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Center for Mathematics, University of Coimbra

Centro de Matemática da Universidade de Coimbra

Reference
UIDP/00324/2020
Project Start Date
2020-01-01
Project End Date
2024-12-31
Principal Investigator
Funding Program
Concurso de avaliação no âmbito do Programa Plurianual de Financiamento de Unidades de I&D (2017/2018) - Financiamento Programático

Abstract

10.2 Summary in English for general dissemination purposes The Centre for Mathematics of the University of Coimbra (CMUC) is a research unit in Mathematics based at the University of Coimbra. It has been in existence, with occasional changes in its organization, for about 40 years. CMUC is actively involved in research as well as in postgraduate education. It hosts more than sixty researchers, currently organized in six research groups covering large areas of Mathematics (Algebra and Combinatorics, Algebra, Logic and Topology, Analysis, Geometry, Numerical Analysis and Optimization, Probability and Statistics). All research groups participate in several international research networks, and have members who are recognized as leaders in their research areas. Transversally to the work of research groups, the Laboratory of Computational Mathematics (LCM) of CMUC, established in 2005, develops interdisciplinary research, high performance computing and numerical software, in collaboration with industry, and provides support for graduate education. CMUC organizes regular seminars within the activities of the research groups and a colloquium. CMUC regularly opens calls for postdoc positions and postgraduate education, and is associated with the UC|UP PhD Program in Mathematics (www.mat.uc.pt/phd_prog), an FCT PhD program based on the research teams at CMUC and at the Centre for Mathematics of the University of Porto. CMUC is also associated with the UAb|UC e-learning PhD Program on Computational Algebra. Beyond the work of its six research groups, CMUC has three lines of activities: Computational Mathematics, Outreach and Popularization of Mathematics, and History of Mathematics. The work on Computational Mathematics is framed by LCM (see www.mat.uc.pt/~cmuc/lcm). It promotes research in computational mathematics and scientific computing, as techniques for the solution of challenging quantitative problems arising in Science, Engineering, Finance, and Management. Outreach activities (see www.mat.uc.pt/~cmuc/OA) include the organization and scientific coordination of the Portuguese Mathematical Olympiads and other types of mathematical competitions (at national and international level), the Delfos School (an informal school founded in 2001 to attract and teach exceptional high school students, and prepare them for international Math Olympiads), and a permanent attention to issues related to Mathematics teaching. Members of the Centre have an intense activity in other areas of popularization, with regular programs of Mathematics talks, exhibitions and debates, mainly in elementary and secondary schools but also addressed to the general public. Our studies in the History of Mathematics, conducted by a small group of researchers, are focused on the study of relevant contributions to the progress of Mathematics made by a few Portuguese mathematicians of the past: see www.mat.uc.pt/~cmuc/HM. Information on CMUC activities is available at the CMUC webpage: http://cmuc.mat.uc.pt 10.3 Summary in English for evaluation Here we list the main topics each research group plans to address. The future research topics of the Algebra and Combinatorics group are as follows. In the area of Algebraic Geometry, we will look at tropical analogues of moduli spaces of curves, abelian varieties and divisors on curves. We will also proceed with the study of the Castelnuovo-Mumford regularity of vanishing ideals for general families of graphs. Within Algebras and Representations, we expect to extend the concept of ring of fractions of a Lie algebra to Lie-Yamaguti algebras. We will classify the Borel-Schur algebras of tame and wild type and will continue work on projective resolutions of Weyl modules. In Combinatorics we will focus on the study of the ribbon Schur full support, and non-symmetric Littlewood formulas in type C. In the area of Linear Algebra, we will work on inverse eigenvalue problems, such as the nonnegative IEP (leading to research on coefficient inequalities for parametrized power series) and multiplicity lists for Hermitian matrices with a given graph. We will explore several types of numerical ranges, and canonical forms for the simultaneous similarity problem. The Algebra, Logic and Topology Group will continue to develop its categorical approach to relevant problems in Algebra and Topology, keeping an eye on possible connections with Logic. Specifically, we envisage to study higher dimensional cohomology and crossed extensions of monoids; factor permutable categories and the Shifting Lemma; the two-way path between semigroups and symbolic dynamics; lax idempotent monads, their categoriesof algebras and the use of lax orthogonal factorisations in Quillen models and categorical Galois theory; regular and exact completions, and generalized equilogical structures; the use of our new results about sublocale lattices to construct completions of uniform structures in a simpler way and to study remainders in compactifications in a pointfree setting. The Analysis Group aims to study problems in analysis and applications of nonlinear partial differential equations, including image processing models, orthogonal polynomials (OPs) and function spaces. Namely, we plan to develop the theory of unbalanced optimal transport and to reveal its relations with other areas of Mathematics such as Metric/Differential Geometry, Hamilton-Jacobi equations, Convex Analysis, or gradient flows on metric spaces. Concerning the design/development of image processing and inversion, we will focus on auxiliary computer-assisted methods for wireless-capsule/conventional endoscopy images and on Biomathematics models to represent and simulate the early stages of colon cancer. A different direction of research is the study of multivariate OPs and the study of connections/applications of OPs in one variable within integrable systems, such as Painlevé equations. Another line of research relates to function spaces. We would like to establish non-smooth atomic decompositions, traces and multipliers in some function spaces, with applications to PDEs. The Geometry Group will study properties of the modular class of a Lie-infinity algebroid and will establish new relations between Lie-infinity algebroids and Courant structures. We will explore applications of Poisson-Haantjes manifolds to integrable systems. We will define Hom-group, to state Lie theorems and a Baker-Campbell-Hausdorff formula in the hom context. We will develop second-order variational methods on Riemannian manifolds to study path planning with collision avoidance for fully actuated, underactuated and multi-agent systems. We will study the Freiheitssatz for minimal meridional generator sets of knot groups, essential surfaces in knot exteriors and free actions on hyperbolic manifolds. We will address hypermap operations, combining topological and algebraic approaches. The NAO's group will pursue the research undertaken in the last years. The focus of the NA subgroup will be multi-physics/multiscale/multi-domain problems defined by systems of elliptic/parabolic/hyperbolic/integro-differerential (possible of fractional type) equations. The main goals are the development of an analytical support for the well-posedness of such problems as well as numerical approaches to solve them. Applications in drug delivery for ophthalmic and cardiovascular diseases, enhanced by electric, magnetic, temperature or ultrasound stimuli, will be addressed in an interdisciplinary environment. The Optimization subgroup investigates the development and analysis of algorithms for a wide range of optimization problems from continuous (nonsmooth, nonlinear, convex) to discrete (combinatorial, network), from single to multiobjective. There is a permanent concern in addressing challenging industrial applications and large-scale instances. Knowledge transfer is among the aims of both subgroups. As in the last decade, they will continue to collaborate in several industrial applications. The Probability and Statistics Group will develop its previous work in each topic. 1. Parametric estimation procedures for general Compound-Poisson INGARCH low order models, and fitting to real data. Characterization of bounds for finite dimensional laws for this class of models. Characterization of distribution of maxima and sums of integer valued models. 2. Automatic selection of tuning parameters for boundary kernels for distribution functions. Application to goodness-of-fit tests. Nonparametric estimation of copula functions. 3. Dependence notions in operatorial framework. Relations between stochastic orders. 4. Model development to measure the disease activity of Systemic lupus (SLE). Modeling of drug effect in treatment of microbioma of diabetic ulcers. Some of these problems will require the expertise of members of different groups, and so we foresee collaboration between the groups of Algebra and Geometry, as well as the groups of Analysis and Probability, the latter one mainly on the realm of our Lab.

Institutions

Main Institutions

  • Universidade de Coimbra Centro de Matemática (CMUC)
  • Universidade de Coimbra (UC)

Funding 845.042,10 €

Fundação para a Ciência e a Tecnologia (FCT) - Portugal

845.042,10 €