## Search

## Topological Quantum Field Theory

Meridional essential surfaces of unbounded Euler characteristics in knot exteriors.

*João Miguel Nogueira*, Universidade de Coimbra.

## Abstract

In this talk we will discuss further the existence of knot exteriors with essential surfaces of unbounded Euler characteristics. More precisely, we show the existence of a knot with an essential tangle decomposition for any number of strings. We also show the existence of knots where each exterior contains meridional essential surfaces of simultaneously unbounded genus and number of boundary components. In particular, we construct examples of knot exteriors each of which having all possible compact surfaces embedded as meridional essential surfaces.

## LisMath

Which parametrized surfaces are Quasi-Ordinary?

*Carlos Sotillo*, Universidade de Lisboa.

## Abstract

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

## Probability and Statistics

Adaptive SVD-based Kalman filtering for state and parameter estimation of linear Gaussian dynamic stochastic models.

*Maria Kulikova*, Center for Computational and Stochastic Mathematics.

## Abstract

In this talk, the recently published results on the robust adaptive Kalman filtering are presented. Such methods allow for simultaneous state and parameter estimation of dynamic stochastic systems. Any adaptive filtering scheme typically consists of two parts: (i) a recursive optimization method for identifying the uncertain system parameters by minimizing an appropriate performance index (e.g. the negative likelihood function, if the method of maximum likelihood is used for parameter estimation), and (ii) the application of the underlying filter for estimating the unknown dynamic state of the examined model as well as for computing the chosen performance index. In this paper we study the gradient-based adaptive techniques that require the corresponding performance index gradient evaluation. The goal is to propose the robust computational procedure that is inherently more stable (with respect to roundoff errors) than the classical approach based on the straightforward differentiation of the Kalman filtering equations.

Our solution is based on the SVD factorization. First, we have designed new SVD-based Kalman filter implementation method in [1]. Next, we have extended the obtained result on the gradient evaluation (with respect to unknown system parameters) and, hence, designed the SVD-based adaptive scheme in [2]. The newly-developed SVD-based methodology is algebraically equivalent to the conventional approach and the previously derived stable Cholesky-based methods, but outperforms them for estimation accuracy in ill-conditioned situations.

(joint work with Julia Tsyganova)

References:

[1] Kulikova M.V., Tsyganova J.V. (2017) Improved discrete-time Kalman filtering within singular value decomposition. IET Control Theory & Applications, 11(15): 2412-2418

[2] Tsyganova J.V., Kulikova M.V.(2017) SVD-based Kalman filter derivative computation. IEEE Transactions on Automatic Control, 62(9): 4869-4875

## LisMath

Periodic Hamiltonian flows on four dimensional manifolds.

*Grace Mwakyoma*, Universidade de Lisboa.

## Abstract

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.

## Geometria em Lisboa

Spectral theory and enumerative geometry.

*Marcos Mariño*, University of Geneva.

## Abstract

In recent years, a surprising correspondence has been found between the spectral theory of certain trace class operators, and the enumerative geometry of certain Calabi-Yau threefolds. This correspondence leads to a new family of exactly solvable operators in spectral theory, as well as to a new point of view on Gromov-Witten theory. In this overview talk I will introduce the conjecture and some developments inspired by it.

## Partial Differential Equations

Penrose’s Weyl Curvature Hypothesis and his Conformal Cyclic Cosmology.

*Paul Tod*, University of Oxford.

## Abstract

Penrose’s Weyl Curvature Hypothesis, which dates from the late 70s, is a hypothesis, motivated by observation, about the nature of the Big Bang as a singularity of the space-time manifold. His Conformal Cyclic Cosmology is a remarkable suggestion, made a few years ago and still being explored, about the nature of the universe, in the light of the current consensus among cosmologists (and the Nobel Committee) that there is a positive cosmological constant. I shall review both sets of ideas within the framework of general relativity, emphasise how the second set solves a problem posed by the first, and say something about predictions of CCC.

## Geometria em Lisboa

To be announced.

*Rui Loja Fernandes*, University of Illinois at Urbana-Champaign.

## Colloquium

Decidir para Ganhar / Decide to Win.

*Cláudia Nunes Philippart*, Instituto Superior Técnico, CEMAT - Universidade de Lisboa.

## Abstract

Há inúmeros problemas financeiros que giram em torno da questão sobre o momento óptimo de agir. Um dos exemplos mais conhecidos é a determinação do momento óptimo para exercer uma opção americana. Mas também na área de tomada de decisão sobre investimentos esta questão também é pertinente. Questões como: quando investir numa nova tecnologia? Quando construir um novo aeroporto? Quando suspender produção? Estes problemas têm impacto na economia e, consequentemente, é necessário tomar a devida atenção.

Nesta sessão aborda-se esta categoria de problemas. São problemas de paragem óptima, intimamente relacionados com problemas de fronteira livre. Uma das ferramentas utilizadas para a sua resolução é uma desigualdade variacional, usualmente conhecida por equação de Hamilton-Jacobi-Bellman (HJB). Assim, numa primeira parte são brevemente apresentados os aspectos fundamentais da paragem óptima e das equações HJB, e numa segunda parte apresenta-se problemas concretos, com solução e discussão.

In the finance world there are many problems related with the optimal time to undertake some action. One of the most common problems is the derivation of the exercise time of an American option. But also in decisions regarding investments this question is essential. Questions like: when to adopt a new technology? When to invest in a new airport? When should suspension out of production occur? These problems have a real impact in the economy, and therefore one needs a proper mathematical formulation and solution for them.

In this talk we address such problems. They are known in the literature as optimal stopping problems, closely related with free boundary problems. One way to solve the optimisation problem is to use a variational inequality, known as the Hamilton-Jacobi-Bellman equation (HJB, for short). In the first part of the talk we present briefly the mathematical formulation and tools to solve such problems, and in the second part we show some applications, providing solution and discussion.

## Partial Differential Equations

To be announced.

*Moritz Reintjes*, Instituto Superior Técnico.