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14/11/2018, 17:00 — 18:00 — Room P9, Mathematics Building
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.
See also
Seminário LisMath_pedro_cardoso.pdf
07/11/2018, 17:00 — 18:00 — Room P9, Mathematics Building
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)
See also
[R. Vega] Topological Strings and Mirror Symmetry.pdf
31/10/2018, 17:00 — 18:00 — Room P9, Mathematics Building
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.
See also
SlidesLocalization Lismath_Rossello.pdf
24/10/2018, 17:00 — 18:00 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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.
See also
Presentation_PauloRocha.pdf
17/10/2018, 17:00 — 18:00 — Room P9, Mathematics Building
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).
See also
LisMathPresentation_schwick.pdf
10/10/2018, 17:00 — 18:00 — Room P9, Mathematics Building
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.
See also
Lismath_Seminar_Carllos_Holanda.pdf
03/10/2018, 17:00 — 18:00 — Room P9, Mathematics Building
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.
See also
LisMath Seminar_Miguel_Duarte.pdf
26/06/2018, 16:40 — 17:10 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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.
26/06/2018, 16:00 — 16:30 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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.
26/06/2018, 15:20 — 15:50 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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”.
26/06/2018, 14:40 — 15:10 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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.
26/06/2018, 14:00 — 14:30 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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.
26/06/2018, 12:00 — 12:30 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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.
26/06/2018, 11:20 — 11:50 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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.
26/06/2018, 10:40 — 11:10 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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).
26/06/2018, 10:00 — 10:30 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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.
29/11/2017, 17:00 — 18:00 — Room P9, Mathematics Building
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.
See also
LisMath Seminar_Grace Mwakyoma.pdf
22/11/2017, 17:00 — 18:00 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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
See also
sotillo.pdf
15/11/2017, 17:00 — 18:00 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
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.
See also
Lismath seminar_Rodrigues.pdf
08/11/2017, 17:00 — 18:00 — Room P9, Mathematics Building
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.
See also
LisMathSeminar - VLorenzo.pdf
