer Science, and Mechanics, room 5070.

## Speakers

### Photo gallery.

## Talks

Knot Theory and 3D TQFT

took place on at the Department of Mathematics, Computer Science, and Mechanics, room 5070.

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.

The second meeting dedicated to
**Saturday, January 23rd, 2010**

at the Department of Mathematics, Computer Science, and Mechanics, room 5070.

## Speakers

### Photo gallery.

## Talks

Topological Quantum Field Theories

and their applications in Algebraic Topology and Algebraic Geometry

took place on and their applications in Algebraic Topology and Algebraic Geometry

at the Department of Mathematics, Computer Science, and Mechanics, room 5070.

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.

The first meeting took place on **Saturday, December 5th, 2009**

at the Department of Mathematics, Computer Science, and Mechanics, room 5070.### Photo gallery.

### From Atiyah to Lurie. 20 years of TQFT.(Krzysztof Kapulkin)

1. Definition of a TQFT as a symmetric monoidal functor n-Cob ---> Vect (everything is to be
defined).

2. n-Cob as a symmetric monoidal category.

3. Frobenius algebras. Theorem: The category of 2D TQFTs is equivalent to the category of commutative Frobenius algebras.

4. Perspective: Extended TQFT, Baez-Dolan conjecture, Lurie theorem.

### Strings – filling gaps between QFT and general relativity theory. (Marek Czarnecki)

1. Problems combining QFT and GRT.

2. What is String Theory about?

3. Further development: what do strings still lack?

###
Classical and quantum mechanics, and field theories: comparison of formal structures. (Jan Liszka-Dalecki)

1. Classical mechanics – takes place on manifolds, the mathematical apparatus used is differential
geometry.

2. Quantum mechanics – takes place in Hilbert spaces, the mathematical apparatus used is functional analysis.

3. Classical field formalism, Feynman path trajectories, Feynman diagrams, field quantisation and QFT.

at the Department of Mathematics, Computer Science, and Mechanics, room 5070.

2. n-Cob as a symmetric monoidal category.

3. Frobenius algebras. Theorem: The category of 2D TQFTs is equivalent to the category of commutative Frobenius algebras.

4. Perspective: Extended TQFT, Baez-Dolan conjecture, Lurie theorem.

2. What is String Theory about?

3. Further development: what do strings still lack?

2. Quantum mechanics – takes place in Hilbert spaces, the mathematical apparatus used is functional analysis.

3. Classical field formalism, Feynman path trajectories, Feynman diagrams, field quantisation and QFT.

Marek Czarnecki (co-founder, organizer 2009–10)

Aleksander Horawa (organizer 2010–11)

Krzysztof Kapulkin (co-founder, organizer 2009–10)

Łukasz Sienkiewicz (organizer 2010–11)

Krzysztof Święcicki (organizer 2010–11)