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

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.

Joint work with Marco Mackaay.

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.

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.

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 $W(S)$.

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.

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.

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.

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.