###
30/09/2013, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Nuno Freitas, *Univ. Bayreuth*

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The Fermat equation over totally real number fields

Jarvis and Meekin have shown that the classical Fermat equation
\(x^p + y^p = z^p\) has no non-trivial solutions over
\(\mathbb{Q}(\sqrt{2})\). This is the only result available over
number fields. Two major obstacles to attack the equation over
other number fields are the modularity of the Frey curves and the
existence of newforms in the spaces obtained after level
lowering.

In this talk, we will describe how we deal with these
obstacles, using recent modularity lifting theorems and level
lowering. In particular, we will solve the equation for infinitely
many real quadratic fields.

###
29/08/2013, 15:00 — 16:00 — Sala P4.35, Pavilhão de Matemática

Travis Willse, *The Australian National University*

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Groups of type ${G}_{2}$ and exceptional geometric structures in
dimensions 5, 6, and 7

Several exceptional geometric structures in dimensions 5, 6, and 7
are related in a striking panorama grounded in the algebra of the
octonions and split octonions. Considering strictly nearly Kähler
structures in dimension 6 leads to prolonging the Killing-Yano (KY)
equation in this dimension, and the solutions of the prolonged
system define a holonomy reduction to a group of exceptional type
${G}_{2}$ of a natural rank-7 vector bundle, which can in turn be
realized as the tangent bundle of a pseudo-Riemannian manifold,
which hence relates this construction to exceptional metric
holonomy. In the richer case of indefinite signature, a suitable
solution $\omega $ of the KY equation can degenerate along a (hence
5-dimensional) hypersurface $\Sigma $, in which case it partitions
the underlying manifold into a union of three submanifolds and
induces an exceptional geometric structure on each. On the two open
manifolds (which have common boundary $\Sigma $), $\omega $ defines
asymptotically hyperbolic nearly Kähler and nearly para-Kähler
structures. On $\Sigma $ itself, $\omega $ determines a generic
$2$-plane field, the type of structure whose equivalence problem
Cartan investigated in his famous Five Variables paper. The
conformal structure this plane field induces via Nurowski's
construction is a simultaneous conformal infinity for the nearly
(para-)Kähler structures.
This project is a collaboration with Rod Gover and Roberto
Panai.

###
20/06/2013, 11:00 — 12:00 — Sala P4.35, Pavilhão de Matemática

John Huerta, *IST, Lisbon*

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QFT V

In the final lecture of our gentle introduction to quantum field
theory, we discuss the renormalization of phi cubed theory at one
loop.

###
14/06/2013, 11:00 — 12:30 — Sala P3.10, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

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QFT IV

We will introduce Feynman diagrams by studying finite-dimensional Gaussian integrals and their perturbations, leading up to phi-cubed theory.

###
24/04/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

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QFT III

Last time, we talked about quantization of the free scalar field by
replacing the modes of the field by quantum oscillators. Now, we
put this field into the form used by physicists, and talk about the
Wightman axioms, which allow a rigorous treatment of free fields.

###
17/04/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

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QFT II

We continue our gentle introduction to quantum field theory for
mathematicians. We discuss the Klein-Gordon equation, and how it
decomposes into oscillators. We quantize this system by quantizing
the oscillators, obtaining the free scalar field, the simplest
quantum field there is.

###
10/04/2013, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

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QFT I

This series of lectures will be a gentle introduction to quantum
field theory for mathematicians. In our first lecture, we give a
lightning introduction to quantum mechanics and discuss the
simplest quantum system: the harmonic oscillator. We then sketch
how this system is used to quantize the free scalar field.

###
03/04/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

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Anomalies IV

We will introduce the notion of stable isomorphism for gerbes, and
talk about how stable isomorphism classes are in one-to-one
correspondence with Deligne cohomology classes. We define WZW
branes and discuss how the basic gerbe on a group trivializes when
restricted to the brane.

###
20/03/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Aleksandar Mikovic, *Univ. Lusófona*

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Categorification of Spin Foam Models

We briefly review spin foam state sums for triangulated
manifolds and motivate the introduction of state sums based on
2-groups. We describe 2-BF gauge theories and the construction of
the corresponding path integrals (state sums) in the case of
Poincaré 2-group.

#### References

- J. F. Martins and A. Mikovic,
*Lie crossed modules and
gauge-invariant actions for 2-BF theories*, Adv. Theor. Math.
Phys. 15 (2011) 1059,
arxiv:1006.0903
- A. Mikovic and M. Vojinovic,
*Poincaré 2-group and quantum
gravity*, Class. Quant. Grav. 291 (2012) 165003, arxiv:1110.4694

###
13/03/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

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Anomalies III

We continue examining Gawedzki and Reis's paper:
WZW branes and gerbes,
http://arxiv.org/abs/hep-th/0205233

We define a gerbe, and show gerbes can be "transgressed" to give
line bundles over loop space. Trivial gerbes give trivial bundles
on loop space, whose sections are thus mere functions. Any compact,
simply connected Lie group comes with a god-given gerbe whose
curvature is the canonical invariant 3-form. Restricting this gerbe
to certain submanifolds, we get trivial gerbes who thus transgress
to trivial line bundles, "cancelling" the anomaly of a nontrivial
line bundle.

###
27/02/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

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Anomalies II

We continue our informal discussion of anomalies by talking about
global anomalies on branes, and their relationship with gerbes.

###
06/02/2013, 14:00 — 15:00 — Sala P5.18, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

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Introduction to anomalies

In physics, an "anomaly" is the failure of a classical symmetry at
the quantum level. Anomalies play a key role in assessing the
consistency of a quantum field theory, and link up with cohomology
in mathematics, a general tool by which mathematicians understand
whether a desired construction is possible. In this informal series
of talks, we aim to understand what physicists mean by an "anomaly"
and their mathematical interpretation.

###
19/12/2012, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática

Sara Tavares, *Univ. of Nottingham*

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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*

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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

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

Jeffrey Morton, *Univ. Hamburg*

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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*

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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.

###
21/11/2012, 14:00 — 15:00 — Sala P4.35, Pavilhão de Matemática

Marco Mackaay, *Univ. Algarve*

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${\mathrm{\U0001d530\U0001d529}}_{3}$ 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
$W(S)$.

###
21/11/2012, 11:00 — 12:00 — Sala P4.35, Pavilhão de Matemática

Volodymyr Mazorchuk, *Univ. Uppsala*

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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*

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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.

###
09/05/2012, 16:15 — 17:15 — Sala P3.10, Pavilhão de Matemática

Anne-Laure Thiel, *Instituto Superior Técnico*

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

Joint work with Marco Mackaay.