# Operator Algebras and Applications – Dedicated to Professor Maria Amélia Bastos

ENSPM 2022 – Tomar – Parallel Session – 18th july 15:30 Website: https://enspm2022.spm.pt/programa-sessoes-paralelas/

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# Linear, Algebraic and Combinatorial Structures.

# Members

# Members

### Ana L. Branco Correia

#### Academia Militar

### Ana Margarida Neto

#### ISEG - U. Lisboa

### Ângela Mestre

#### Bolseira FCT

### Carlos A. M. André

#### Faculdade de Ciências, ULisboa

### Carlos Pereira dos Santos

#### Faculdade de Ciências, ULisboa

### Fernando C. Silva

#### Faculdade de Ciências, ULisboa

### José Agapito Ruiz

#### Bolseiro FCT

### Laura Azevedo

#### ISEL

### Maria Amélia Fonseca

#### Faculdade de Ciências, ULisboa

### Maria Antónia Duffner

#### Faculdade de Ciências, ULisboa

### Maria da Purificação Coelho

#### Faculdade de Ciências, ULisboa

### Maria Manuel Torres

#### Faculdade de Ciências, ULisboa

### Marija Dodig

#### Investigadora FCT, ULisboa

### Owen J. Brison

#### Faculdade de Ciências, ULisboa

### Raquel Simões

#### Bolseira FCT

### Sonia Carvalho

### Susana Furtado

#### Faculdade de Economia, U. Porto

# Collaborators

### Filipe Gomes

#### PhD Student, FCUL (LisMath)

### Inês Legatheaux Martins

#### PhD Student, FCUL

### João Dias

#### PhD Student, FCUL (LisMath)

### Jocelyn Lochon

#### PhD Student, FCUL (LisMath)

### Maria da Graça Pereira

#### PhD Student, FCUL

### Pedro J. Freitas

#### Faculdade de Ciências, ULisboa

### Pedro Matos

#### PhD Student, FCUL (LisMath)

### Tânia Zaragoza Silva

#### PhD Student, FCUL (LisMath)

The Group conducts research primarily on Representations of Groups, Matrix Theory and Linear Systems. It also pursues ongoing research on related topics: Hom-configurations, Sequence Subgroups in Finite fields, Rees Algebras of Modules, Symmetry of Tensors and Related Combinatorics, Cellular Automata and Non-Linear Dynamical Systems, Polytopes and Polygon Spaces, Riordan Matrices and Symbolic Calculus.

- Representation of Groups
We use methods of Algebraic, Geometric and Combinatorial Representation Theory to study Character and Supercharacter Theories of Finite and Infinite (Topological) Groups. On the other hand, methods from Functional Analysis, Ergodic Theory and Topological Dynamics are used to study extreme characters and supercharacters of infinite groups which occur as inductive limits of finite groups. We also study the structure of Hopf algebra on the space of superclass functions defined on all finite unitriangular groups which is isomorphic to the combinatorial Hopf algebra of symmetric functions on noncommuting indeterminates, and explore applications in Algebra, Number Theory, Statistics and Algebraic Combinatorics.

- Matrix Equations and Matrix Completion Problems
We study the existence of solutions of matrix equations and systems of matrix equations with some prescribed properties, and consider generalizations in order to study the stabilization of linear systems with input variables.

We also consider the existence of certain matrices with prescribed entries and properties, extend our research to operators in spaces of infinite dimension and to matrices over noncommutative rings, and consider engineering applications. We also study how the properties of a matrix vary under perturbations on some of its entries.

- Matrix pencil completion problems
We search for an explicit and constructive solution for the general matrix pencil completion problem: describe the possible Kronecker invariants of a pencil with a prescribed subpencil. A related problem is to study small perturbations on pencils. We also explore purely combinatorial aspects of matrix completion problems, study the close relationship between completion problems of matrix pencils and the representation theory of quivers, and consider matrix problems arising from engineering areas, as computer vision and signal processing.

- Symmetries of Tensors
Symmetries of tensors arise naturally in connection to the irreducible characters of the symmetric group. Schur-Weyl dualities provide a fruitful approach to the study of classical problems in different situations, namely in the study of symmetries of harmonic tensors, or in the setting of the infinite symmetric group. A related problem concerns about generalized matrix functions and their directional derivatives, and opens new lines of research.

- Calabi-Yau triangulated categories
These appear in many branches of mathematics and physics: conformal field theory and string theory in theoretical physics, homological mirror symmetry in algebraic and symplectic geometry, and cluster-tilting theory in representation theory. Particular research is devoted to finding appropriate representation-theoretic categories associated to Riemann surfaces.

- Sequence Subgroups in Finite fields
We describe of the non-standard configurations of f-sequence subgroups when f(t) is any polynomial.

- Rees Algebras of Modules
We study several asymptotic and arithmetical properties of certain classes of modules arising in connection with Algebraic Geometry, such as equimultiple modules.

- Cellular Automata and Non-Linear Dynamical Systems
We use timewise updating and spacewise updating to simulate complex systems with cellular automata, and derive certain exact analytical formulae to quantify the algorithmic complexity of spacewise evolution of certain cellular automata.

- Combinatorial Game Theory