Seminário de Teoria Quântica do Campo Topológica  

19/12/2012, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática
Sara Tavares, Univ. of Nottingham

Observables in 2D BF theory

BF theory in two dimensions has been the subject of intensive study in the last twenty five years. I will readdress it by highlighting the TQFT interpretation of the spinfoam approach to its quantisation. I will also introduce the mathematical model that allows us to treat surfaces with inbuilt topological defects and how we expect them to relate to operators in the quantum field theory.

28/11/2012, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática
Louis H. Kauffman, Univ. of Illinois at Chicago

Non-Commutative Worlds and Classical Constraints

This talk shows how discrete measurement leads to commutators and how discrete derivatives are naturally represented by commutators in a non-commutative extension of the calculus in which they originally occurred. We show how the square root of minus one ($i$) arises naturally as a time-sensitive observable for an elementary oscillator. In this sense the square root of minus one is a clock and/or a clock/observer. This sheds new light on Wick rotation, which replaces $t$ (temporal quantity) by $it$. In this view, the Wick rotation replaces numerical time with elementary temporal observation. The relationship of this remark with the Heisenberg commutator $[P,Q]=i\hslash$ is explained. We discuss iterants - a generalization of the complex numbers as described above. This generalization includes all of matrix algebra in a temporal interpretation. We then give a generalization of the Feynman-Dyson derivation of electromagnetism in the context of non-commutative worlds. This generalization depends upon the definitions of derivatives via commutators and upon the way the non-commutative calculus mimics standard calculus. We examine constraints that link standard and non-commutative calculus and show how asking for these constraints to be satisfied leads to some possibly new physics.

Ver também

https://www.math.ist.utl.pt/seminars/qci/index.php.en?action=show&id=3243

23/11/2012, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática
Jeffrey Morton, Univ. Hamburg

Classifying Extended TQFT and the Cobordism Hypothesis

An overview of the concept of extended field theories, and a look at the role of the Cobordism Hypothesis (now more accurately the Cobordism Theorem) in classification of such theories. Given time the talk will touch on Jacob Lurie's proof of the Cobordism Hypothesis.

21/11/2012, 15:30 — 16:30 — Sala P4.35, Pavilhão de Matemática
Louis-Hadrien Robert, Institut Mathématique de Jussieu, Paris

Indecomposable modules over a Kuperberg-Khovanov algebras

The ${\mathrm{𝔰𝔩}}_3$ polynomial is a quantum invariant for knots. It has been categorified by Khovanov in 2004 in a TQFT fashion. The natural way to extend this categorification to an invariant of tangle is construct a $0+1+1$ TQFT. From this construction emerge some algebras called Khovanov-Kuperberg algebras (or ${\mathrm{𝔰𝔩}}_3$-web algebras) and some particular projective modules called web-modules over these algebras. I will give a combinatorial caracterisation of indecomposable web-modules.

21/11/2012, 14:00 — 15:00 — Sala P4.35, Pavilhão de Matemática
Marco Mackaay, Univ. Algarve

${\mathrm{𝔰𝔩}}_3$ web algebras, cyclotomic KLR algebras and categorical quantum skew Howe duality

I will introduce ${\mathrm{𝔰𝔩}}_3$ web algebras $K(S)$, which involve Kuperberg's ${\mathrm{𝔰𝔩}}_3$ web space $W(S)$ and Khovanov ${\mathrm{𝔰𝔩}}_3$ foams with boundary in $W(S)$. These algebras are the ${\mathrm{𝔰𝔩}}_3$ analogues of Khovanov's ${\mathrm{𝔰𝔩}}_2$ arc algebras. I will show how the $K(S)$ are related to cyclotomic Khovanov-Lauda-Rouquier algebras (cyclotomic KLR algebras, for short) by a categorification of quantum skew Howe duality. This talk is closely related to the next one by Robert. In particular, I will show that the Grothendieck group of $K(S)$ is isomorphic to $W(S)$ and that, under this isomorphism, the indecomposable projective $K(S)$-modules, which Robert constructs explicitly, correspond precisely to the dual canonical basis elements in $W(S)$.

21/11/2012, 11:00 — 12:00 — Sala P4.35, Pavilhão de Matemática
Volodymyr Mazorchuk, Univ. Uppsala

The endomorphism category of a cell 2-representation

Fiat 2-categories are 2-analogues of finite dimensional algebras with involutions. Cell 2-representations of fiat 2-categories are most appropriate analogues for simple modules over finite dimensional algebras. In this talk I will try to describe (under some natural assumptions) a 2-analogue of Schur's Lemma which asserts that the endomorphism category of a cell 2-representation is equivalent to the category of vector spaces. This is applicable, for example to the fiat category of Soergel bimodules in type A. This is a report on a joint work with Vanessa Miemietz.

28/09/2012, 14:00 — 15:00 — Sala P4.35, Pavilhão de Matemática
Nuno Freitas, Univ. Barcelona

Fermat-type equations of signature \((13,13,p)\) via Hilbert cuspforms

In this talk I will give an introduction to the modular approach to Fermat-type equations via Hilbert cuspforms and discuss how it can be used to show that certain equations of the form $x^{13}+y^{13}=C{z}^p$ have no solutions $(a,b,c)$ such that $\gcd (a,b)=1$ and $13\nmid c$ if $p>4992539$. We will first relate a putative solution of the previous equation to the solution of another Diophantine equation with coefficients in $Q(\sqrt{13})$. Then we attach Frey curves $E$ over $Q(\sqrt{13})$ to solutions of the latter equation. Finally, we will discuss on the modularity of $E$ and irreducibility of certain Galois representations attached to it. These ingredients enable us to apply a modular approach via Hilbert newforms to get the desired arithmetic result on the equation.

09/05/2012, 16:15 — 17:15 — Sala P3.10, Pavilhão de Matemática
Anne-Laure Thiel, Instituto Superior Técnico

Diagrammatic categorification of extended Hecke algebra and quantum Schur algebra of affine type A

Joint work with Marco Mackaay.

09/05/2012, 14:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Marco Mackaay, Univ. Algarve

\(\mathfrak{sl}_3\) web algebras

This is joint work with Weiwei Pan and Daniel Tubbenhauer from Gottingen University, Germany.

12/01/2012, 11:30 — 12:30 — Sala P12, Pavilhão de Matemática
Jeffrey C. Morton, Instituto Superior Técnico

Groupoidification and Khovanov's Categorification of the Heisenberg Algebra

The aim of this talk is to describe the connection between two approaches to categorification of the Heisenberg algebra. The groupoidification program of Baez and Dolan has been used to give a representation of the quantum harmonic oscillator in the category Span(Gpd) where the Fock space is represented by the groupoid of finite sets and bijections. This naturally gives a combinatorial interpretation of the (one-variable) Heisenberg algebra in the endomorphisms of this groupoid. On the other hand, Khovanov has given a categorification in which the integral part of the (many variable) Heisenberg algebra is recovered as the Grothendieck ring of a certain monoidal category described in terms of a calculus of diagrams. I will describe how an extension of the groupoidification program to a 2-categorical form of Span(Gpd) recovers the relations used by Khovanov's construction, and how to interpret them combinatorially in terms of the groupoid of finite sets.

06/01/2012, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
Nuno Freitas, Universitat de Barcelona

From Fermat's Last Theorem to some generalized Fermat equations

The proof of Fermat's Last Theorem was initiated by Frey, Hellegouarch, Serre, further developed by Ribet and ended with Wiles' proof of the Shimura-Tanyama conjecture for semi-stable elliptic curves. Their strategy, now called the modular approach, makes a remarkable use of elliptic curves, Galois representations and modular forms to show that $a^p+b^p=c^p$ has no solutions, such that $(a,b,c)=1$ if $p\ge 3$. Over the last 17 years, the modular approach has been continually extended and allowed people to solve many other Diophantine equations that previously seemed intractable. In this talk we will use the equation $x^p+2^{\alpha }{y}^p=z^p$ as the motivation to introduce informally the original strategy ($\alpha =0$) and illustrate one of its first refinements (for $\alpha =1$). Then we will discuss some further generalizations that recently led to the solution of equations of the form $x^5+y^5=d{z}^p$.

06/12/2011, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
John Huerta, Australian National University, Canberra

Superstrings, higher gauge theory, and division algebras

Recent work on higher gauge theory suggests the presence of 'higher symmetry' in superstring theory. Just as gauge theory describes the physics of point particles using Lie groups, Lie algebras and bundles, higher gauge theory is a generalization that describes the physics of strings and membranes using categorified Lie groups, Lie algebras and bundles. In this talk, we will summarize the mathematics of a higher gauge theory. We then show how to construct the categorified Lie algebras relevant to superstring theory by a systematic use of the normed division algebras. At the end, we will touch on how this leads to a categorified supergroup extending the Poincare supergroup in the mysterious dimensions where the classical superstring makes sense — 3, 4, 6 and 10.

09/11/2011, 14:00 — 15:00 — Sala P4.35, Pavilhão de Matemática
Pedro Vaz, Instituto Superior Técnico

Categorified $q$-Schur algebra and the BMW algebra

In 1989 François Jaeger showed that the the Kauffman polynomial of a link $L$ can be obtained as a weighted sum of HOMFLYPT polynomials on certain links associated to $L$. In this talk I will explain how to use a version of Jaeger's theorem to stablish a connection between the $\mathrm{SO}(2N)-\mathrm{BMW}$ and the $q$-Schur algebras. I will then present a subcategory of the Schur category which categorifies the $\mathrm{SO}(2N)-\mathrm{BMW}$ algebra (joint with E. Wagner).

28/10/2011, 10:30 — 11:30 — Sala P3.10, Pavilhão de Matemática
Jeffrey Morton, Instituto Superior Técnico

Extended TQFT in a Bimodule 2-Category

I will describe an extended (2-categorical) topological QFT with target 2-category consisting of C*-algebras and bimodules. The construction is explained as factorizable into a classical field theory valued in groupoids, and a quantization functor, as in the program of Freed-Hopkins-Lurie-Teleman. I will explain the Lagrangian action functional in terms of cohomological twisting of the groupoids in the classical part of the theory, and describe how this is incorporated into the quantization functor. This project is joint work with Derek Wise.

28/07/2011, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática
Andreas Döring, Univ. of Oxford

Towards Noncommutative Gel'fand Duality

Gel'fand-Naimark duality (1943) between the categories of unital commutative $C^{*}$-algebras and compact Hausdorff spaces is a key insight of 20th century mathematics, providing an enormously useful bridge between algebra on the one hand and topology and geometry on the other. Many generalisations and related dualitites exist, in logical, localic and constructive forms. Yet, all this is for commutative algebras (and distributive lattices of projections, or opens), while in quantum theory and in a large variety of mathematical situations, noncommutative algebras play a central role. A good, generally useful notion of spectra of noncommutative algebras is still lacking. Clearly, such spectra will be of considerable interest for physics and Noncommutative Geometry.

I will report on recent progress towards defining spectra of noncommutative operator algebras, mostly for von Neumann algebras. This work comes from the approach using topos theory to reformulate quantum physics (C. Isham, AD), where a presheaf or sheaf topos is assigned to each noncommutative operator algebra, together with a distinguished spectral object. It will be shown that this assignment is functorial, and that the spectral object determines the algebra up to Jordan isomorphisms (J. Harding, AD). Progress on characterising the action of the unitary group of an algebra - relating to Lie group and Lie algebra aspects - is presented. Moreover, recently established connections with Zariski geometries from geometric model theory will be sketched (B. Zilber, AD).

This is joint work with John Harding, Boris Zilber, and Chris Isham.

28/07/2011, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Yasuyoshi Yonezawa, Univ. of Bonn

A specialized Kauffman polynomial using 4-valent planar diagrams

I want to discuss about specialized Kauffman polynomial using 4-valent planar diagrams. A problem is can we categorify this polynomial.

13/07/2011, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática
João Faria Martins, Univ. Nova de Lisboa

Categorifying the Knizhnik-Zamolodchikov connection

(Joint with Lucio Simone Cirio, Max Planck Institute for Mathematics)
In the context of higher gauge theory, we categorify the Knizhnik-Zamolodchikov connection in the configuration space of $n$ particles in the complex plane by means of a differential crossed module of (totally symmetric) horizontal 2-chord diagrams, categorifying the 4-term relation.

We carefully discuss the representation theory of differential crossed modules in chain-complexes of vector spaces, inside which we formulate the notion of infinitesimal 2-R matrix, an infinitesimal counterpart of some of the relations satisfied by braid cobordisms.

We present several open problems.

13/07/2011, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Stephen Huggett, University of Plymouth

Seifert graphs, Eulerian subgraphs, and partial duality

I will describe how to characterize the Seifert graphs of link diagrams, and how, using the notion of partial duality in ribbon graphs, this leads to a generalization of the classical result that a plane graph is Eulerian if and only if its geometric dual is bipartite. This is joint work with Iain Moffatt and Natalia Virdee.

08/06/2011, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática
Marco Mackaay, Univ. of the Algarve

The Schur quotient of the Khovanov-Lauda categorification of quantum ${\mathrm{sl}}_n$ and colored HOMFLY homology

In my talk I will first remind everyone of the relation between quantum $\mathrm{sln}$, the $q$-Schur algebra and colored HOMFLY homology. After that I will explain how these objects and the relation between them have been categorified.

08/06/2011, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alistair Savage, Univ. of Ottawa, Canada

Hecke algebras and a categorification of the Heisenberg algebra

In this talk, we will present a graphical category in terms of certain planar braid-like diagrams. The definition of this category is inspired by the representation theory of Hecke algebras of type A (which are certain deformations of the group algebra of the symmetric group). The Heisenberg algebra (in infinitely many generators), which plays an important role in the description of certain quantum mechanical systems, injects into the Grothendieck group of our category, yielding a "categorification" of this algebra. We will also see that our graphical category acts on the category of modules of Hecke algebras and of general linear groups over finite fields. Additionally, other algebraic structures, such as the affine Hecke algebra, appear naturally.

We will assume no prior knowledge of Hecke algebras or the Heisenberg algebra. The talk should be accessible to graduate students. This is joint work with Anthony Licata and inspired by work of Mikhail Khovanov.