10.2 Summary in English for general dissemination purposes CMAFcIO intends to play a key role in the research in mathematics done in Portugal and abroad. To this end, we set high standards of quality in all areas of our activity. Having common interests with researchers in other units and many national and international collaborations, we will give special attention to the features that shape the team’s identity. We wish to preserve the character of a center where fundamental and applied research coexist, with a handful of quite visible strong areas but where special domains may progress and evolve even in small subgroups. In the project for 2018-22 the team deals with mathematical problems currently attracting attention in several important areas. Moreover, some members will continue to tackle applications to real world problems. This feature and the areas involved both in theoretical and applied research define a profile for CMAFcIO which is singular among similar units. Members working in NONLINEAR ANALYSIS AND DIFFERENTIAL EQUATIONS will do research in ordinary and partial differential equations, with interest in nonlinear operator theory or the Calculus of Variations, and in several applications in Mathematical Physics and in Biomathematics. Among topics of research are specific equations, such as Navier-Stokes, Schrödinger and Kac-Boltzman, together with Kepler-type problems, functional-differential equations, kinetic theory of gases, phase field models, elasto-plasticity, free boundary problems, shape optimisation, renormalisation and signal processing. The research planned under OPERATIONS RESEARCH AND OPTIMISATION will concern topics like integer linear programming formulations based on convex-hull reformulations of subproblems for node/arc routing and network design; efficient dual decomposition techniques based upon the relaxation of the non-anticipativity constraints to solve models for network design; an exact algorithm for bi-objective integer linear problems based on the Tchebycheff metric. Our research in the areas of LOGIC (proof theory, model theory, o-minimality), GEOMETRY (D-modules, Wilmore surfaces, Hodge structures), DYNAMICAL SYSTEMS (Lyapunov exponents of linear cocycles) will be pursued in smaller but very active subgroups. By their scope and reach, these contributions substantiate the profile of the Center in what concerns mathematical competences. A part of the research has a clearly applied or interdisciplinary character. It is the case of some members’ involvement in mathematical studies of Epidemiology, the interest on theoretically anchored contributions to materials science, or on Optimization in Services and Industry (e. g. health, decision support system for rescue operations, fishery surveys, forest management, disjunctive programming in Chemical Engineering). CMAFcIO has a Coordinator and a Board of Directors, elected by the scientific council (that is, all the integrated PhDs members) which reflects the diversity of the scientific areas presented above. 10.3 Summary in English for evaluation Our main goals for 2018-22 are to contribute to new, cutting-edge research, to deepen our expertise in applications of mathematics and to strengthen connections with the industry, as well as to continue the engagement in advanced training and in the organization of selected international scientific events. A) Topics of future research plans NONLINEAR ANALYSIS AND DIFFERENTIAL EQUATIONS Nonlinear PDEs: qualitative study of solutions of problems arising in physics and in the sciences such as: Complex Ginzburg-Landau equation (blowup phenomena); Nonlinear Schrödinger equations; concentration, regularity of solutions, free boundaries, shape optimization; equations of Navier-Stokes type; nonlocal operators and unilateral problems; quantum Master equations, namely quantum analogs of Kac´s N-particle system; chain polymers modeled by fractional Brownian motion paths. Ordinary Differential Equations: study of the Kepler problem with nonlinear dissipation; positive periodic solutions of delayed DE; Lotka-Volterra Models, persistence and permanence; travelling waves in presence of nonlinear diffusion. LOGIC, GEOMETRY, DYNAMICAL SYSTEMS Model Theory: geometric consequences NIP in the context of Banach spaces and in neural networks; transfer cohomology tools from o-minimal linear structures with a point at infinity to the Hrushovski-Loeser non Archimedean setting; descriptive complexity of logical derivations. Geometry: complex integral transform and D-modules; constrained Willmore spectral deformation; mixed Hodge structures, Hodge-Euler polynomials; moduli spaces of Higgs bundles and quivers defined on fundamental groups of surfaces and of torus knots and links. Dynamical systems: study of continuity and large deviation problems about Lyapunov exponents of linear cocycles; dynamics of ode’s on polytopes in the context of Evolutionary Game Theory. OPERATIONS RESEARCH AND OPTIMISATION Development of: new and stronger integer linear programming formulations based on convex-hull reformulations for network design problems combined with decomposition methods (e.g. Benders and Dantzig-Wolfe); efficient dual decomposition techniques for network design problems with uncertainty features; an exact algorithm for bi-objective integer linear problems based on a variant of the Tchebycheff metric; global optimization algorithms for MIQCPs based on a two-stage MILP-NLP decomposition with piecewise relaxation techniques. B) List of some commitments in APPLIED & INDUSTRIAL MATHEMATICS Dengue and vaccination: due to the imperfection of the presently only licensed vaccine (see 4.2), vaccine developers as Sanofi-Pasteur, Butantan and Takeda are already contacting us for collaboration of vaccine trial analysis and modeling, including data analysis of the present dengue situation. Data agreements are in process. Materials Science: development, analysis and numerical implementation of a) a novel model of elasto-plasticity: topological sensitivity modelling of the plastic evolution laws, numerical simulations in synergy with experimental results; b) a phase field model for hydraulic fracturing. Optimization in Services&Industry: i) Health: Exact and heuristic approaches to a) elective surgeries planning and scheduling with public and private hospitals b) assignment and reallocation of emergency vehicles for INEM; ii) Routing: Decision support system for mass rescue operations in cooperation with the Portuguese Navy. Exact approaches and heuristics for a) the TSP with selective cities arising in fishery surveys for IPMA b) designing dissimilar routes (in collaboration with EMEL) and waste collection in collaboration with a Lisbon district council; iii) Forest Management: model integrating harvesting road building/maintenance and timber transportation taking into account EU environmental and energy policies; iv) Food - Solving a tri-objective supply chain design problem motivated by the design of a multi-echelon food bank supply chain network operated by the Portuguese Federation of Food Banks; v) Chemical engineering: use of Generalized Disjunctive Programming to incorporate complex constraints in scheduling models in petroleum distribution. C) ADVANCED TRAINING We shall open calls for grants at several levels, and take advantage of the FCT “Incentive to scientific employment” in order i) to enrol PhD students - namely in doctoral programs at FCUL and ISEG - and ii) to attract younger and promising researchers with PhD. D) INTERNATIONALIZATION Some events in which the center will be involved: Topics in Nonlinear Analysis and Nonlinear PDEs - yearly workshops at FCUL planned to start in 2018. ESGI140 European Study Group with Industry, Setúbal, June 2018. European Conference on Mathematical and Theoretical Biology July 2018, celebrating the year of Mathematical Biology - FCUL, co-organized by EMS & ESMTB. ICM 2018 Satellite Conference on Nonlinear PDEs - Fortaleza, Brazil. Thematic yearly meeting (school+workshop) on Dynamical Systems - FCUL, starting in 2018. EURO 2019 Conference of Operations Research - Dublin (3000 participants). The PC chair is a member of the center. ENOG Winter Schools in Network Optimization - series of annual schools in Lisbon, 8th edition 2019. 10th Editions of DynSystems Applied to Biology and Natural Sciences 2019 - yearly event. Vector Bundles on Algebraic Curves - and annual conference on Algebraic Geometry whose 2020 or 2021 edition will take place in Lisbon. Days in Logic 2020 - a biennial meeting directed to graduate students. Optimisation 2020 - Aveiro. Some research networks during 2018-22, namely a COST action. Members will apply to FCT and ERC Grants and involvement in H2020 and FP9. E) OUTREACH All members will be encouraged to put their work available on the CMAFcIO repository (through the RCCAAP network) and on arXiv. Other actions in 2018, Mathematics and Literature III and exhibition associated to Mathematical Biology year, are being prepared. O CMAFcIO pretende assumir um papel de relevo na investigação que atualmente se faz em matemática. Procuraremos padrões de qualidade em todas as áreas da nossa atividade. Apresentando interesses comuns com investigadores de outros centros e muitas colaborações a nível nacional e internacional, daremos especial atenção às características que moldam a identidade da equipa. Preservaremos o caráter de um centro em que investigação fundamental e aplicada coexistem, com áreas fortes bastante visíveis, mas onde domínios especiais podem progredir e evoluir mesmo em pequenos subgrupos. No projeto para 2018-22 a equipa lida com problemas matemáticos atuais e pertinentes em várias áreas importantes. Para além disso, alguns membros estão envolvidos em aplicações da matemática a problemas do mundo real. Esta característica e as áreas envolvidas tanto na investigação teórica como na investigação aplicada definem um perfil para o CMAFcIO que é único entre unidades semelhantes. A investigação em ANÁLISE NÃO-LINEAR E EQUAÇÕES DIFERENCIAIS abrange equações diferenciais ordinárias e parciais, teoria de operadores não-lineares e Cálculo de Variações, e várias aplicações (por ex. em Física-Matemática ou Biomatemática). Entre os tópicos de trabalho estarão equações específicas, como Navier-Stokes, Schrödinger e Kac-Boltzman; problemas de tipo Kepler; equações diferenciais funcionais; teoria cinética dos gases, modelos de campo de fase; elasto-plasticidade; problemas de fronteiras livres, otimização de forma; renormalização e processamento de sinais. No âmbito da INVESTIGAÇÃO OPERACIONAL E OPTIMIZAÇÃO abordar-se-ão, entre outros temas, formulações de programação linear inteira com base em reformulações para problemas de desenho de rotas e de desenho topológico de redes; técnicas de decomposição dual baseadas no relaxamento das restrições de não-anticipatividade para resolver modelos de desenho de rede; um algoritmo exato para problemas lineares inteiros bidimensionais com base na métrica de Tchebycheff. Nas áreas de LÓGICA (teoria de demonstração, teoria de modelos, o-minimalidade), GEOMETRIA (D-módulos, estruturas de Hodge), SISTEMAS DINÂMICOS (expoentes de Lyapunov de co-ciclos lineares) trabalharão núcleos de investigadores menos numerosos mas muito ativos. Pelo seu alcance e qualidade, essas contribuições substanciam o perfil do Centro no que diz respeito a competências matemáticas. Parte da investigação tem caráter aplicado ou interdisciplinar. É o caso do envolvimento de alguns membros em estudos matemáticos de Epidemiologia, trabalhos teóricos com aplicação à ciência dos materiais, ou sobre Otimização em Serviços e Indústria (por exemplo em saúde, sistema de apoio à decisão para operações de resgate, pesca, floresta, Engenharia Química). O CMAFcIO tem um Coordenador e uma Comissão Diretiva, eleitos pelo Conselho Científico (formado pelos membros integrados doutorados) que de algum modo reflete a diversidade das áreas científicas referidas.