Abstract: We start with a brief introduction to the classical Extreme Value Theory for i.i.d. sequences and stationary stochastic processes. We address new developments having the application to dynamical systems in mind. In particular, we consider mixing conditions designed to prove the existence of Extreme Value Laws (EVLs).

We introduce stochastic processes arising from dynamical systems and show how one can use information regarding decay of correlations (or, in other words, the memory loss of the system) in order to check the mixing conditions and hence obtain EVLs.

We establish and explain the connection between the existence of EVLs and the recurrence properties of the systems, which pertain to the asymptotic distribution of the elapsed time before hitting or returning to some critical regions of the phase space.

We discuss the relation between the clustering of extreme observations and the occurrence periodic phenomena. We make a brief introduction to the theory of point processes and its application to the study of dynamically generated extremal processes both in the presence and absence of clustering.

The course is quite introductory and should be reasonable to follow for someone with some background in probability theory and analysis.

[1] V. Lucarini, D. Faranda, A. C. M. Freitas, J. M. Freitas, M. Holland, T. Kuna, M. Nicol, and S. Vaienti. Extremes and Recurrence in Dynamical Systems. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley, Hoboken, NJ, 2016.

[2] A. C. M. Freitas, J. M. Freitas, and M. Todd. Hitting time statistics and extreme value theory. Probab. Theory Related Fields, 147(3):675–710, 2010.

[3] A. C. M. Freitas, J. M. Freitas, and M. Todd. The extremal index, hitting time statistics and periodicity. Adv. Math., 231(5):2626–2665, 2012.

Resumen: La criptografía es el arte de guardar y compartir secretos. En este curso aprenderemos sobre qué es la criptografía, conoceremos algunos criptosistemas clásicos y veremos las amenazas que implica la llegada de los computadores cuánticos. Introduciremos también la teoría de corrección de errores y presentaremos un criptosistema post-cuántico basado en teoría de códigos.

Prerrequisitos: Álgebra lineal y si saben programar mejor, pero no lo necesitan.

The purpose of this talk is to present quantitative criteria for a dynamically convex Reeb flow on the 3-sphere to be right-handed in the sense of Ghys. Once right-handedness is checked, the following interesting conclusions for the dynamics can be deduced: (a) every periodic orbit is a fibered knot, and (b) every finite collection of periodic orbits spans a global surface of section for the flow. As an application, we find an explicit pinching constant 0 < d < 0.7225 such that if a Riemannian metric on the 2-sphere is pinched by at least d then its geodesic flow lifts to a right-handed flow on the 3-sphere. This is joint work with Anna Florio.

A grosso modo, la Teoría de Brouwer explica cómo para homeomorfismos del plano que preservan orientación "cualquier tipo de recurrencia implica punto fijo". En esta charla trataremos de dejar impregnada esta idea en la audiencia, desde una perspectiva histórica, repasando pruebas y aplicaciones, con el énfasis en su relevancia para la dinámica topológica de superficies.

En biología, se busca entender las relaciones evolutivas entre las especies, y estas están comúnmente representadas por un árbol filogenético. Los criterios de algunos de estos algoritmos forman desigualdades polinomiales, y por tanto, descomponen el espacio de posibles inputs en conos semi-algebraicos. En esta charla presentaré de manera sencilla un trabajo donde estudiamos estos conos para el algoritmo Neighbor-Joining, que es uno de los más populares en Biología.

In the last years, thorough research have been conducted in order to understand which properties of graphs can be translated in terms of intrinsic properties of the associated right-angled Artin group, i.e. a group whose relations are given by commutation of generators. After presenting a brief report of the state of the art, in this talk we will show how to describe graph hamiltonicity in terms of the cohomology of the associated right-angled Artin group. As a byproduct, we will obtain an interesting criterion to decide when a square matrix is invertible.

Nesta palestra, apresentaremos alguns exemplos de problemas reais do setor produtivo e um problema da área da saúde. Discutiremos as estratégias de resolução e apresentaremos alguns resultados obtidos por meio dos métodos desenvolvidos. No final, apontaremos algumas direções futuras relacionadas à interação universidade-indústria.

O estudo de existência e continuidade de atratores para problemas parabólicos em relação a perturbação de contorno é um assunto bastante abordado na literatura. Em geral, encontra-se uma extensa variedade de trabalhos que tratam perturbação de contorno em domínios suaves. Nesta palestra consideraremos uma família de problemas parabólicos semi-lineares com condição de fronteira Neumann não linear, definidos em domínios com fronteira Lipschitz. Esses domínios são obtidos considerando uma família de perturbações do quadrado que dependem de um parâmetro, e que convergem para a identidade na norma C^1. Utilizando técnicas de perturbação de contorno provaremos a existência de atrator global para o semigrupo associado.

Referências

[1] Henry, D. B.; Perturbation of the boundary in boundary-value problems of partial diferential equations, Cambridge Univ. Press, Cambrigde, 2005.

[2] Henry, D. B.; Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin, 1981.

[3] Grisvard, P.; Elliptic problems in nonsmooth domains. In: Classics in Applied Mathematics, vol. 69, SIAM, Philadelphia, 2011.

[4] Barbosa, P.S., Pereira, A.L. and Pereira, M.C., Continuity of attractors fora family of C^1 perturbations of the square, Annali di Matematica Pura ed Applicata, v. 196, p.1365-1398, (2017).

[5] Barbosa, P.S., Pereira, A. L., Continuity of attractors for C^1 perturbations of a smooth domain. Electronic Journal of Differential Equations, v. 2020, p. 1-31, (2020).

A widely used assumption in the study of network reliability is that failures occur independently between components and with the same probability.

Under this hypothesis, reliability problems have a reach structure that has allowed us to understand them from different perspectives. Nevertheless, evidence shows that this hypothesis is unrealistic and that models without this assumption are needed to capture essential phenomenons.

The Marshall-Olkin Multivariate models have emerged as a parametric family that can capture dependence between failures while maintaining tractability. Efficient simulations are available for these models, and they can capture geographical failure correlations. In this talk, I will present the Marshall-Olkin and show some interesting results and properties we have found, and discuss other open questions that remain open. (Work in collaboration with Héctor Cancela, Mario Estrada, Guido Lagos, Omar Matus, Eduardo Moreno, and Gerardo Rubino)

Las álgebras de Hopf son objetos matemáticos que poseen una gran cantidad de estructuras relacionadas entre sí y diagramáticamente simétricas. Si bien aparecieron por primera vez en estudios sobre topología algebraica y teoría de grupos algebraicos son hoy en día objetos estudiados con interés en sí mismos, siendo el estudio de su clasificación un problema central en el área. Entre las aplicaciones del estudio de las álgebras de Hopf se pueden mencionar la teoría de nudos, combinatoria, teoría conforme de campos, teoría de categorías, entre otras.

En esta charla daremos nociones generales sobre álgebras de Hopf y discutiremos algunos aspectos del problema de clasificación de las mismas y sus representaciones. En particular, nos enfocaremos en el caso de la subfamilia de álgebras de Hopf punteadas, es decir, aquellas cuyo corradical es el álgebra de grupo de un dado grupo G.

Para ilustrar estos conceptos mencionaremos algunos resultados recientes sobre la clasificación y sobre las representaciones de álgebras de Hopf punteadas para cuando G es un grupo diedral.

La estadística algebraica es la aplicación del álgebra conmutativa y la geometría algebraica a la estadística. Particularmente, se ocupa del desarrollo de técnicas para abordar problemas en estadística y sus aplicaciones.

Muchos de los modelos estadísticos más utilizados son las familias de dimensiones finitas de distribuciones de probabilidad parametrizadas por aplicaciones algebraicas, las cuales se denominan como modelos estadísticos algebraicos. Algunos de estos son por ejemplo los modelos gaussianos, las familias exponenciales, los modelos ocultos de Markov, los modelos de árboles filogenéticos, los modelos de grafos dirigidos y no dirigidos, entre otros modelos discretos.

El punto clave de la estadística algebraica es que muchos modelos estadísticos de interés son conjuntos semialgebraicos, es decir, son conjuntos de puntos definidos por las igualdades y desigualdades polinómicas. En este aspecto la estadística algebraica no solo se ocupa de comprender la geometría y el álgebra del modelo estadístico subyacente, sino también de aplicar este conocimiento para mejorar el análisis de los procedimientos estadísticos y diseñar nuevos métodos para analizar datos. En esta charla introduciré algunos conceptos básicos de esta área y algunos problemas de la Biología que se pueden abordar.

[1] Pachter, L., & Sturmfels, B. (Eds.). (2005). Algebraic Statistics for Computational Biology.

Cambridge: Cambridge University Press. doi:10.1017/CBO9780511610684

[2] Seth Sullivant. Algebraic Statistics. American Mathematical Society, Graduate Studies in Mathematics, 194,

2018, 490pp.

A Dyck path is a lattice path in the first quadrant of the xy-plane that starts at the origin and ends on the x-axis. It consists of the same number of North-East (U) and South-East (D) steps. A pyramid is a subpath of the form U^nD^n. A valley is a subpath of the form DU. The height of a valley is the y-coordinate of its lowest point. A Dyck path is called non-decreasing if the heights of its valleys form a non-decreasing sequence from left to right.

In this talk, we count several aspects of non-decreasing Dyck paths. We count, for example, the number and weight of pyramids and numbers of primitive paths. Throughout the talk, we give connections (bijective relations) between non-decreasing Dyck paths with other objects of the combinatorics. Some examples are, words, trees, polyominoes. In particular, we give a relation between non-decreasing Dyck paths and Riordan arrays. In the end of the talk, we introduce the concept of symmetric pyramids and count them. This is a joint work with Eva Czabarka, José L. Ramírez, and Leandro Junes.

Discontinuous differential equations have been used for many years to model physical and mechanical systems. The mathematical theory behind these models has developed a great deal recently, largely based on the use of the Filippov convention and the Sotomayor-Teixeira regularization. In this talk we will present an overview of recent developments within the theory of smooth differential equations by parts, considering the Filippov convention. Our focus will be mainly in the use of Vishik's normal form for studying local aspects of these systems, plus some global results involving the chaotic behavior of piecewise smooth differential equations defined on torus.

In private-key ciphers the same key is used for encryption and decryption. Thus, the key exchange between the two parts involved in the communication plays an important role. There are two types of private-key ciphers: stream ciphers and block ciphers; depending on whether they encrypt individual bits or blocks of bits, respectively. Our work focuses on the first cipher type. Indeed, stream ciphers are the fastest among the encryption procedures so they are implemented in many technological applications. Assume the messages are binary sequences, then stream ciphers encrypt bits individually. The ciphertext is obtained XOR-ing the message and a keystream sequence satisfying certain characteristics of pseudorandomness. Decryption is performed in the same way; XOR-ing the ciphertext and the same keystream sequence in order to recover the original message. The central problem in stream ciphers is to generate a keystream sequence that looks as random as possible by using a short key. Algorithms which generate keystreams from a secret key are called keystream generators. Among the current keystream generators used in symmetric cryptography, the class of irregularly decimated generators is one of the most popular. The underlying idea is the irregular decimation of a sequence according to the bits of another sequence. The result of this decimation process is a new sequence with good cryptographic properties that will be used as keystream sequence in stream ciphers