### INTRODUCTION

LOGIN INFORMATION TO ZOOM: The lectures will be streamed in Zoom and later uploaded to this web page.
The access information is the following:

Link: https://u-bordeaux-fr.zoom.us/j/84518699079?pwd=TkNUZGtKbkVSdk5nZ2lzeUlyUC94QT09

Meeting ID: 845 1869 9079

Password: WBGRTP

This workshop is meant to give a formal introduction to
relevant topics in geometry, representation theory and
its applications to physics. In this Workshop, young
researcher in these domains will give 3 to 4 hour long
lectures during the last week of July 2022. In between
lectures, there will be special speakers that will talk
about more specific subjects like the geometry of AdS or
classification problems in Algebraic Geometry.
This free-to-enter workshop is targeted towards young
researchers in mathematics, mathematical physics or
theoretical physics but can be attended by anyone
interested on theses domain.

Sign up to attend presencially is no longer available. If you are interested check the online information.

LOCATION: The workshop will take place at
the IMUB's room, in the last floor of the Facultat de Matemàtiques i Informàtica.

Edifici Històric UB.

Gran Via de les Corts Catalanes, 585, 08007 Barcelona

Poster for the event

### LECTURERS

#### Ivano Basile. String theory and characters

Ivano Basile obtained his PhD at Scuola Normale Superiore,
Pisa, under the supervision of Augusto Sagnotti. He worked
on supersymmetry breaking in string theory. Currently he is
a postdoctoral researcher at UMONS, and will join the string
theory group at LMU and Max Planck Institute in Munich in
the fall. His academic interest string theory, holography,
swampland program.
The plan of this talk is to introduce strings in flat spacetime,
focusing on the representation theory of the free spectrum.
The consistency conditions on these spectra are reflected in
the remarkable modular properties of the corresponding partition
functions, and lead to just a handful of possible consistent models,
among which the celebrated superstrings in ten dimensions.

#### Thomas Basile. Introduction to deformation quantization of Poisson manifolds

Thomas Basile is starting his secod PostDoc on Higher spin Gravity
at the University of Mons. His interests include algebraic aspects
of higher spin gravity, deformation quantization, and its relation
to topological sigma models.
The lectures that he will provide is about deformation quantizaton
of Poisson maniolds. The latter consists in deforming the structure
of associative algebra of the algebra of functions on a Poisson
manifold. He will also discuss the so-called formality theorem.
More precisely, we will discuss the precise content of the formality
theorem, and explain how it implies the existence of a deformation
quantization for all Poisson manifolds.

#### Anna Blanco Cabanillas. Classification of plane Cremona maps

She graduated with a double Bachelor’s in mathematics and physics at
the University of Barcelona last February. Her interests are p-adic
representations of Galois groups, algebraic methods in number theory,
and applications of Langland's program in physics.
Plane cremona maps are mappings between projective planes with some special properties.
The classification of those type of maps are an important tool in algebraic geometry. In this talks we will
introduce key ideas in the field of algebraic geometry and then dive a little more into the particular case
of plane Cremona maps.

#### Roberto Emparan. Geometric Aspects in AdS/CFT

Roberto Emparan is ICREA research professor at the Institute of
Cosmos Sciences of the University of Barcelona since 2003. He
received his PhD from the University of the Basque Country in
1995, and then held postdoctoral positions at UC Santa Barbara,
Durham University, and CERN. His research lies at the interface
between General Relativity and String Theory, focusing on the
classical, quantum, and stringy properties of black holes in
different numbers of dimensions, subjects on which he has
authored more than 120 articles. In 2016 he received an
Advanced Grant of the European Research Council. In 2022
he has been elected Fellow the International Society of
General Relativity and Gravitation. He is a member of
the editorial board of Living Reviews in Relativity, Journal
of High Energy Physics, and Advances in Theoretical and
Mathematical Physics.
In his talk he will describe some of the geometric features
that underlie the AdS/CFT correspondence, which provides a
holographic quantum theory of gravity. He will discuss, at a
very elementary level, how several particular properties of
the geometry of Anti-deSitter space suggest the possibility
that a theory with conformal invariance exists at its boundary
– not a proof of the AdS/CFT correspondence, but a necessary
condition for it. I will then discuss how the long-distance
(infinite size) divergences in AdS can be interpreted as
short-distance (ultraviolet) divergences in a quantum conformal
theory.

#### Josh'O'Connor. Kac-Moody algebras and physics

Josh O’Connor is a PhD student of theoretical physics at the
University of Mons in Belgium. His previous studies focused
on pure mathematics and, later, theoretical physics. Some of
his interests include Kac-Moody algebras, dualisation,
mixed-symmetry fields and algebraic structure in physics.
He will provide lectures which will focus on the elementary
structure theory of Lie algebras and, more generally, Kac-Moody
algebras. Also, some relationships between physics and these
algebras will be discussed.

#### David Mateos. Dynamical aspects in AdS/CFT

David Mateos obtained his PhD from the University of Barcelona
in 2000. After that he held postdoctoral positions at Cambridge
University, Perimeter Institute and the University of California
at Santa Barbara. He has been an ICREA Research Professor at the
Institute of Cosmos Sciences at the University of Barcelona since
July 2008. In 2012 he was awarded a Starting Grant from the
European Research Council. His main goal is to understand the
physics of the Universe at the most fundamental level. This
requires a unified, quantum theory of all interactions. For this
reason he works on string theory, a quantum theory that has the
potential to describe all forces and particles in Nature in a
single, unified framework.
In this talk he will describe some dynamical aspects of the
AdS/CFT correspondence, possibly the most important development
in theoretical physics in the last 25 years.

#### Ignasi Mundet. Groups acting on manifolds: finite groups vs Lie groups

Ignasi Mundet is Associate Professor at the University of Barcelona.
A recurrent topic in his research is the notion of group actions
on manifolds. Group actions have quite different flavors depending on
the sort of group that is acting (compact connected Lie group, complex
reductive Lie group, gauge group, finite group, etc) and on the geometry of
the manifold that supports the action (topological, smooth, Kaehler, or
Riemannian manifolds, algebraic varieties, etc). Trying to compare actions
of different sorts of groups on the same manifold usually leads to nice
problems. In my talk I plan to discuss the relation between finite group
actions and actions of compact connected Lie groups, mainly (but not only)
on topological manifolds. If a manifold supports an action of the circe,
then it also supports an action of any finite cyclic group, because any
finite cyclic group is a subgroup of the circle. Is the converse
true? In general no. However, I will explain some results which roughly
speaking imply that if a manifold supports actions of finite groups which
are very big in an appropriate sense, then it also supports actions of tori.

#### Joan Carles Naranjo. Rationality problems in Algebraic Geometry

Joan Carles Naranjo is a permanent researcher at the University of
Barcelona since 1993. He is also a member of the Centre de Recerca
Matemàtica. His area of expertise is Algebraic Geometry, more
specifically he works on Abelian varieties, algebraic curves,
irregular varieties and cycle theory.
A rational variety is an algebraic variety which is birational to
an affine space. It is a hard and fundamental problem to recognize
which varieties are rational. Historically, rationality problems
have been of great importance in Algebraic Geometry and have
motivated fundamental developments, as the theory of Riemann
surfaces, and the Abel–Jacobi map. Castelnuovo’s solution of the
so-called Lüroth problem in dimension 2 and the counterexamples
to the Lüroth problem in dimension 3 are milestones in the
development of the area. The aim of the talk is to give an
elementary survey to some old and new methods used in this
context.

#### Simon Pekar. Higher-spin diffeomorphisms

Simon Pekar is a PhD student at University of Mons. His academic
interest consists in algebra, geometry, asymptotic symmetries,
holography, higher-spin theories and Carrollian physics.
During the lectures we will discuss the symmetries of higher-spin
gravity theories, which are classical theories of gravity,
including gauge fields with spin strictly greater than two.
In particular, we will take a look at higher-spin diffeomorphisms
that are gauge transformations of those fields. After a review
of higher-spin theories and the representation of their algebra
of infinitesimal gauge transformations, we discuss in details
the obstructions to the lift of these representations to those
of a group. We also explain how to bypass this problem by
elaborating on results from symplectic geometry and deformation
quantization.

#### Mattia Serrani. Higher Spin algebras

Mattia Serrani did his Master in Theoretical Physics at the University
of Pisa in December 2020. Currently he is a PhD Student at University
of Mons. His interest is mainly in physics and pure math, more
precisely, in Higher spin algebras, Constrained Hamiltonian Systems,
AdS/CFT and Geometry.
The aim of this talk is to create a bridge between math and physics
regarding the study of higher spin algebras. First, I will give some
Physical motivation for the interest on these algebras. Secondly, we
will define this algebras in various dimensions, with a particular
attention in the case of 3 and 4 dimensions. Also, we will interpret
them physically as global symmetries or as a gauge symmetry if we
add one more dimension to our system. Lastly, we will see explicitly
how to construct them and illustrate it with examples.

#### Enric Solé-Farré. Mathematical applications of Gauge Theory

Enric Solé-Farré is a PhD student at the London School of Geometry and Number
Theory, under the supervision of Lorenzo Foscolo and Simon Donaldson. His
research interests are mainly mathematical gauge theory and special holonomy.
In his talk, he will discuss some well-known applications of gauge theory
in geometry and outline some modern conjectures related to these results.

#### Tung Tran. Deformation of complex structures

Tung Tran his currently a PostDoc in theoretical physics at the
University of Mons in Belgium. His academic interst is mainly
about Higher-spin theories, twistor theory, and deformation theory.
In his talk, we will study the deformation of complex structures
on complex manifolds based on the Spencer-Kodaira theory, which
has significant applications in twistor theory. In particular, we
will show how the deformation of twistor geometry can lead to
various self-dual theories on spacetime via the Penrose transform.