IME-USP

# Research Areas in Applied Mathematics

#### Computational Applied Mathematics

Study of techniques of mathematical and computational modeling aiming the realistic simulation of natural phenomena through computer graphics. Currently, we focus on some applications in Biology.

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Development of numerical methods for partial differential equations of fluid mechanics and applications, specially in multiphase flows and problems related to numerical weather and climate prediction.

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Optimization is the area of study that deals with the problem of finding values for variables that, among all values that satisfy a given constraint, minimize (or maximize) an objective function. We are interested in the nonlinear optimization problem with real variables, where we study analytical properties fulfilled by the solutions, in particular, the ones that can be used to guide an iterative process to solve the problem. The group also works in the area of shape optimization, where the variable is the geometry, or shape, of subsets of Rn, and which is in particular concerned with partial differential equations constraints. Optimization is a cross-disciplinary field which relies on mathematical tools from different disciplines such as differential geometry, topology, optimal control, numerical analysis and applied linear algebra. The interests of this research group ranges from theoretical problems to computational implementation and applications in real problems in Physics, Chemistry, Statistics, Economics, Engineering and Industrial Mathematics. The group has a fruitful collaboration with the Department of Computer Science from IME-USP.

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#### Differential Equations and Applications

We consider fundamental aspects of the qualitative behavior of infinite dimensional dynamical systems, derived from partial differential equations or from functional equations. These systems play an important role in the mathematical modeling of natural phenomena related to several areas, as Physics, Biology, Chemistry, Economics, Engineering and Ecology.

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To study the Hamiltonian dynamics of mechanical systems with emphasis on applications to celestial mechanics and to the motion of solid bodies inside fluids. To analyze the dynamical consequences of forces that dissipate energy.

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Study of problems of geometric mechanics, celestial mechanics and of Control theory.

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Tmain purpose of the project is to continue the work undertaking by the research team of Projeto Temático 2003/12206-0 in the fields of Linear Partial Differential Equations and Multidimensional Complex Analysis as well as to increase our activities on supervision of graduate students research work in these areas. The main topics to be studied are: (a) Local, semi-global and global solvability for linear differential operators and involutive systems of complex vector fields; (b) Regularity properties of the solutions: $C^\infty$, analytic and Gevrey hypoellipticity; (c) General properties of the approximate solutions to involutive systems of complex vector fields; (d) The theory of Hardy spaces for solutions of non-elliptic vectohe r fields; (e) The extension of the F. and M. Riesz and Rudin-Carleson theorems to complex vector fields.

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Studies of stability of stationary waves.

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The main topics studied are smooth or non-smooth vector fields, implicit differential equations, and reversible vector fields. From the geometrical point of view, for differential equations the special curves in surfaces (main curvature lines, asymptotic lines, Darboux curves) and other questions that relate dynamics and geometry are studied. For vector fields, the proposed general lines in the Thom-Smale program are studied: determination of boundary cycle quotas, global phenomena (periodic connections and orbits), singular perturbation, asymptotic behavior, dynamics and bifurcations. In summary, the aim is to study techniques from the Qualitative Theory of Ordinary Differential Equations and apply them to Differential Geometry and to problems of different areas such as control theory, physics, biology, engineering, etc.

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#### Dynamical Systems

The modern theory of dynamical systems began with the work of Poincaré in the early twentieth century and since then has grown and matured, becoming an important and active area of mathematics, with several sub-areas. The main research themes of this group are: Dynamics in dimension 2 (dynamics of homeomorphisms and diffeomorphisms of the torus, topological dynamics on surfaces, Hénon maps); Teichmüller theory and its connections with dynamics and geometry in low dimensions; Endomorphisms of the interval, critical circle maps, renormalization and parameter space; Pseudo-holomorphic curves and symplectic dynamics; Complex dynamics in dimensions 1 and 2. The group has a fruitful collaboration with the Department of Mathematics from IME-USP. For more information, access the group website

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The focus of the Ergodic Theory is the study of invariant measures for a dynamic or group actions. Besides being one of the main branches of the theory of dynamical systems, the results produced are used as a tool and are important to researchers from several other areas from both pure and applied mathematics: probabilists, specialists in rigorous statistical mechanics, in symbolic dynamics, in amenable groups and others. The most frequent research lines are Ergodic Optimization and Thermodynamic Formalism. More specifically: the study of maximizing measures, Gibbs and equilibrium measures, grounds states and phase transitions.

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#### Mathematical Modeling and Applications

Full Bayesian Significance Test (FBST) is a new statistical procedure to access the likelihood of precise hypothesis. This procedure solve several problems from similar frequentist statistical methods, as p-values, or from orthodox bayesian statistics, like bayesian factors.

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The principal aim of this project is the study of local and global human mobility and their influence on the spreading of contagious diseases. The spatial dynamics and choice of parameters should take into account the seasonal variations. The risks of contamination and epidemiological thresholds shall also be evaluated. Stochastic modeling is adequate for mobility. Due to difficulties in obtaining sufficient data, Monte Carlo techniques will be use to generate data for initial tests.

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We consider mathematical models as an aid in the interpretation of empirical evidence in social systems. We are interested specifically in the emergence and sustaining of altruism in large scale in human societies; in relations between cognitive limitations, social structure and opinion dynamics; and opinion dynamics under imitation processes, adaption, self-references and reputation effects. We employ analytic and simulation techniques from statistical mechanics of disordered systems, as well as frameworks from evolutionary games theory and complex networks. Rigorous results do not constitute our main focus, but are quite desirable whenever possible. This project is part of the Center for Natural and Artificial Information Processing Systems, CNAIPS-USP, which receives financial support from our University.

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Algebraic model for the genetic code. Mathematical modeling of gene expression.

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#### Mathematical Physics

Development of a general formalism (Lagrangian and Hamiltonian), symmetries and conservation laws, Geometric models (general relativity, caliber theories, …)

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Rigorous study of short and long-range models of statistical mechanics, such as the Ising type models, in the classic case, as well as the Hubbard and BCS types, in the quantum case. The primary objective is to produce theorems to get a better understanding of them. In general, we are looking for results about the behavior of the correlations, the description of the DLR states in the classical case and the KMS states in the quantum case, the existence or not of phase transitions, the characterization of the ground states, the properties of the pressure, among others. The mathematical tools used are probability, graph theory and combinatorics, functional analysis, C*-algebras and von Neumann algebras, convex analysis, measure theory, ergodic theory, etc.

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