# Seminars and short courses

Seminars, for informal dissemination of research results, exploratory work by research teams, outreach activities, etc., constitute the simplest form of meetings at a Mathematics research centre.

CAMGSD has recorded its seminars for a long time, this page serving both as a means of public announcement of forthcoming activities but also as a historic record.

For a full search interface see the Mathematics Department seminar page. Here you will be restricted to lists of forthcoming CAMGSD seminars for the next two weeks or to a given year.

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### 20/12/2013, Friday

#### 11:30, Geometria em Lisboa

Nuno Romão, University of Göttingen.

I will describe joint work with M. Boekstedt and C. Wegner aiming at uncovering fundamental features of $$N=(2,2)$$ supersymmetric quantum mechanics on moduli spaces of vortices on compact Riemann surfaces, in analogy with the spectrum of quantum dyon-monopole bound states that emerged in connection with Sen's S-duality conjectures in the 1990s. My focus in this talk will be on the (topological A-twisted) supersymmetric Abelian Higgs model coupling to local systems, for both linear and nonlinear targets; the corresponding ground states can be investigated by means of the theory of $$L^2$$-invariants applied to the natural Kaehler metrics on the moduli spaces. I shall explain why the quanta of such Abelian gauge theories can nontrivially realize non-Abelian statistics in particular examples, and motivate a precise conjecture regarding the nonlinear superposition of ground states.

### 18/12/2013, Wednesday

#### 16:30, Topological Quantum Field Theory

Marko Vojinovic, Grupo de Fisica Matemática, Universidade de Lisboa.

We will give an overview of the renormalization procedure in Quantum Field Theory. The emphasis will be on the general idea of constructing a finite QFT from the one plagued by divergencies, in the standard perturbative approach, and discussing the uniqueness of the resulting QFT. The lecture does not assume much background knowledge in QFT, and should be accessible to a wide audience.

### 11/12/2013, Wednesday

#### 17:00, Topological Quantum Field Theory

Carlos Guedes, AEI, Golm-Potsdam.

The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. In quantum mechanics on the Euclidean space, the standard Fourier transform gives a unitary map between the position representation -- functions on the configuration space -- and the momentum representation -- functions on the corresponding cotangent space. That is no longer the case for systems whose configuration space is a more general Lie group. In this talk I will introduce a notion of Fourier transform that extends this duality to arbitrary Lie groups.

arXiv:1301.7750

### 10/12/2013, Tuesday

#### 16:30, Geometria em Lisboa

João Pimentel Nunes, Instituto Superior Técnico.

We will describe an approach, inspired by geometric quantization, to the complexification of real analytic Hamiltonian flows on Kahler manifolds. The flows correspond to geodesics on the space of Kahler metrics, as suggested by Semmes and Donaldson. We will look at some examples, such as cotangent bundles of compact Lie groups. Based on joint work with J. Mourão.

### 09/12/2013, Monday

#### 16:30, String Theory

Kiril Hristov, Università degli Studi di Milano-Bicocca.

The microscopic description of the 4 dimensional supersymmetric (or BPS) black holes in flat space has already been well understood via the AdS/CFT correspondence on the black hole horizon. In this talk I address the similar problem of finding the dual description for BPS black holes in $$AdS_4$$. The gravity picture of a flow between asymptotic $$AdS_4$$ and $$AdS_2 \times S^2$$ on the horizon can be understood as a renormalization group (RG) flow between a 3d and a 1d superconformal field theory. I discuss in some detail both the supergravity and the field theory side, providing evidence for their precise match. At the end I present a proposal for the 1d CFT states that make up the black hole entropy.

### 04/12/2013, Wednesday

#### 16:30, Topological Quantum Field Theory

Quantum mechanics in phase space: The Schrödinger and the Moyal representations.

I will present some recent results on the dimensional extension of pseudo-differential operators. Using this formalism it is possible to generalize the standard Weyl quantization and obtain, in a systematic way, several phase space (operator) representations of quantum mechanics. I will present the Schrodinger and the Moyal phase space representations and discuss some of their properties, namely in what concerns the relation with deformation quantization.

### 04/12/2013, Wednesday

#### 15:00, Operator Theory, Complex Analysis and Applications

Antti Perälä, University of Helsinki, Finland.
Optimal bounds for analytic projections.

We discuss some recent advances related to size estimates of analytic projections and the possible uses for such estimates in applications. The spaces considered include Hardy, Bergman, Bloch, Besov and Segal-Bargmann spaces. We study in detail the case of Bergman projection onto the maximal and minimal Möbius invariant spaces.

### 02/12/2013, Monday

#### 16:30, String Theory

David Berman, Queen Mary College London.
Duality Symmetric String and M-Theory.

We review recent developments in duality symmetric string theory. We begin with the world sheet doubled formalism which describes strings in an extended space time with extra coordinates conjugate to winding modes. This formalism is T-duality symmetric and can accommodate non-geometric T-fold backgrounds which are beyond the scope of Riemannian geometry. Vanishing of the conformal anomaly of this theory can be interpreted as a set of spacetime equations for the background fields. These equations follow from an action principle that has been dubbed Double Field Theory (DFT). We review the aspects of generalised geometry relevant for DFT. We outline recent extensions of DFT and explain how, by relaxing the so-called strong constraint with a Scherk Schwarz ansatz, one can obtain backgrounds that simultaneously depend on both the regular and T-dual coordinates. This provides a purely geometric higher dimensional origin to gauged supergravities that arise from non-geometric compactification. We then turn to M-theory and describe recent progress in formulating an $$E_{n(n)}$$ U-duality covariant description of the dynamics. We describe how spacetime may be extended to accommodate coordinates conjugate to brane wrapping modes and the construction of generalised metrics in this extend space that unite the bosonic fields of supergravity into a single object. We review the action principles for these theories and their novel gauge symmetries. We also describe how a Scherk Schwarz reduction can be applied in the M-theory context and the resulting relationship to the embedding tensor formulation of maximal gauged supergravities.

### 27/11/2013, Wednesday

#### 16:30, Topological Quantum Field Theory

John Huerta, Instituto Superior Técnico.
What can higher categories do for physics?

We describe Baez and Dolan's cobordism hypothesis - a deep connection between topological quantum field theory, higher categories, and manifolds. Physically, this encodes the idea that quantum field theories, even "topological" ones, should be local: no matter how we cut up the spacetime on which they are defined in order to perform the path integral, the net result must be the same. Recently, this hypothesis was formulated and proved by Jacob Lurie using the tools of homotopy theory. We describe the version of the hypothesis he proved. Finally, we touch on Freed, Hopkins, Lurie and Teleman's recent work on Chern-Simons theory, and on Urs Schreiber's ideas for using Lurie's toolkit in full-fledged quantum field theory.

### 25/11/2013, Monday

#### 16:30, String Theory

Paul Richmond, University of Oxford.
Localization on Three-Manifolds.

We consider supersymmetric gauge theories on Riemannian three-manifolds with the topology of a three-sphere. The three-manifold is always equipped with an almost contact structure and an associated Reeb vector field. We show that the partition function depends only on this vector field, giving an explicit expression in terms of the double sine function. In the large $$N$$ limit our formula agrees with a recently discovered two-parameter family of dual supergravity solutions. We also explain how our results may be applied to prove vortex-antivortex factorization. Finally, we comment on the extension of our results to three-manifolds with non-trivial fundamental group.

### 19/11/2013, Tuesday

#### 16:30, Geometria em Lisboa

Alfonso Zamora, Instituto Superior Técnico.
GIT characterizations of Harder-Narasimhan filtrations.

In a moduli space, usually, we impose a notion of stability for the objects and, when constructing the moduli space by using Geometric Invariant Theory, another notion of GIT stability appears, showing during the construction of the moduli that both notions do coincide at the stable and semistable level. For an object which is unstable (this is, contradicting the stability condition) there exists a canonical unique filtration, called the Harder-Narasimhan filtration. Onthe other habd, GIT stability is checked by 1-parameter subgroups, by the classical Hilbert-Mumford criterion, and it turns out that there exists a unique 1-parameter subgroup giving some notion of maximal unstability in the GIT sense. We show how to prove that this special 1-parameter subgroup can be converted into a filtration of the object and coincides with the Harder-Narasimhan filtration, hence both notions of maximal unstability are the same, for the moduli problem of classifying coherent sheaves on a smooth complex projective variety (cf. [GSZ]). A similar treatment can be used to prove the analogous correspondence for holomorphic pairs, Higgs sheaves, rank 2 tensors and finite dimensional quiver representations.

[GSZ] T. Gómez, I. Sols, A. Zamora, A GIT characterization of the Harder-Narasimhan filtration, arXiv:1112.1886v2, (Preprint 2012)

### 14/11/2013, Thursday

#### 14:00, Operator Theory, Complex Analysis and Applications

Abdelhamid Boussejra, Université Ibn Tofail, Kenitra, Morocco.
The Hua operators on homogeneous line bundles over bounded symmetric domains of tube type.

Let $𝒟=G/K$ be a bounded symmetric domain of tube type. We show that the image of the Poisson transform on the degenerate principal series representation of $G$ attached to the Shilov boundary of $𝒟$ is characterized by a $K$- covariant differential operator on a homogeneous line bundle over $𝒟$. As a consequence of our result we get the eigenvalues of the Casimir operator for Poisson transforms on homogeneous line bundles over $G/K$. This extends a result of Imemura and all on symmetric domains of classical type to all symmetric domains. Also we compute a class of Hua type integrals generalizing an earlier result of Faraut and Koranyi.

### 12/11/2013, Tuesday

#### 15:30, Partial Differential Equations

Edgard Pimentel, Instituto Superior Técnico.
Existence of classical solutions for time dependent mean-field games.

In this we consider time dependent mean-field games with a power-like dependence on the measure. We establish existence of smooth solutions under a certain set of conditions depending both on the growth of the Hamiltonian as well as on the dimension. This is done by combining a Gagliardo-Niremberg type of argument and the non-linear adjoint method with a new class of polynomial estimates for the Fokker-Planck equation.

This is a joint work with D. A. Gomes and H. Sanchez-Morgado.

### 06/11/2013, Wednesday

#### 15:00, Operator Theory, Complex Analysis and Applications

Sergey Naboko, University of Kent and St.Petersburg State University.
Spectral analysis of Jacobi Matrices and asymptotic properties of orthogonal polynomials.

We review basic features of the spectral theory of Hermitian Jacobi operators. The analysis is based on asymptotic properties of the related orthogonal polynomials at infinity for fixed spectral parameter. We discuss various examples of bounded and unbounded Jacobi matrices. This talk is meant to give an introduction to the theory of Jacobi matrices and orthogonal polynomials.

### 04/11/2013, Monday

#### 16:30, String Theory

Álvaro Osório, Instituto Superior Técnico.
Regularizing extremal black branes in gauged supergravity.

We consider extremal solutions in four-dimensional $$N=2$$ gauged supergravity. The main focus is on solutions for which no $$\operatorname{AdS}_2 \times \mathbb{R}^2$$ horizon can be found, but that instead possess pathological near-horizon geometries. By adding quantum corrections, we show that a regular horizon develops; furthermore we show that these horizons correspond to black brane solutions with $$\operatorname{AdS}_4$$ asymptotics. Finally we develop the appropriate entropy function formalism to incorporate the effect of higher-curvature corrections on similar solutions.

### 31/10/2013, Thursday

#### 16:30, Geometria em Lisboa

Tiago Fonseca, Laboratoire d'Annecy-le-Vieux de Physique Théorique, Université de Savoie.
The D-Kadomtsev–Petviashvili and its grassmannian description.

The Kadomtsev–Petviashvili (KP) equation is an important equation in fluid dynamics: it describes sea waves. It is an integrable model, meaning that we can find "all" solutions. In fact, the solutions are described by a purely geometric object, the grassmaniann. In the case of the equation D-KP (the orthogonal variation of KP) the solutions are described by isotropic grassmannians. In this talk, I'll explain how from an isotropic grassmannian we can build a solution for this equation.

### 29/10/2013, Tuesday

#### 16:30, Geometria em Lisboa

Gonçalo Oliveira, Imperial College.
Monopoles in Higher Dimensions.

The Monopole (Bogomolnyi) equations are Geometric PDEs in 3 dimensions. In this talk I shall introduce a generalization of the monopole equations to both Calabi Yau and $$G_2$$ manifolds. I will motivate the possible relations of conjectural enumerative theories arising from "counting" monopoles and calibrated cycles of codimension 3. Then, I plan to state the existence of solutions and sketch how these examples are constructed.

### 21/10/2013, Monday

#### 16:30, String Theory

Boris Pioline, CERN Geneva.
Quantum Hypermultiplet Moduli Spaces in $$N=2$$ String Vacua.

The hypermultiplet moduli space $$M_H$$ in type II string theories compactified on a Calabi-Yau threefold $$X$$ is largely constrained by supersymmetry (which demands quaternion-Kählerity), S-duality (which requires an isometric action of $$SL(2, \mathbb{Z})$$) and regularity. Mathematically, $$M_H$$ ought to encode all generalized Donaldson-Thomas invariants on $$X$$ consistently with wall-crossing, modularity and homological mirror symmetry. We review recent progress towards computing the exact metric on $$M_H$$, or rather the exact complex contact structure on its twistor space.

### 14/10/2013, Monday

#### 16:30, String Theory

Óscar Dias, Instituto Superior Técnico.
Gravitational Turbulence.

Anti-de Sitter (AdS) spacetime is linearly stable, but non-linearly unstable to a weak gravitational turbulent instability. Regardless of how weak the initial scalar or gravitational perturbation of AdS is, this instability forces the system to transfer energy from low to high frequency modes. This energy cascade is similar to what happens with the familiar process of turbulence. In a full time evolution, the scalar instability leads to the formation of a black hole and a similar endpoint is conjectured for the turbulent gravitational instability. In the context of the gravity/gauge theory correspondence, this gravitational instability of AdS provides a holographic description of quantum turbulence in the dual field theory.

### 30/09/2013, Monday

#### 15:00, Topological Quantum Field Theory

Nuno Freitas, Univ. Bayreuth.
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, Thursday

#### 15:00, Topological Quantum Field Theory

Travis Willse, The Australian National University.
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.

### 23/07/2013, Tuesday

#### 16:00, Colloquium

Univalent Foundations of Mathematics.

I will outline the main ideas of the new approach to foundations of practical mathematics which we call univalent foundations. Mathematical objects and their equivalences form sets, groupoids or higher groupoids. According to Grothendieck's idea higher groupoids are the same as homotopy types. Therefore mathematics may be considered as studying homotopy types and structures on them. Homotopy type theories, the underlying formal deduction system of the univalent foundations allows one to reason about such objects directly.

### 23/07/2013, Tuesday

#### 14:30, Geometria em Lisboa

The topology of a toric symplectic manifold can be read directly from its orbit space (a.k.a. moment polytope), and much the same is true of the topological generalizations of toric symplectic manifolds and projective toric varieties: quasitoric manifolds, topological toric manifolds and torus manifolds. An origami manifold is a manifold endowed with a closed 2-form with a very mild degeneracy along a hypersurface, and in the toric case its orbit space is an "origami polytope". In this talk we examine how the topology of a toric origami manifold can be read from its orbit space and how these results hold for the appropriate topological generalization of the class of toric origami manifolds, which includes quasitoric manifolds, and some torus manifolds. These results are from ongoing joint work with Tara Holm.

### 18/07/2013, Thursday

#### 18:00, Operator Theory, Complex Analysis and Applications

Carl Cowen, Indiana University-Purdue University Indianapolis, USA.
Rota's Universal Operators and Invariant Subspaces in Hilbert Spaces.

Rota showed, in 1960, that there are operators $T$ that provide models for every bounded linear operator on a separable, infinite dimensional Hilbert space, in the sense that given an operator $A$ on such a Hilbert space, there is $\lambda \ne 0$ and an invariant subspace $M$ for $T$ such that the restriction of $T$ to $M$ is similar to $\lambda A$. In 1969, Caradus provided a practical condition for identifying such universal operators. In this talk, we will use the Caradus theorem to exhibit a new example of a universal operator and show how it can be used to provide information about invariant subspaces for Hilbert space operators. Of course, Toeplitz operators and composition operators on the Hardy space ${H}^{2}\left(𝔻\right)$ will play a role!

This talk describes work in collaboration with Eva Gallardo-Gutiérrez, Universidad Complutense de Madrid, done there this year during the speaker's sabbatical.

### 18/07/2013, Thursday

#### 16:30, Geometria em Lisboa

Pavel Etingov, Massachusetts Institute of Technology.
D-modules on Poisson varieties and Poisson traces.

Let $$V$$ be an affine symplectic algebraic variety over $$\mathbb{C}$$, and $$G$$ a finite group of automorphisms of $$V$$ (for example, $$V$$ is a symplectic vector space, and $$G$$ is a subgroup of $$\mathop{Sp}(V)$$). Let $$A$$ be the algebra of regular functions on $$V/G$$, and $$E$$ be the space of linear functionals on $$A$$ which are invariant under Hamiltonian vector fields on $$V/G$$ (so called Poisson traces). It turns out that $$E$$ is finite dimensional. I will explain how to prove and generalize this statement, using the theory of D-modules, and will also describe some applications to noncommutative algebra. This is joint work with Travis Schedler.

### 18/07/2013, Thursday

#### 16:30, Operator Theory, Complex Analysis and Applications

David Krejcirik, Nuclear Physics Institute ASCR, Czech Republic.
The Brownian traveller on manifolds.

We study the inﬂuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the ﬂat case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the ﬂat case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

1. Martin Kolb and David Krejcirik: The Brownian traveller on manifolds, J. Spectr. Theory, to appear; preprint on arXiv:1108.3191 [math.AP].

### 17/07/2013, Wednesday

#### 16:30, Geometria em Lisboa

Marta Batoreo, UC Santa Cruz.
On hyperbolic points and periodic orbits of symplectomorphisms.

In this talk, we discuss a variant of the Conley conjecture which asserts the existence of infinitely many periodic orbits of a symplectomorphism if it has a fixed point which is unnecessary in some sense. More specifically, we discuss a result claiming that, for a certain class of closed monotone symplectic manifolds, any symplectomorphism isotopic to the identity with a hyperbolic fixed point must necessarily have infinitely many periodic orbits as long as the symplectomorphism satisfies some constraints on the flux.

### 17/07/2013, Wednesday

#### 16:30, Algebra

Anthony Blanc, Université de Montpellier 2.
Topological K-theory of complex non-commutative Spaces.

It was known for some time by Bondal and Toën that an appropriate notion of topological K-theory of dg-categories will furnish a candidate for a rational structure on the periodic cyclic homology of a smooth and proper dg-category. The main motivation comes from the conjecture by Katzarkov-Kontsevich-Pantev that there exists a pure non-commutative Hodge structure on the periodic homology of a smooth and proper dg-algebra. I will present a meaningful definition of topological K-theory of dg-categories over the complex, using the topological Betti realization functor. This definition is based on non-trivial results involving a generalization of Deligne's proper cohomological descent. Finally I will talk about the case of finite dimensional algebras.

### 17/07/2013, Wednesday

#### 15:00, Algebra

Peter Trapa, University of Utah.
Unitary representations of reductive Lie groups.

Unitary representations of Lie groups appear in many parts of mathematics: in harmonic analysis (as generalizations of the sines and cosines appearing in classical Fourier analysis); in number theory (as spaces of modular and automorphic forms); in quantum mechanics (as "quantizations" of classical mechanical systems); and in many other places. They have been the subject of intense study for decades, but their classification has only recently recently emerged. Perhaps surprisingly, the classification has inspired connections with interesting geometric objects (equivariant mixed Hodge modules on flag varieties). These connections have made it possible to extend the classification scheme to other related settings. The purpose of this talk is to explain a little bit about the history and motivation behind the study of unitary representations and offer a few hints about the algebraic and geometric ideas which enter into their study. This is based on a recent preprint with Adams, van Leeuwen, and Vogan.

### 16/07/2013, Tuesday

#### 16:30, Geometria em Lisboa

Eric Sommers, University of Massachusetts.
Properties of some resolutions of Schubert varieties.

Schubert varieties are certain closed subvarieties of the flag variety of a complex algebraic group $$G$$. They are indexed by elements of the Weyl group $$W$$ of $$G$$. In general they are singular and some of the structure of their singularities can be understood via the resolutions known as the Bott-Samelson resolutions and the resulting interplay between the intersection cohomology of the Schubert variety and the ordinary cohomology of the fibers of these resolutions.

This talk focuses on joint work with my student, Jennifer Koonz, on some generalizations of the Bott-Samelson resolutions, which provide some new information about the singularities of Schubert varieties. Special cases of these resolutions have already appeared in work by Polo, Wolper, Ryan, and Zelevinsky in the case where $$G$$ is the general linear group $$\operatorname{GL}(n)$$ and $$W$$ is the symmetric group $$S_n$$. I will explain the Bott-Samelson resolutions and these new resolutions in some small examples for the general linear group and perhaps finish with a couple of propositions that are valid for arbitrary $$G$$.

### 11/07/2013, Thursday

#### 16:30, String Theory

Ricardo Couso Santamaria, Santiago de Compostela.
Resurgence and the Topological String.

Topological string theory is simple enough to be solved perturbatively, yet it is able to compute amplitudes in string theory and it also enjoys large N dualities. These gauge theory duals, sometimes in the form of matrix models, can be solved past perturbation theory by plugging transseries ansätze into the so-called string equation. Based on the mathematics of resurgence, developed in the 80's by J. Ecalle, this approach has been recently applied with tremendous success to matrix models and their double-scaling limits (Painlevé I, etc). A natural question is if something similar can be done in the stringy side. In this seminar I will show how the holomorphic anomaly equations of Bershadsky-Cecotti-Ooguri-Vafa (BCOV) provide the starting point to derive a master equation which can be solved with a transseries ansatz. I will review the perturbative sector of the solutions, its structure and how it generalizes for higher instanton sectors. Some resurgence in the guise of large-order behavior of the perturbative sector will be used to derive the holomorphicity of the instanton actions that control the asymptotics of the perturbative sector. This work will appear shortly in a paper together with J.D. Edelstein, R. Schiappa and M. Vonk.

### 11/07/2013, Thursday

#### 14:30, String Theory

Ricardo Vaz, Stony Brook.
The Resurgent Quartic Matrix Model: A Progress Report.

In this talk we will review some recent developments on the quartic matrix model, and its use as a testing ground for ideas of so-called "resurgence". Without going into extreme technical detail we will try to present the relations that were studied and tested, the methods to extract predictions and data in the quartic matrix model (both in the one- and two-cut phases), as well as some natural interpretations of some of the new features. Finally, we will mention some open problems and future directions.

### 11/07/2013, Thursday

#### 11:30, String Theory

The Conformal Bootstrap Program in $$d=4$$.

In this talk we review recent progress in the conformal bootstrap program that started with the pioneering work of Rattazzi, Rychkov, Tonni and Vichi (arXiv: 0807.004), mostly focusing on four dimensions. The idea behind this program is to find which values of operator dimensions and operator product expansion coefficients are compatible with the constraints imposed by unitarity and crossing symmetry on four-point correlation functions. In this way one can obtain bounds on these quantities, even if numerically, that must be satisfied in order to have a consistent CFT.

### 08/07/2013, Monday

#### 11:30, Geometria em Lisboa

David Martínez Torres, Utrecht.
Non-contractible loops in the diffeomorphism group of coadjoint orbits.

A compact connected semisimple Lie group G acts in a Hamiltonian fashion on its coadjoint orbits, i.e. G maps into the group of Hamiltonian transformations of the coadjoint orbit. McDuff and Tolman showed that the induced map on fundamental groups is injective, answering a question of A. Weinstein. In this talk we will show that this is not quite a symplectic phenomenon, but a topological one, since the (finite) fundamental group of G already injects in the fundamental group of the group of diffeomorphisms. This is joint work with I. Mundet.

### 03/07/2013, Wednesday

#### 15:00, Geometria em Lisboa

J. C. Naranjo and G. P. Pirola, Univ. Barcelona and Univ. Pavia.
Isogenies between Jacobians.

In this series of two lectures we will report about the following Theorem: Let $Z$ be a subvariety of codimension $k>0$ of the moduli space ${ℳ}_{g}$ of curves of genus $g$ with $g>3k+4$ and let $\chi :JC\prime \to JC$ an isogeny between two Jacobians, where $C$ is generic in $Z$. Then $C$ and $C\prime$ are isomorphic and $\chi$ is the multiplication by an integer. The statement is also true for $k=1$ and $g\ge 5$. This extends a result by Bardelli and Pirola for the case $Z={ℳ}_{g}$ and $g\ge 4$.

There are two natural approaches to this problem. The first one, which will be the content of the first talk, uses infinitesimal variations of Hodge structures in order to translate the statement into a problem concerning the quadrics containing a canonical curve, probably of independent interest. The second one consists in to degenerate to some special (mainly singular stable) curves. This second approach, much more subtle, works for $g\ge 5$. However it needs a good control of the intersection of the closure of $Z$ with the boundary of $\overline{{ℳ}_{g}}$, and for this we have to restrict to $k=1$. This procedure will be explained in the second lecture.

This is a joint work of the speakers with V. Marcucci.

### 02/07/2013, Tuesday

#### 14:00, Partial Differential Equations

Leonard Monsaingeon , Carnegie Mellon University, USA.
Relaxation to Planar Travelling Waves in Inertial Confinement Fusion.

The models of ICF usually couple hydrodynamical features in plasmas with reaction-diffusion equations for the temperature. One of the main challenges is the non-linear heat diffusion of the porous media type, i.e. $$\nabla\cdot(\lambda \nabla T)$$ with $$\lambda=\lambda(T)=T^{m-1}$$ for some conductivity exponent $$m>1$$. In this talk we will consider an approximate nonlinear parabolic equation for which there exist planar wave solutions having a physical meaning. Stability of these waves with respect to wrinkled perturbations is crucial for the ICF process to be effective. We show here that a diffusion-induced mechanism forces relevant perturbations to become planar exponentially fast in the long time regime, and also derive an asymptotic dispersion relation relating this rate of relaxation to some small temperature ratio $$\varepsilon=T_{min}/T_{max}\to 0$$. In this limit the problem degenerates into a free boundary one, and the proof involves a rescaled singular principal eigenvalue problem.

### 01/07/2013, Monday

#### 16:30, Geometria em Lisboa

Nick Sheridan, Institute for Advanced Study and Princeton University.
Homological mirror symmetry.

Mirror symmetry is a conjectural relationship between complex and symplectic geometry, and was first noticed by string theorists. Mathematicians became interested in it when string theorists used it to predict counts of curves on the quintic three-fold (just as there are famously 27 lines on a cubic surface, there are 2875 lines on a quintic three-fold, 609250 conics, and so on). Kontsevich conjectured that mirror symmetry should reflect a deeper equivalence of categories: his celebrated 'Homological Mirror Symmetry' conjecture. Most of the talk will be an overview of mirror symmetry with a focus on the symplectic side, leading up to Kontsevich's conjecture. Finally I will describe a proof of Kontsevich's conjecture for the quintic three-fold, and more generally for a Calabi-Yau hypersurface in projective space of any dimension. If time permits I will draw lots of pictures in the one-dimensional case.

### 28/06/2013, Friday

#### 14:30, IST courses on Algebraic Geometry

Geometry of Higgs Bundles.

### 28/06/2013, Friday

#### 11:30, IST courses on Algebraic Geometry

Steven Bradlow, University of Illinois at Urbana-Champaign.
Geometry of Higgs Bundles.

### 28/06/2013, Friday

#### 11:30, String Theory

Oscar Varela, University of Utrecht.

The field theory defined on a stack of $$N$$ M2-branes is thought to correspond to that first introduced by BLG/ABJM. At large $$N$$, an important sector of this theory can be described, holographically, by the $$SO(8)$$-gauged maximal supergravity in four dimensions of de Wit and Nicolai. Since its inception, the latter has been tacitly assumed to be unique. Recently, however, a one-parameter family of $$SO(8)$$ gaugings of maximal supergravity has been discovered, the de Wit-Nicolai theory being just a member in this class. I will explain how this overlooked family of $$SO(8)$$-gauged supergravities is deeply related to electric/magnetic duality in four dimensions. I will then discuss some predictions that can be made about the possible family of holographic dual field theories, focusing on the structure of conformal phases and the RG flows between them.

### 27/06/2013, Thursday

#### 16:00, IST courses on Algebraic Geometry

Steven Bradlow, University of Illinois at Urbana-Champaign.
Geometry of Higgs Bundles.

### 27/06/2013, Thursday

#### 14:30, IST courses on Algebraic Geometry

Geometry of Higgs Bundles.

### 26/06/2013, Wednesday

#### 16:00, IST courses on Algebraic Geometry

Steven Bradlow, University of Illinois at Urbana-Champaign.
Geometry of Higgs Bundles.

### 26/06/2013, Wednesday

#### 16:00, Summer Lectures in Geometry

Song Sun, Imperial College.
Kahler-Einstein metrics and stability - IV.

In these lectures we will explain the recent proof by X-X. Chen, S. K. Donaldson and the speaker on Yau's conjecture relating existence of Kahler-Einstein metrics on Fano manifolds to algebro-geometric K -stability. We will describe the main problem, outline the strategy and highlight some technical aspects involved in the proof.

### 26/06/2013, Wednesday

#### 14:30, IST courses on Algebraic Geometry

Geometry of Higgs Bundles.

### 26/06/2013, Wednesday

#### 11:00, Summer Lectures in Geometry

Song Sun, Imperial College.
Kahler-Einstein metrics and stability - III.

In these lectures we will explain the recent proof by X-X. Chen, S. K. Donaldson and the speaker on Yau's conjecture relating existence of Kahler-Einstein metrics on Fano manifolds to algebro-geometric K -stability. We will describe the main problem, outline the strategy and highlight some technical aspects involved in the proof.

### 25/06/2013, Tuesday

#### 16:00, IST courses on Algebraic Geometry

Steven Bradlow, University of Illinois at Urbana-Champaign.
Geometry of Higgs Bundles.

Higgs bundles appear in several guises including (a) as solutions to gauge-theoretic equations for connections and sections of a bundle (b) as holomorphic realizations of fundamental group representations or, equivalently, local systems and (c) as special cases of principal bundles with extra structure (principal pairs). Each point of view leads to a construction of a moduli space, i.e. a geometric object whose points parametrize equivalence classes of Higgs bundles. The first part of this mini course will explain these different points of view and describe how they are related. We will then explore some key topological and geometric features of the moduli spaces. We will confine attention to Higgs bundles over closed Riemann surfaces.

### 25/06/2013, Tuesday

#### 14:30, IST courses on Algebraic Geometry

Geometry of Higgs Bundles.

Higgs bundles appear in several guises including (a) as solutions to gauge-theoretic equations for connections and sections of a bundle (b) as holomorphic realizations of fundamental group representations or, equivalently, local systems and (c) as special cases of principal bundles with extra structure (principal pairs). Each point of view leads to a construction of a moduli space, i.e. a geometric object whose points parametrize equivalence classes of Higgs bundles. The first part of this mini course will explain these different points of view and describe how they are related. We will then explore some key topological and geometric features of the moduli spaces. We will confine attention to Higgs bundles over closed Riemann surfaces.

### 25/06/2013, Tuesday

#### 11:00, Summer Lectures in Geometry

Song Sun, Imperial College.
Kahler-Einstein metrics and stability - II.

In these lectures we will explain the recent proof by X-X. Chen, S. K. Donaldson and the speaker on Yau's conjecture relating existence of Kahler-Einstein metrics on Fano manifolds to algebro-geometric K -stability. We will describe the main problem, outline the strategy and highlight some technical aspects involved in the proof.

### 24/06/2013, Monday

#### 16:00, Summer Lectures in Geometry

Song Sun, Imperial College.
Kahler-Einstein metrics and stability - I.

In these lectures we will explain the recent proof by X-X. Chen, S. K. Donaldson and the speaker on Yau's conjecture relating existence of Kahler-Einstein metrics on Fano manifolds to algebro-geometric K-stability. We will describe the main problem, outline the strategy and highlight some technical aspects involved in the proof.

### 20/06/2013, Thursday

#### 16:30, Operator Theory, Complex Analysis and Applications

Berezin Calculus over Weighted Bergman Spaces of Polyanalytic type.

Starting from the Poincaré metric $d{s}^{2}=\frac{1}{2\pi i}{\left(1-\mid z{\mid }^{2}\right)}^{-2}d\stackrel{‾}{z}\phantom{\rule{thickmathspace}{0ex}}dz$ on the the unit disk $𝔻$, we will study the range of the Berezin transforms generated from the normalized kernel function ${K}_{\zeta }^{n}\left(z\right)={K}^{n}\left(z,\zeta \right){K}^{n}\left(\zeta ,\zeta {\right)}^{-\frac{1}{2}}$ regarding the weighted polyanalytic Bergman spaces ${A}_{n}^{\alpha }\left(𝔻\right)$ of order $n$. Special emphasize will be given to the invariance of the range of the Berezin transformation under the action of the Möbius transformations ${\phi }_{\zeta }\left(z\right)=\frac{z-a}{1-\stackrel{‾}{\zeta }z}$. Connection between Berezin calculus over weighted Bergman spaces of polyanalytic type on the disk $𝔻$ and on the upper half space ${ℂ}^{+}$ will also be discussed along the talk.

### 20/06/2013, Thursday

#### 16:30, Geometria em Lisboa

Manuel Araújo, Instituto Superior Tecnico.
Symplectic embeddings into $$\mathbb{C}P^\infty$$.

Let $\left(M,\omega \right)$ be a compact symplectic manifold with $\left[\omega \right]$ integral. It is a Theorem of Gromov (1970) and Tischler (1977) that $\left(M,\omega \right)$ symplectically embeds into $ℂ{P}^{n}$ with the Fubini-Study symplectic form, for $n$ large enough. Let ${\beta }_{1}\left(M\right)$ be the first Betti number of $M$. We refine this result of Gromov and Tischler by showing that the weak homotopy type of the space of symplectic embeddings of such a symplectic manifold into $ℂ{P}^{\infty }$ is $\left({S}^{1}{\right)}^{{\beta }_{1}\left(M\right)}×ℂ{P}^{\infty }$.

### 20/06/2013, Thursday

#### 11:00, Topological Quantum Field Theory

John Huerta, IST, Lisbon.
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, Friday

#### 11:00, Topological Quantum Field Theory

John Huerta, IST, Lisbon.
QFT IV.

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

### 11/06/2013, Tuesday

#### 16:30, String Theory

Richard Szabo, Heriot-Watt University Edinburgh.
Quantization of non-geometric flux backgrounds.

We describe the emergence of nonassociative geometries probed by closed strings in non-geometric flux compactifications of string theory. We show that these non-geometric backgrounds can be geometrised through the dynamics of membranes propagating in the phase space of the target space compactification. Quantization of the membrane sigma-model leads to a proper quantization of the nonassociative background, which we relate to Kontsevich's formalism of global deformation quantization. We construct Seiberg-Witten type maps between associative and nonassociative backgrounds, and show how they may realise a nonassociative deformation of gravity. We also explain how this approach is related to the quantization of certain Lie 2-algebras and cochain twist quantization using suitable quasi-Hopf algebras.

### 11/06/2013, Tuesday

#### 15:00, Working Seminar on Symplectic/Contact Geometry/Topology

Daniele Sepe, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos.
From semi-toric systems to Hamiltonian ${S}^{1}$-spaces.

Semi-toric integrable systems on closed four dimensional manifolds, introduced by Vu Ngoc, lie at the intersection of integrable Hamiltonian systems and Hamiltonian torus actions. In particular, they are integrable Hamiltonian systems which have one integral that generates an effective Hamiltonian ${S}^{1}$-action. Vu Ngoc showed that these systems share an important property with symplectic toric manifolds, i.e. it is possible to associate a family of convex polygons to each of them. On the other hand, considering the manifold only with the Hamiltonian ${S}^{1}$-action, they give rise to examples of Hamiltonian ${S}^{1}$-spaces, which have been classified by Karshon. The aim of this talk is to illustrate how, given a semi-toric system, Karshon's invariants of the underlying Hamiltonian ${S}^{1}$-space can be reconstructed from one (and hence any) convex polygon associated to the system. Time permitting, we will consider how to construct examples of these systems starting from symplectic toric manifolds.

This is joint work with Sonja Hohloch and Silvia Sabatini.

### 07/06/2013, Friday

#### 16:30, Geometria em Lisboa

Dennis The, Australian National University.
The gap phenomenon in parabolic geometries.

Many geometric structures (such as Riemannian, conformal, CR, projective, systems of ODE, and various types of generic distributions) admit an equivalent description as Cartan geometries. For Cartan geometries of a given type, the maximal amount of symmetry is realized by the flat model. However, if the geometry is not (locally) flat, how much symmetry can it have? Understanding this "gap" between maximal and submaximal symmetry in the case of parabolic geometries is the subject of this talk. I'll describe how a combination of Tanaka theory, Kostant's version of the Bott-Borel-Weil theorem, and a new Dynkin diagram recipe led to a complete classification of the submaximal symmetry dimensions in all parabolic geometries of type $\left(G,P\right)$, where $G$ is a complex or split-real simple Lie group and $P$ is a parabolic subgroup. (Joint work with Boris Kruglikov.)

### 07/06/2013, Friday

#### 11:30, String Theory

Veselin Filev, IAS Dublin.
Magnetic Catalysis in compact spaces.

I will describe the properties of a quantum field theory in compact space subjected to an external magnetic field. In the context of the AdS/CFT correspondence this is realized by introducing flavour branes to the ${\mathrm{AdS}}_{5}×{S}^{5}$ geometry in global coordinates. The dual field theory lives on a round three sphere. The theory has a finite Casimir free energy having dissociating effect on the fundamental condensate of the theory. This competes with the pairing effect of the magnetic field leading to an interesting phase structure.

### 06/06/2013, Thursday

#### 16:30, Working Seminar on Symplectic/Contact Geometry/Topology

Daniele Sepe, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos.
Singular integral affine structures and integrable Hamiltonian systems.

An important outstanding question in the theory of integrable Hamiltonian systems is their classification, i.e. the construction of some (hopefully computable!) invariants which completely determine these systems up to a suitable notion of equivalence. In this talk, a weaker (but still considerably hard) problem is going to be introduced, namely that of determining when two integrable Hamiltonian systems are equivalent. In the absence of singularities, i.e. equilibria, this problem can be solved using integral affine geometry, which studies symmetries of Euclidean space fixing the standard integral lattice therein; this was first observed by Duistermaat and Dazord and Delzant. One way to extend this result to include singularities is to define an appropriate notion of singular integral affine geometry, which is the aim of this talk. Time permitting, some low dimensional explicit examples will be discussed. This is joint ongoing work with Rui Loja Fernandes.

### 04/06/2013, Tuesday

#### 16:30, Working Seminar on Symplectic/Contact Geometry/Topology

Daniele Sepe, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos.
On complete isotropic realisations of Poisson manifolds.

Poisson geometry can be viewed as the study of those manifolds which admit a (possibly singular) foliation by symplectic manifolds (e.g. $$\mathbb{R}^3$$ foliated by spheres centred at the origin of increasing radius and the radius). A complete symplectic realisation of a Poisson manifold is a desingularisation, in the sense that it consists of a symplectic manifold together with a submersion onto the original Poisson manifold which reflects its Poisson structure. A beautiful theorem of Crainic and Fernandes proves that existence of such a desingularisation is equivalent to the Poisson manifold (viewed as an infinitesimal object, the analogue of a Lie algebra) admitting a global integration (the analogue of a Lie group). In this talk, we consider complete isotropic realisations of Poisson manifolds, which are desingularisations as above of minimal dimension; these are related to non-commutative integrable Hamiltonian systems. The fundamental driving question behind this talk is the following: what can be said about Poisson manifolds that admit such desingularisations? Some results in this direction, both old and new, will be presented. This is ongoing joint work with Ioan Marcut.

### 04/06/2013, Tuesday

#### 15:00, Analysis, Geometry, and Dynamical Systems

Christian Le Merdy, Université de Franche-Comté.
Dilation of operators on ${L}^{p}$-spaces.

Let $1, let $\left(\Omega ,\mu \right)$ be a measure space and let $T:{L}^{p}\left(\Omega \right)\to {L}^{p}\left(\Omega \right)$ be a bounded operator. We say that it admits a dilation (in a loose sense) when there exist another measure space $\left(\Omega \prime ,\mu \prime \right)$, an invertible operator $U$ on ${L}^{p}\left(\Omega \prime \right)$ such that $\left\{{U}^{n}:n\in ℤ\right\}$ is bounded and two bounded operators $J:{L}^{p}\left(\Omega \right)\to {L}^{p}\left(\Omega \prime \right)$ and $Q:{L}^{p}\left(\Omega \prime \right)\to {L}^{p}\left(\Omega \right)$ such that ${T}^{n}=Q{U}^{n}J$ for any integer $n\ge 0$. When $p=2$, this property is equivalent to $T$ being similar to a contraction. The main question considered in this talk is to characterize operators with this property when $p\ne 2$. Our results give partial answers and strong connections with functional calculus properties. The talk will include motivation for this dilation question. (Joint work with C. Arhancet.)

### 04/06/2013, Tuesday

#### 14:00, Partial Differential Equations

Farid Bozorgnia, IST, Lisbon.
Optimal partitions for first eigenvalue; numeric and some related problems.

Given a bounded domain in dimension two, we aim to approximate partitions of the given domain minimizing the sum of first eigenvalues of Dirichlet Laplacian. To do this, a new idea to approximate the second eigenfunction and second eigenvalue is presented. Also we present numerical approximation for Fucik spectrum. We use the qualitative properties of the minimization problem to construct a numerical algorithm to approximate optimal configurations. Moreover, we discuss the numerical implementation of the resulting approach and present computational tests confirming the expected asymptotic behavior of optimal partitions with large numbers of partitions.

### 03/06/2013, Monday

#### 16:30, String Theory

$$D$$-brane geometries and black hole microstates.

String amplitude computations indicate that the geometry sourced by a bound state of $$D$$-branes differs from the "naive" black hole solution with the same $$D$$-brane charges. We will show how such computations can be applied to the construction of black hole microstates and could shed light on the black hole information paradox.

### 21/05/2013, Tuesday

#### 16:30, Geometria em Lisboa

Carlos Florentino, Instituto Superior Técnico.
Irreducibility of character varieties of abelian groups.

The description of the space of commuting elements in a compact Lie group is an interesting algebro-geometric problem with applications in Mathematical Physics, notably in Supersymmetric Yang Mills theories.

When the Lie group is complex reductive, this space is the character variety of a free abelian group. Let $$K$$ be a compact Lie group (not necessarily connected) and $$G$$ be its complexiﬁcation. We consider, more generally, an arbitrary finitely generated abelian group $$A$$, and show that the conjugation orbit space $\mathrm{Hom}\left(A,K\right)/K$ is a strong deformation retract of the character variety $\mathrm{Hom}\left(A,G\right)/G$.

As a Corollary, in the case when $$G$$ is connected and semisimple, we obtain necessary and sufficient conditions for $\mathrm{Hom}\left(A,G\right)/G$ to be irreducible. This is also related to an interesting open problem about irreducibility of the variety of $$k$$ tuples of $$n$$ by $$n$$ commuting matrices.

### 20/05/2013, Monday

#### 16:30, String Theory

Diego Bombardelli, University of Porto.
Thermodynamic Bethe Ansatz and double-wrapping corrections for non-supersymmetric deformations of AdS/CFT.

We study finite-size corrections of the ${\gamma }_{i}$-deformed AdS/CFT vacuum energy/anomalous dimension. In particular, we compute the leading (at large volume) and next-to-leading order (NLO) Luescher-like corrections, corresponding to single- and double-wrapping diagrams respectively. On the other hand, we solve to the NLO the twisted Thermodynamic Bethe Ansatz equations describing exactly the ground-state energy of the theory; then we compare the results of the two approaches and find exact agreement. Next, we evaluate explicitly LO and NLO corrections up to six loops at weak coupling.

Finally, I will show some work in progress about the possible conjecture of a Luescher-like formula for the double-wrapping corrections of undeformed excited states energy.

### 16/05/2013, Thursday

#### 16:30, Operator Theory, Complex Analysis and Applications

Sérgio Mendes, Instituto Universitário de Lisboa, ISCTE-IUL.
Noncommutative summands of the ${C}^{*}$-algebra ${C}_{r}^{*}{\mathrm{SL}}_{2}\left({𝔽}_{2}\left(\left(\varpi \right)\right)\right)$.

Let ${𝔽}_{2}\left(\left(\varpi \right)\right)$ denote the Laurent series in the indeterminate $\varpi$ with coefficients over the finite field with two elements ${𝔽}_{2}$. This is a local nonarchimedean field with characteristic $2$. We show that the structure of the reduced group ${C}^{*}$-algebra of ${\mathrm{SL}}_{2}\left({𝔽}_{2}\left(\left(\varpi \right)\right)\right)$ is determined by the arithmetic of the ground field. Specifically, the algebra ${C}_{r}^{*}{\mathrm{SL}}_{2}\left({𝔽}_{2}\left(\left(\varpi \right)\right)\right)$ has countably many noncommutative summands, induced by the Artin-Schreier symbol. Each noncommutative summand has a rather simple description: it is the crossed product of a commutative ${C}^{*}$-algebra by a finite group. The talk will be elementary, starting from the scratch with the definition of ${C}_{r}^{*}{\mathrm{SL}}_{2}$.

### 16/05/2013, Thursday

#### 16:30, Geometria em Lisboa

Matias del Hoyo.
On the linearization of certain smooth structures.

The linearization theorem for proper Lie groupoids, whose prove was completed by Zung a few years ago, generalizes various results such as Ehresmann theorem for submersions, Reeb stability for foliations, and the Tube Theorem for proper actions. In a work in progress with R. Fernandes we show that this linearization can be achieved by means of the exponential flow of certain metrics, providing both a stronger theorem and a simpler proof. In this talk I will recall the classic linearization theorems, discuss its groupoid formulation, and present our work on riemannian structures for Lie groupoids.

### 14/05/2013, Tuesday

#### 15:00, Analysis, Geometry, and Dynamical Systems

Filippo Cagnetti, University of Sussex.
A new method for large time behavior of convex Hamilton-Jacobi equations.

We introduce a new method to study the large time behavior for general classes of Hamilton-Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.

### 14/05/2013, Tuesday

#### 10:30, Colloquium

Alessio Figalli, University of Texas at Austin.
Stability results for sumsets in $$\mathbb{R}^n$$.

Given a Borel set $$A$$ in $$\mathbb{R}^n$$ of positive measure, one can consider its semisum $$S=(A+A)/2$$. It is clear that $$S$$ contains $$A$$, and it is not difficult to prove that they have the same measure if and only if $$A$$ is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is stable: if the measure of $$S$$ is close to the one of $$A$$, is $$A$$ close to his convex hull? More in general, one may consider the semisum of two different sets $$A$$ and $$B$$, in which case our question corresponds to proving a stability result for the Brunn-Minkowski inequality. When $$n=1$$, one can approximate a set with finite unions of intervals to translate the problem onto $$\mathbb{Z}$$, and in the discrete setting this question becomes a well studied problem in additive combinatorics, usually known as Freiman's Theorem. In this talk I'll review some results in the one-dimensional discrete setting, and show how to answer to this problem in arbitrary dimension.

### 06/05/2013, Monday

#### 16:30, String Theory

Ivo Sachs, LMU Munich.
Homotopy Algebras and String Field Theory.

We revisit the existence, background independence and uniqueness of bosonic- and topological string field theory using the machinery of homotopy algebra. In a theory of classical open- and closed strings, the space of inequivalent open string field theories is shown to be isomorphic to the space of classical closed string backgrounds. We then discuss obstructions of these moduli spaces at the quantum level.

### 29/04/2013, Monday

#### 16:30, String Theory

Jorge Russo, Universitat de Barcelona.
Evidence for Large $$N$$ phase transitions in $$N=2^\ast$$ theory.

Using localization, we solve for the large-$$N$$ master field of $$N=2^\ast$$ super-Yang-Mills theory and calculate expectation values of large Wilson loops and the free energy. At weak coupling, these observables only receive non-perturbative contributions. The analytic solution holds for a finite range of the 't Hooft coupling and terminates at the point of a large-$$N$$ phase transition. We find evidence that as the coupling is further increased the theory undergoes an infinite sequence of similar transitions that accumulate at infinity.

### 24/04/2013, Wednesday

#### 15:00, Partial Differential Equations

Diego Marcon Farias, IST, Lisbon.
A quantitative log-Sobolev inequality for a two parameter family of functions.

We prove a sharp, dimension-free stability result for the classical logarithmic Sobolev inequality for a two parameter family of functions. Roughly speaking, our family consists of a certain class of log ${C}^{1,1}$ functions. Moreover, we show how to enlarge this space at the expense of the dimensionless constant and the sharp exponent. As an application we obtain new bounds on the entropy. (with E. Indrei)

### 24/04/2013, Wednesday

#### 11:30, Topological Quantum Field Theory

John Huerta, Instituto Superior Técnico.
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.

### 18/04/2013, Thursday

#### 16:30, Operator Theory, Complex Analysis and Applications

Gabriel Cardoso, Instituto Superior Técnico.
A light introduction to supersymmetry.

We give a brief introduction to supersymmetric quantum mechanics.

### 17/04/2013, Wednesday

#### 15:00, Partial Differential Equations

Ana Ribeiro, Universidade Nova de Lisboa.
Existence of solutions for non level-convex problems in the supremal form.

It is well known that lower semicontinuity of functionals in the supremal form $F\left(u\right)={esssup}_{x\in \Omega }f\left(\nabla u\left(x\right)\right)$ is related to the level-convexity of the supremand $$f$$ . We adress the problem of existence of solutions for the Dirichlet boundary problem in the lack of this convexity condition relating it with some differential inclusion problem. This is a joint work with E. Zappale.

### 17/04/2013, Wednesday

#### 11:30, Topological Quantum Field Theory

John Huerta, Instituto Superior Técnico.
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.

### 16/04/2013, Tuesday

#### 15:00, Analysis, Geometry, and Dynamical Systems

Florin Radulescu, Università di Roma - Tor Vergata.

### 10/04/2013, Wednesday

#### 11:30, Topological Quantum Field Theory

John Huerta, Instituto Superior Técnico.
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.

### 08/04/2013, Monday

#### 16:30, String Theory

Dimitrios Zoakos, University of Porto.
Holographic flavor in Chern-Simons-matter theories.

After reviewing the gravity dual of $$N=6$$ Chern-Simons-matter theory, we will analyze the addition of backreacted flavors. We will then construct the corresponding flavored black hole and study the thermodynamic properties of brane probes and of the meson melting transition that they undergo at a certain critical temperature.

### 03/04/2013, Wednesday

#### 11:30, Topological Quantum Field Theory

John Huerta, Instituto Superior Técnico.
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.

### 02/04/2013, Tuesday

#### 15:00, Analysis, Geometry, and Dynamical Systems

Jorge Ferreira, Universidade Federal Rural de Pernambuco.
On the asymptotic behaviour of nonlocal nonlinear problems.

This lecture deals with nonlocal nonlinear problems. Our main results concern existence, uniqueness and asymptotic behavior of the weak solutions of a nonlinear parabolic equation of reaction-diffusion nonlocal type by an application of the Faedo-Galerkin approximation and Aubin-Lions compactness result. Moreover, we prove continuity with respect to the initial values, the joint continuity of the solution and a result on the existence of the global attractor for the problem $\left\{\begin{array}{l}{u}_{t}-a\left(l\left(u\right)\right)\Delta u+\mid u{\mid }^{\rho }u=f\left(u\right)\phantom{.}\text{in}\phantom{.}\Omega ×\left(0,T\right),\\ u\left(x,t\right)=0\phantom{.}\text{on}\phantom{.}\partial \Omega ×\left(0,T\right),\\ u\left(x,0\right)={u}_{0}\left(x\right)\phantom{.}\text{in}\phantom{.}\Omega ,\end{array}$ when $0<\rho \le 2/\left(n-2\right)$ if $n\ge 3$ and $0<\rho <\infty$ if $n=1,2$, where $u=u\left(x,t\right)$ is a real valued function, $\Omega \subset {ℝ}^{n}$ is a bounded smooth domain, $n\ge 1$ with regular boundary $\Gamma =\partial \Omega$, $p\ge 2$. Moreover, $a$ and $f$ are continuous functions satisfying some appropriate conditions and $l:{L}^{2}\left(\Omega \right)\to ℝ$ is a continuous linear form.

### 26/03/2013, Tuesday

#### 16:30, Geometria em Lisboa

Leonardo Macarini, Universidade Federal do Rio de Janeiro.
Two periodic orbits on the standard three-sphere.

We prove that every contact form on the tight three-sphere has at least two geometrically distinct periodic orbits. This result was obtained recently by Cristofaro-Gardiner and Hutchings using embedded contact homology but our approach instead is based on cylindrical contact homology. An essential ingredient in the proof is the notion of a symplectically degenerate maximum for Reeb flows whose existence implies infinitely many prime periodic orbits (in any dimension). This is joint work with V. Ginzburg, D. Hein and U. Hryniewicz.

### 21/03/2013, Thursday

#### 16:30, Operator Theory, Complex Analysis and Applications

Cristina Câmara, Instituto Superior Técnico.
A Riemann-Hilbert approach to Toeplitz operators and the corona theorem.

Together with differential operators, Toeplitz operators (TO) constitute one of the most important classes of non-self adjoint operators , and they appear in connection with various problems in physics and engineering. The main topic of my presentation will be the interplay between TOs and Riemann-Hilbert problems (RHP), and the relations of both with the corona theorem. It has been shown that the existence of a solution to a RHP with $2×2$ coefficient $G$, satisfying some corona type condition, implies – and in some cases is equivalent to – Fredholmness of the TO with symbol $G$. Moreover, explicit formulas for an appropriate factorization of $G$ were obtained, allowing to solve different RHPs with coefficient $G$, and to determine the inverse, or a generalized inverse, of the TO with symbol $G$. However, those formulas depend on the solutions to 2 meromorphic corona problems. These solutions being unknown or rather complicated in general, the question whether the factorization of $G$ can be obtained without the corona solutions is a pertinent one. In some cases, it already has a positive answer; how to solve this question in general is open, and all the more so in the case of $n×n$ matrix functions $G$, for which the results regarding the $2×2$ case have recently been generalized.

### 20/03/2013, Wednesday

#### 11:30, Topological Quantum Field Theory

Aleksandar Mikovic, Univ. Lusófona.
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, Wednesday

#### 11:30, Topological Quantum Field Theory

John Huerta, Instituto Superior Técnico.
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.

### 12/03/2013, Tuesday

#### 16:30, Geometria em Lisboa

Gromov width of a symplectic manifold $M$ is a supremum of capacities of balls that can be symplectically embedded into $M$. The definition was motivated by the Gromov's Non-Squeezing Theorem which says that maps preserving symplectic structure form a proper subset of volume preserving maps.

Let $G$ be a compact connected Lie group, $T$ its maximal torus, and $\lambda$ be a point in the chosen positive Weyl chamber.

The group $G$ acts on the dual of its Lie algebra by coadjoint action. The coadjoint orbit, $M$, through $\lambda$ is canonically a symplectic manifold. Therefore we can ask the question of its Gromov width.

In many known cases the width is exactly the minimum over the set of positive results of pairing $\lambda$ with coroots of $G$:

$\mathrm{min}\left\{⟨{\alpha }_{j}^{\vee },\lambda ⟩;{\alpha }_{j}\text{a coroot},⟨{\alpha }_{j}^{\vee },\lambda ⟩>0\right\}.$

For example, this result holds if $G$ is the unitary group and $M$ is a complex Grassmannian or a complete flag manifold satisfying some additional integrality conditions.

We use the torus action coming from the Gelfand-Tsetlin system to construct symplectic embeddings of balls. In this way we prove that the above formula gives the lower bound for Gromov width of all $U\left(n\right)$ and most of $\mathrm{SO}\left(n\right)$ coadjoint orbits.

In the talk I will describe the Gelfand-Tsetlin system and concentrate mostly on the case of regular $$U(n)$$ orbits.

### 07/03/2013, Thursday

#### 02:30, Algebra

Bob Oliver, Université Paris XIII.
Local equivalences between finite Lie groups.

Fix a prime $p$. Two finite groups $G$ and $H$ will be called $p$-locally equivalent if there is an isomorphism from a Sylow $p$-subgroup $S$ of $G$ to a Sylow $p$-subgroup $T$ of $H$ which preserves all conjugacy relations between elements and subgroups of $S$ and $T$.

Martino and Priddy proved that if the $p$-completed classifying spaces ${\mathrm{BG}}_{p}$ and ${\mathrm{BH}}_{p}$ are homotopy equivalent, then $G$ and $H$ are $p$-locally equivalent. They also conjectured the converse, a result which has since been proven, but only by using the classification theorem of finite simple groups.

Anyone who works much with finite groups of Lie type (such as linear, symplectic, or orthogonal groups over finite fields) notices that there are many cases of $p$-local equivalences between them. For example, if $q$ and $q\prime$ are two prime powers such that ${q}^{2}-1$ and $\left(q\prime {\right)}^{2}-1$ have the same 2-adic valuation, then ${\mathrm{SL}}_{2}\left(q\right)$ and ${\mathrm{SL}}_{2}\left(q\prime \right)$ are 2-locally equivalent.

In joint work with Carles Broto and Jesper Møller, we proved, among other results, the following very general theorem about such $p$-local equivalences between finite Lie groups.

Theorem: Fix a prime $p$, a connected, reductive group scheme $G$ over $Z$, and a pair of prime powers $q$ and $q\prime$ both prime to $p$. Then $G\left(q\right)$ and $G\left(q\prime \right)$ are $p$-locally equivalent if $\stackrel{‾}{⟨q⟩}=\stackrel{‾}{⟨q\prime ⟩}$ as closed subgroups of ${Z}_{p}^{×}$.

Our proof of this theorem is topological: we show that the $p$-completed classifying spaces have the same homotopy type, and then apply the theorem of Martino and Priddy mentioned above. The starting point is a theorem of Friedlander, which describes the space $\mathrm{BG}\left(q{\right)}_{p}$ as a “homotopy fixed space” of a some self map of $\mathrm{BG}\left(C{\right)}_{p}$ of a certain type (an “unstable Adams operation”). This is combined with a theorem of Jackowski, McClure, and Oliver that classifies more precisely the self maps of $\mathrm{BG}\left(C{\right)}_{p}$; and with a result of Broto, Møller, and Oliver which says that under certain hypotheses on a space $X$, the homotopy fixed space of a self equivalence $f$ of $X$ depends (up to homotopy type) only on the closed subgroup $\stackrel{‾}{⟨f⟩}$ in the group $\mathrm{Out}\left(X\right)$ of all homotopy classes of self equivalences of $X$.

Currently, no other proof seems to be known of this purely algebraic theorem.

### 27/02/2013, Wednesday

#### 11:30, Topological Quantum Field Theory

John Huerta, Instituto Superior Técnico.
Anomalies II.

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

### 25/02/2013, Monday

#### 16:00, Partial Differential Equations

Carlo Mariconda, Università degli Studi di Padova.
Non occurrence of the Lavrentiev phenomenon for scalar multi-dimensional variational problems.

Let $f:{ℝ}^{n}\to ℝ$ be a convex function, $\Omega$ be an open and bounded subset of ${ℝ}^{n}$. We consider the functional $I\left(u\right)={\int }_{\Omega }f\left(\nabla \left(u\left(x\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}dx\phantom{\rule{2em}{0ex}}u\in {W}^{1,1}\left(\Omega \right).$ It is known [2] that if $\Omega$ is star-shaped then the Lavrentiev phenomenon does not occur if one does not consider a fixed boundary datum, i. e. $\mathrm{inf}\left\{I\left(u\right)\phantom{\rule{thinmathspace}{0ex}}:\phantom{\rule{thinmathspace}{0ex}}u\in {W}^{1,1}\left(\Omega \right)\right\}=\mathrm{inf}\left\{I\left(u\right)\phantom{\rule{thinmathspace}{0ex}}:\phantom{\rule{thinmathspace}{0ex}}u\in {W}^{1,\infty }\left(\Omega \right)\right\}$ The importance of the non occurrence of the Lavrentiev phenomenon is due to the fact that only in that case, the methods of numeric analysis allow to approximate the infimum value of the operator (finite elements method). When the boundary datum is taken into account, in spite of the paradigm saying that the Lavrentiev phenomenon should not occur, there are just a few results corroborating the statement, apart the obvious case where some “natural growth conditions” are assumed: in a recent paper Cellina and Bonfanti [1] proved that if the lagrangian is radial and both the boundary datum and the domain are of class ${𝒞}^{2}$ then the Lavrentiev phenomenon does not occur.

In a work in progress, jointly with Pierre Bousquet and Giulia Treu, we make a considerable step forward in favor of the above conjecture and take into account a wider class of domains and lagrangians, with a minimum set of assumptions (in particular no growth conditions!): its description is the main argument of the lecture.

References:

[1] G. Bonfanti and A. Cellina, On the non-occurrence of the lavrentiev phenomenon, Adv. Calc. Var. 6 (2013), 93–121.

[2] G. Buttazzo and M. Belloni, A survey on old and recent results about the gap phenomenon in the calculus of variations, Recent developments in well-posed variational problems, 1995, pp. 1–27.

### 21/02/2013, Thursday

#### 16:30, Geometria em Lisboa

I will discuss a generalization of the classical Clifford's theorem to singular curves, reducible or non reduced. I will prove that for 2-connected curves a Clifford-type inequality holds for a vast set of torsion free rank one sheaves. I intend to show that our assumptions on the sheaves are the most natural when working with this kind of results. I will moreover show that this result has many applications to the study of the canonical morphism of a singular curve, in particular that it implies a generalization of the classical Noether's theorem to 3-connected curves. This is a joint work with M. Franciosi.

### 19/02/2013, Tuesday

#### 16:30, Geometria em Lisboa

Jacopo Stoppa, Università di Pavia.
Refined curve counting, quivers, and wall-crossing.

I will sketch some aspects of an interesting Gromov-Witten theory on weighted projective planes introduced by Gross, Pandharipande and Siebert. It admits a very special expansion in terms of tropical counts (called the tropical vertex), as well as a conjectural BPS structure. Then I will describe a refinement or "$q$-deformation" of the expansion using Block-Goettsche invariants, motivated by wall-crossing ideas. This leads naturally to a definition of a class of putative $q$-deformed BPS counts. We prove that this coincides with another natural $q$-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined (joint with S. A. Filippini).

### 18/02/2013, Monday

#### 14:00, Operator Theory, Complex Analysis and Applications

Generalized invertibility in rings: some recent results.

The theory of generalized inverses has its roots both on semigroup theory and on matrix and operator theory. In this seminar we will focus on the study of the generalized inverse of von Neumann, group, Drazin and Moore-Penrose in a purely algebraic setting. We will present some recent results dealing with the generalized inverse of certain types of matrices over rings, emphasizing the proof techniques used.

### 14/02/2013, Thursday

#### 15:00, Partial Differential Equations

We introduce a new method to study the large time behavior for general classes of Hamilton-Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.

This is a joint work with D. Gomes, H. Mitake and H. V. Tran.

### 06/02/2013, Wednesday

#### 14:00, Topological Quantum Field Theory

John Huerta, Instituto Superior Técnico.
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.

### 17/01/2013, Thursday

#### 16:30, Operator Theory, Complex Analysis and Applications

Spectral analysis of some non-self-adjoint operators.

We give an introduction to the study of one particular class of non-self-adjoint operators, namely $𝒫𝒯$-symmetric ones. We explain briefly the physical motivation and describe the classes of operators that are considered. We explain relations between the operator classes, namely their non-equivalence, and mention open problems.

In the second part, we focus on the similarity to self-adjoint operators. On the positive side, we present results on one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. Using functional calculus, closed formulas for the similarity transformation and the similar self-adjoint operator are derived in particular cases. On the other hand, we analyse the imaginary cubic oscillator, which, although being $𝒫𝒯$-symmetric and possessing real spectrum, is not similar to any self-adjoint operator. The argument is based on known semiclassical results.

1. P. Siegl: The non-equivalence of pseudo-Hermiticity and presence of antilinear symmetry, PRAMANA-Journal of Physics, Vol. 73, No. 2, 279-287,
2. D. Krejcirík, P. Siegl and J. Zelezný: On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators, Complex Analysis and Operator Theory, to appear,
3. P. Siegl and D. Krejcirík: On the metric operator for imaginary cubic oscillator, Physical Review D, to appear.

### 08/01/2013, Tuesday

#### 11:00, Algebra

Luke Wolcott, University of Western Ontario.
Bousfield lattices, quotients, ring maps, and non-Noetherian rings.

Given an object $X$ in a compactly generated tensor triangulated category $C$ (such as the derived category of a ring, or the stable homotopy category), the Bousfield class of $X$ is the collection of objects that tensor with $X$ to zero. The set of Bousfield classes forms a lattice, called the Bousfield lattice $\mathrm{BL}\left(C\right)$. First, we will look at examples of when a functor $F:C\to D$ induces a lattice map $\mathrm{BL}\left(C\right)\to \mathrm{BL}\left(D\right)$, and will describe several lattice quotients and lattice isomorphisms. Second, we will focus on homological algebra; a ring map $f:R\to S$ induces, via extension of scalars, a functor $D\left(R\right)\to D\left(S\right)$, and this induces a map on Bousfield lattices. Third, we specialize to a specific map between some interesting non-Noetherian rings.