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Nuno Freitas, Universitat de Barcelona

From Fermat's Last Theorem to some generalized Fermat equations

Topological Quantum Field Theory Seminar

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 ${x}^{5}+{y}^{5}=d{z}^{p}$.

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.

Marco Mackaay, Univ. Algarve

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

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

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.

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 $\mathrm{gcd}(a,b)=1$ 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.

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.

Marco Mackaay, Univ. Algarve

${\mathrm{\U0001d530\U0001d529}}_{3}$ web algebras, cyclotomic KLR algebras and
categorical quantum skew Howe duality

I will introduce ${\mathrm{\U0001d530\U0001d529}}_{3}$ web algebras $K(S)$, which
involve Kuperberg's ${\mathrm{\U0001d530\U0001d529}}_{3}$ web space $W(S)$ and Khovanov
${\mathrm{\U0001d530\U0001d529}}_{3}$ foams with boundary in $W(S)$. These algebras are
the ${\mathrm{\U0001d530\U0001d529}}_{3}$ analogues of Khovanov's ${\mathrm{\U0001d530\U0001d529}}_{2}$ 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
$W(S)$.

Louis-Hadrien Robert, Institut Mathématique de Jussieu, Paris

Indecomposable modules over a Kuperberg-Khovanov algebras

The ${\mathrm{\U0001d530\U0001d529}}_{3}$ 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
${\mathrm{\U0001d530\U0001d529}}_{3}$-web algebras) and some particular projective
modules called web-modules over these algebras. I will give a
combinatorial caracterisation of indecomposable web-modules.

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.

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.

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.