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) except for 26/6/2018, *LisMath Seminar Day*, when all sessions will be held at the former location.

Maria Silva, Instituto Superior Técnico.

Decomposition of time series.

A time series is a realization of a stochastic process. In general, time series are decomposed into their natural components (trend, seasonal, cyclical and irregular components) before further analysis. Neverthless, a decomposition into unusual components can also be useful in several areas of research. In this talk, different methods for time series decomposition into unusual components will be present. The Empirical Mode Decomposition (EMD) and the Independent Component Analysis (ICA) are two examples. Moreover, some of these methods will be tested with a simulated time series and the results will be discussed.

**Bibliography:**

[1] Cleveland, R.B., Cleveland, W.S., and Terpenning, I. (1990). STL: A seasonal-trend decomposition procedure based on loess. Journal of Official Statistics 6(1):3.

[2] Mijovic, B., De Vos, M., Gligorjevic, I., Taelman, J., and Van Huffel S. (2010). Source separation using single-channel recordings by combining empirical-mode decomposition and independent components analysis. IEEE Transactions on biomedical engineering, 57(9):2188-2196.

Pedro Filipe, LisMath, Instituto Superior Técnico.

Ordering protoalgebraic logics.

The Leibniz hierarchy arose as an attempt to rank which types of logics are more amenable to be studied from an algebraic point of view and plays a central role in modern abstract algebraic logic. Within this hierarchy, the class of protoalgebraic logics resides at the very bottom, not being included in any other class. In this sense, protoalgebraicity in one of the weakest properties of logics that makes them amenable to most of the standard methods in algebra. In this seminar we will study the order properties of this lattice of logics with some interesting results.

**Bibliography:**

[1] Josep Maria Font, Ordering protoalgebraic logics, Journal of Logic and Computation 26 (2016)

[2] Josep Maria Font, Abstract Algebraic Logic - An Introductory Textbook, 2016

Salvatore Baldino, Instituto Superior Técnico.

String theory and integrability.

The goal of this seminar is to introduce the concept of integrability, showing how it can be applied to obtain solutions for dynamical systems. We will then apply the tools that we build in the first part of the talk to attack a problem that is relevant in topological string theory, the problems of the KP and KdV hierarchy. We will introduce those problems in a self-contained way, explore the use of the technique of the Lax pairs to solve them and finally we will see how those solutions are of use in matrix models, that appear in minimal string theories of interest.

**Bibliography:**

[1] E. Witten, Two-dimensional gravity and intersection theory on moduli space, Survey in Differential Geometry, 243-310 (1990)

[2] R. Dijkgraaf, Intersection theory, integrable hierarchies and topological field theory

[3] P. van Moerbecke, Integrable foundations of string theory, In: Lec- tures on Integrable Systems, ed by O. Babelon, P. Cartier, Y. Kosmann- Schwarzbach (World Sci. Publishing, River Edge, NJ, 1994) pp 163-267

[4] R. Dijkgraaf, E. Witten, Developments in topological gravity

Augusto Pereira, LisMath, Instituto Superior Técnico.

The twistor correspondence and the ADHM construction on $S^4$.

We shall give a brief overview of Yang-Mills theory, discussing some of its applications to other areas of geometry. We then focus on the construction of holomorphic bundles corresponding, via the twistor transform, to instanton solutions of the Yang-Mills equation.

**Bibliography:**

[1] M. F. Atiyah, N. J. Hitchin, I. M. Singer - Self-duality in four-dimensional Riemannian geometry (1978)

[2] R. S. Ward, R. O. Wells Jr. - Twistor Geometry and Field Theory (1991)

[3] M. F. Atiyah - Geometry of Yang-Mills Fields (1979)

[4] N. J. Hitchin - Linear field equations on self-dual spaces (1980)

Pedro Cardoso, LisMath, Instituto Superior Técnico.

Spectral Gap of Markov Chains.

The aim of this talk is to present the spectral gap of reversible Markov chains and to study some techniques that give bounds on the eigenvalues in order to estimate the spectral gap.

**Bibliography:**

[1] Persi Diaconis and Laurent Saloff-Coste. Comparison theorems for reversible Markov chains. Ann. Appl. Probab., 3(3):696-730, 1993.

[2] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov chains and mixing times. American Mathematical Society, Providence, RI, 2009.

[3] Roger A. Horn and Charles R. Johnson. Matrix analysis. Cambridge University Press, Cambridge, second edition, 2013.

Roberto Vega, LisMath, Instituto Superior Técnico.

Topological strings and mirror symmetry.

Mirror Symmetry is a conjecture that suggests the connection between the structures of two mirror manifolds. In this talk, we will present an introduction to this symmetry, first through the Strominger-Yau-Zaslow conjecture and then in more general terms. Finally, we will mention the origin of Mirror Symmetry in the context of Topological Strings and we will make some comments on the topological A and B models.

**Bibliography:**

[1] E. Witten, Topological sigma models, Comm. Math. Phys. 118, 441-449 (1988)

[2] E. Witten, Mirror manifolds and topological field theory

[3] B. Greene, String theory on Calabi-Yau manifolds

[4] K. Hori, C. Vafa, Mirror symmetry

[5] M. Jinzenji, Classical mirror symmetry, SpringerBriefs in Math. Phys. (2018)

Martí Rosselló, LisMath, Instituto Superior Técnico.

Localization in supersymmetric quantum field theories.

Supersymmetric localization is an effective technique to obtain exact results in certain supersymmetric quantum field theories. It can be seen as an extension of the localization formula of equivariant cohomology. A brief introduction to both topics will be given.

**Bibliography:**

[1] Stefano Cremonesi, An Introduction to Localisation and Supersymmetry in Curved Space, Ninth Modave Summer School in Mathematical Physics, 2013.

[2] Localization techniques in quantum field theories. Special volume.

[3] N. Berline and M. Vergne, Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539-541

[4] M. Atiyah and R. Bott, The Moment map and equivariant cohomology, Topology 23 (1984) 1-28.

[5] E. Witten, Mirror manifolds and topological field theory.

Paulo Rocha, LisMath, Faculdade de Ciências.

An introduction to PT-Symmetric Quantum Theory.

Traditionally in quantum mechanics it is assumed that the Hamiltonian must be Hermitian in order to obtain real energy levels and unitary time evolution. Here we will show that the requirement of Hermiticity may be replaced by space-time reflection (PT-symmetry) without losing any of the essential physical features of quantum mechanics. In this seminar we will give an introduction to PT-symmetric quantum theory and work with some examples.

**Bibliography:**

[1] Carl M. Bender, Introduction to PT-Symmetric Quantum Theory.

[2] Carl M. Bender and Javad Komijani, Painlevé Transcendents and PT-Symmetric Hamiltonians.

[3] Dorje C. Brody, Consistency of PT-symmetric quantum mechanics.

Maximilian Schwick, LisMath, Instituto Superior Técnico.

Resurgence.

Resurgence is a method used to solve differential equations with a wide range of applications. It is based on the so called alien calculus. The talk will give a brief insight on what resurgence is used for. Then, via example, a short introduction to alien calculus is given.

**Bibliography:**

[1] D. Sauzin, Introduction to 1-Summability and Resurgence, in Divergent Series, Summability and Resurgence I: Monodromy and Resurgence, Lec. Notes Math. 2153 (2016).

Carllos Holanda, LisMath, Instituto Superior Técnico.

Applications of ergodic theory to number theory.

Ergodic theory can be described as the study of measurable maps and flows preserving a certain measure. Some emphasis is given to the study of the recurrence properties and stochastic properties of the dynamics. It turns out that there are many nontrivial applications of ergodic theory to number theory. As an illustration, we shall consider fractional parts of polynomials and continued fractions.

**Bibliography:**

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

[2] H. Weyl, Ueber die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313-352.

Miguel Duarte, LisMath, Instituto Superior Técnico.

Singularity theorems of Hawking and Penrose.

Singularity theorems in General Relativity: proof, relevance, and open questions.

**Bibliography:**

[1] S. W. Hawking, R. Penrose, The singularities of gravitational collapse and cosmology, Proc. Roy. Soc. Lond. A 314, 529-548 (1970).

[2] S. W. Hawking and G. Ellis, The large scale structure of space-time, Cambridge University Press, 1995.

[3] J. Senovilla, D. Garfinkle, The 1965 Penrose singularity theorem.

[4] J. Senovilla, Singularity theorems in General Relativity: achievements and open questions.

Sílvia Reis, LisMath, Faculdade de Ciências, Universidade de Lisboa.

Generically Stable Types and Banach Spaces.

We discuss the notion of generically stable types in the framework of dependent theories in continuous first order logic. We will also mention some applications of this framework to structures arising in functional analysis, Banach spaces in particular.

Fábio Silva, LisMath, Faculdade de Ciências, Universidade de Lisboa.

Patience Sorting monoids and their combinatorics.

Monoids arising from combinatorial objects have been intensively studied in recent years. Important examples include the plactic, the sylvester, the Chinese, the hypoplactic, the Baxter, and the stalactic monoids, which are, respectively, associated to the following combinatorial objects: Young tableaux, binary trees, Chinese staircases, quasi-ribbon tableaux, pairs of twin binary trees, and stalactic tableaux.

In this talk we present two monoids which arise in a similar way, the left Patience Porting monoid (lPS monoid), also known in the literature as the Bell monoid, and the right Patient Sorting monoid (rPS monoid), that are, respectively, associated to lPS tableaux and rPS tableaux.

Several properties regarding the monoids mentioned in the first paragraph have been considered. Naturally, we pose the same kind of questions for the lPS and rPS monoids. In this seminar, we will discuss some of our results, which include:

- presentations, growth, identities and automacity regarding both monoids;
- Robinson-Schensted-Knuth-type correspondences for the two types of tableaux;
- formulas to count both the number of each type of tableaux for given evaluations, as well as the Bell numbers, together with a hook length formula;
- the cyclic shift graph of the finitely ranked rPS monoids and the diameter of their connected components.

Juan Pablo Quijano, LisMath, Instituto Superior Técnico.

Sheaves and functoriality of groupoid quantales.

This talk has two main aims, one being the study of functoriality of groupoid quantales, which is accomplished in the étale case (in a sense completing the previously ongoing program concerning quantales of étale groupoids), and the other being to provide steps for addressing a similar program for quantales of non-étale groupoids, in this case studying sheaves for a suitable subclass of open groupoids, namely those with “étale covers”.

Pedro Pinto, LisMath, Faculdade de Ciências, Universidade de Lisboa.

The Bounded Functional Interpretation and Proof Mining.

Proof mining is the research program that aims to analyse proofs of mathematical theorems in order to extract hidden quantitative information — such as rates of convergence, rates of metastability and rates of asymptotic regularity. Proof theoretical tools like Kohlenbach's monotone functional interpretation ([1]), a variant of Gödel’s Dialectica, are of standard use. A newer functional interpretation was introduced by Ferreira and Oliva in 2005 ([2]), dubbed the bounded functional interpretation (BFI). The focus of my research was the better understanding of the BFI in the context of proof mining.

I will show a general technique that allows the elimination of weak sequential compactness arguments in the analysis of certain types of proofs. It also gives a better understanding of previous quantitative results done Kohlenbach ([3]) where this argument was already eliminated. This technique was also employed to produce a first quantitative version of Bauschke's theorem ([4]).

Other results, in the context of the proximal point algorithm ([5], [6]), were also analysed with the BFI and their first quantitative versions were obtained.

These results are new and the first practical application of the BFI in the proof mining program.

**Bibliography:**

[1] Kohlenbach, Ulrich. Applied proof theory: proof interpretations and their use in mathematics. Springer Science & Business Media, 2008.

[2] Ferreira, Fernando, and Paulo Oliva. Bounded functional interpretation. Annals of Pure and Applied Logic 135.1-3 (2005): 73-112.

[3] Kohlenbach, Ulrich. On quantitative versions of theorems due to FE Browder and R. Wittmann. Advances in Mathematics 226.3 (2011): 2764-2795.

[4] Bauschke, Heinz H. The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications 202.1 (1996): 150-159.

[5] H. K. Xu, Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66(1) (2002): 240-256.

[6] Boikanyo, Oganeditse A., and G. Morosanu. Inexact Halpern-type proximal point algorithm. Journal of Global Optimization 51.1 (2011): 11-26.

Hillal M. Elshehabey, LisMath, Instituto Superior Técnico.

Mathematical Modelling and Numerical Simulation of an Anaerobic Digester.

Anaerobic digestion is a bacterial process, carried out in the absence of oxygen, used to convert the organic fraction of large volumes of slurries and sludge into biogas and a digested product. The objective of this work is to perform a numerical modeling of the fluid dynamics process inside an anaerobic digestion tank and numerical simulations of the model, which might indicate properly sized extra piping and pumping systems, in order to minimize the deposition of inert materials. This research is being developed within a consulting project for Valorlis - Valorização e Tratamento de Resíduos Sólidos, SA.

In this seminar, we begin by presenting the mathematical model which describes the behavior of the pseudo-plastic fluid in the tank, where parameters such as temperature and total solids content are compatible with several experimental cases reported in the literature and have been validated by Valorlis. The influence of such parameters in the fluid behavior will be discussed in simpler, classical geometries.

Following [1], we propose alternative conditions for outflow. The benefits of using the directional do-nothing boundary condition comparing with the classical one [1] will be presented for the proposed non-Newtonian model and for some benchmark problems, including a comparison with the Newtonian model.

**Bibliography:**

[1] M. Braack and P. B. Mucha, Directional do-nothing condition for the Navier-Stokes equations, Journal of Computational Mathematics, 32, No.5 (2014), 507-521.

Filipe Gomes, LisMath, Faculdade de Ciências, Universidade de Lisboa.

Supercharacter Theories and Multiplicative Ramification Graphs.

Supercharacter theories are generalizations of the usual character theory of a group. In this talk, we construct graded graphs using restriction and superinduction of supercharacters and use them to determine the extreme supercharacters of direct limits of certain groups. We mention the infinite unitriangular group as a particularly important example of this construction.

João Dias, LisMath, Faculdade de Ciências, Universidade de Lisboa.

Supercharacters for algebra groups and their geometric relations.

Given any algebra group over any finite field one has a supercharacter theory constructed by P. Diaconis and I. M. Isaacs. And we may ask three questions:

- How does the supercharacter theory behave with respect to change of field (i. e. finite field extensions)?
- Does there exist an object that contains all supercharacter theory for all changes of field?
- If the answer to the second question is positive, does there exist a group and a supercharacter theory that has the information given by that object?

In this talk I will give a brief introduction to the supercharacter theory and give the answer to the questions above.

Alexandra Symeonides, LisMath, Faculdade de Ciências, Universidade de Lisboa.

Invariant and quasi-invariant measures for Euler equations.

We will discuss how invariant (or quasi-invariant) probability measures can be used to show existence of statistical solutions for the two-dimensional Euler equation (or a slight modification of it), both in the periodic and non periodic case. For initial data in the support of the measures, these solutions are globally defined in time and they are unique. This is joint work with Ana Bela Cruzeiro (IST-UL).

Pedro Oliveira, LisMath, Instituto Superior Técnico.

Cosmic no-hair in spherically symmetric black hole spacetimes.

We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein-Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approaches that of a subextremal Reissner-Nordstrm-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray parallel to the cosmological horizon (“near” $i^+$), in such a way that $r=+\infty$ is, in an appropriate sense, a spacelike hypersurface. We also prove a version of the Cosmic No-Hair Conjecture by showing that in the past of any causal curve reaching infinity both the metric and the Riemann curvature tensor asymptote those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.