LISMATH

Lisbon Mathematics PhD

LISMATH

Lisbon Mathematics PhD

The purpose of the LisMath seminar is to provide an initiation into research as well as to make students train their oral skills. Each LisMath student will be asked to give a talk, based on research papers, chosen from a list covering a variety of research topics. The seminar should be comprehensible to everyone. The student will be asked to make an effort to explain why he/she finds the topic interesting, and how it fits into the broader research picture. The LisMath seminar will thus help broadening the students training.

The LisMath seminar takes place on a weekly basis in the Spring semester. Attendance is mandatory for LisMath students.

Venue: Wednesday 17h-18h, alternating between FCUL (seminar room 6.2.33 of the Department of Mathematics) and IST (seminar room P9 of the Department of Mathematics).

Grace Mwakyoma, Universidade de Lisboa.

Periodic Hamiltonian flows on four dimensional manifolds.

In this seminar, I would like to present the paper of Yael Karshon on *Periodic Hamiltonian flows on four dimensional manifolds*. We will explore the classification of periodic Hamiltonian flows on compact symplectic 4-manifolds through the use of labelled graphs and show that two such spaces are isomorphic if and only if they have the same graph. Moreover, if time allows we will also see that all these spaces are Kähler manifolds.

**Bibliography:**

[1] Y. Karshon, Periodic Hamiltonian flows on four dimensional manifolds,

https://arxiv.org/abs/dg-ga/9510004

[2] Y. Karshon, Hamiltonian actions of Lie groups, Ph.D. thesis, Harvard University, April 1993.

[3] A. Cannas da Silva, Lectures on Symplectic Geometry, Springer-Verlag Berlin Heidelberg, 2008.

[4] T. Broecker and K. Janich, Introduction to differential topology, Cambridge University Press, 1982.

[5] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, 2nd edition, Oxford Univ. Press, 1998.

Carlos Sotillo, Universidade de Lisboa.

Which parametrized surfaces are Quasi-Ordinary?

A germ of a singular surface $S$ is QO if there is a finite projection $p:S\to \mathbb C^2$ such that its discriminant is normal crossings. A QO surface admits a very special parametrization, that is a natural generalization of the Puiseux parametrization of a plane curve. The purpose of this seminar is to find criteria to determine if a surface $S$ with a parametrization $(u,v) \mapsto (x(u,v),y(u,v),z(u,v))$ is QO. For instance, if the semigroup of the surface $S$ is the semigroup of a QO surface, is the surface $S$ QO?

**Bibliography:**

[1] Gonzalez Perez, The semigroup of a quasi-ordinary hypersurface, http://www.mat.ucm.es/~pdperezg/semi4

[2] Gonzalez Perez, Quasi-ordinary Singularities via toric Geometry, http://www.mat.ucm.es/~pdperezg/PhD-Es-gonzalez.pdf

[3] Patrick Popescu-Pampu, On the analytic invariance of the semigroup of a QO hyperface singularity, http://math.univ-lille1.fr/~popescu/04-Duke.pdf

Inês Rodrigues, Universidade de Lisboa.

Quasisymmetric Schur functions.

Quasisymmetric functions are a generalization of symmetric functions, i.e., functions that are invariant under a certain action of the symmetric group. Like the latter, quasi symmetric functions form a graded algebra, over a commutative ring. The aim of this seminar is to introduce the basic notions on these functions, as well as some bases for this algebra. Within the basie, we highlight the quasisymmetric Schur functions, a recent and natural refinement of the classic Schur functions.

**Bibliography:**

[1] J. Haglund, K. Luoto, S. Mason, S. van Willigenburg, 'Quasisymmetric Schur functions', Journal of Combinatorial Theory, Series A, 118 (2) (2011), 463-490.

[2] K. Luoto, S. Mykytiuk, S. van Willigenburg, 'An Introduction to Quasisymmetric Schur Functions', Springer, 2013.

Vicente García, Universidade de Lisboa.

The canonical map and the canonical ring of algebraic curves.

The aim of this seminar is to describe the behaviour of the canonical and pluricanonical maps of algebraic curves, and to explain the structure of the so-called canonical ring of curves.

**Bibliography:**

[1] B. Saint-Donat, On Petri's Analysis of the Linear System of Quadrics through a Canonical Curve, Mathematische Annalen 206 (1973): 157-176.

[2] E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of Algebraic Curves - Volume I, Springer-Verlag.

Pedro Aniceto, Universidade de Lisboa.

Jones polynomial and gauge theory.

We present a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable, and we describe how Khovanov homology can emerge upon adding a fifth dimension.

**Bibliography:**

[1] Edward Witten, Two lectures on the Jones Polynomial and Khovanov Homology

Matthias Hofmann, Universidade de Lisboa.

Chegger constants and partition problems.

The Cheeger constant of a manifold (or domain), first introduced about 50 years ago, is a quantitative measure of how easily the manifold can be partitioned into two pieces. It is also closely related to the first non-trivial eigenvalue of the Laplace-Beltrami operator on the manifold and thus links partitioning problems with spectral geometry. The proposed seminar talk explores some of these connections, as well as contemporary research results giving generalisations to graphs, higher-order partitions and eigenvalues.

**Bibliography:**

[1] P. Buser, Ann. Sci. École Norm. Sup. (4) 15 (1982), 213-230.

[2] J. Cheeger, Problems in Analysis, Princeton Univ. Press (1970), 195-199.

[3] B. Kawohl and V. Fridman, Comm. Math. Univ. Carol. 44 (2003), 659-667.

[4] C. Lange, S. Liu, N. Peyerimhoff and O. Post, Calc. Var. PDE 54 (2015), 4165-4196.

[5] J. Lee, S. Oveis Gharan and L. Trevisan, Proc. 2012 ACM Symposium on Theory of Computing, ACM, NY (2012), 1117-1130.

[6] L. Miclo, Invent. Math. 200 (2015), 311-343.

Renato Ricardo de Paula, Universidade de Lisboa.

Porous medium model in contact with slow reservoirs.

This seminar is dedicated to the study of the porous medium model with slow reservoirs and to, heuristically, obtain the hydrodynamic equations for this model, depending on the parameter that rules the slowness of the reservoirs. The slow boundary means that particles can be born or die only at the boundary with rate proportional to $N^{-\theta}$, where $\theta \geq 0$ and $N$ is the scale parameter, while in the bulk the particle's exchange rate is either equal to $1$ or $2$, depending on the local configuration of the system. So, the goal is to explain how we can study the hydrodynamic limit of this interacting particle system, which guarantees that the evolution of the density of particles of this model is described by the weak solution of the corresponding hydrodynamic equation, namely, the porous medium equation with Dirichlet boundary conditions (when $\theta \in [0,1)$), with Robin boundary conditions (when $\theta = 1$) and Neumann boundary conditions (when $\theta \in (1, \infty)$). This presentation is based on the methods initially proposed in [3] and which are adapted to slow boundaries in [1] and [2].

**Bibliography**

[1] Bernardin, C.; Gonçalves, P; Oviedo, B.: Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps. https://arxiv.org/abs/1702.07216 (2017).

[2] Baldasso, R.; Menezes, O.; Neuman, A.; Souza, R. R.: Exclusion process with slow boundary. https://arxiv.org/abs/1407.7918 (2016).

[3] Franco, T.; Gonçalves, P.; Neumann, A.: Hydrodynamical behavior of symmetric exclusion with slow bonds. Ann. Inst. H. Poincaré Probab. Statist. 49, 2 (2013), 402-427.

João Rijo, Universidade de Lisboa.

Topological pressure and dimension.

Introduction to the thermodynamic formalism, including the notions of topological entropy and topological pressure. Variational principle with sketch of the proof in the particular case of topological entropy. Description of some applications to the dimension theory of hyperbolic dynamics, after introducing the relevant notions of dimension theory and of repeller, with formulation (without proof) of the result giving the dimension of a conformal repeller via the topological pressure.

**Bibliography**

[1] L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Birkhäuser, 2008.

[2] L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory, Birkhäuser, 2011.

[3] L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Springer, 2012.

[4] L. Barreira and C. Valls, Dynamical Systems, Springer, 2013.