Algebra and Algebraic Geometry

Algebra research at IMPA has been carried out mainly in the areas of Algebraic Geometry and Representation Theory.

Algebraic Geometry studies objects defined by polynomial equations in several variables. It is one of the most traditional areas of mathematics, with numerous connections to other fields, such as Number Theory, Mathematical Physics, Cryptography, Complex Geometry, among others. The main research topics in Algebraic Geometry currently being developed at IMPA include:

Birational geometry: Cremona group, Minimal Models Program, Sarkisov Program, rationality problems and measures of irrationality.

Special geometries: Fano varieties, K3 surfaces and Calabi-Yau varieties, abelian varieties.

Algebraic curves: moduli, linear series, limit linear series, compactified Jacobians.

Representation Theory seeks to study abstract algebraic structures through their representations as symmetries in linear spaces. This area has wide applications in various branches of mathematics and other sciences, from Fourier Analysis and Algebraic Geometry to Number Theory and Automorphic Forms, especially through the Langlands Program. The main research topics in Representation Theory currently being developed at IMPA are:

Lie algebras: nilpotent orbits and their quantizations, Kac-Moody algebras and their representations.

Vertex algebras: structure and classification of vertex algebras, moonshine and automorphic forms, tensor categories, conformal blocks and chiral homology.

Sigma models and their symmetries: arc spaces, loop algebras, chiral de Rham complex, Courant algebroids and generalized geometry.

Analysis and Partial Differential Equations

Analysis developed from the process of moving to the limit in Differential and Integral Calculus. One of its main objectives is to solve differential and integral equations, characterizing the solution space and ensuring the convergence of solution methods by approximation. Many of its techniques were later unified in Functional Analysis, in which function spaces are considered in an abstract way.

Analysis is an indispensable tool in physics, geometry, engineering and practically all fields of mathematical applications.
The main areas of research currently being carried out at IMPA are:

Partial Differential Equations in Mathematical Physics

Non-linear evolution equations are studied, such as those of Korteweg-de Vries, Benjamin-Ono, Navier-Stokes and Euler, and aspects such as the existence of solutions, uniqueness, dependence on initial data and asymptotic behavior are addressed. Another important topic is the Schrödinger equation with time-dependent Hamiltonian functions, which is studied through the spectral properties of the associated operators.

Inverse Problems and Applications

The theory of inverse problems is dedicated to the determination of parameters or functions that enter into physical models based on properties or observations of the solutions of the equations that characterize these models. In general, the models considered lead to partial differential equations, the solution of which requires the use of numerical methods in conjunction with analytical techniques. The area of inverse problems has been the subject of much recent activity and has multidisciplinary interfaces with applications, for example, in computerized tomography, geophysics, semiconductors and quantitative finance.

Solitons and Nonlinear Analysis

Solitons are high-amplitude waves that propagate in non-linear media and interact without substantial changes in their shape. This theory developed sharply from the 1970s onwards, seeking to understand the surprising robustness of this phenomenon and to develop its numerous applications, ranging from optical engineering to signal transmission.

Data Science

The term “Data Science” refers to an interdisciplinary field that integrates statistics, computer science and applied mathematics, with the aim of extracting knowledge and identifying patterns from data of different kinds. In addition to practical applications, the area advances academically by proposing new methods of modeling, inference and analysis in contexts characterized by large volumes, diversity and uncertainty of information.

The main research directions in Data Science at IMPA are:

 

  • Deep learning and artificial intelligence;
  • Partial differential equations and neural networks inspired by physics;
  • Mathematical statistics and probability in high dimensions (concentration of measure, limit theorems, sub-Gaussian estimators);
  • Network models and random graphs;
  • Uncertainty quantification in machine learning;
  • Monte-Carlo simulation;
  • Computer vision;
  • Machine learning theory;
  • Transporte ideal e aprendizado de máquina.

Our group has important connections and collaborations with other groups at IMPA, such as Visual Computing, Fluid Dynamics and Probability. We are also involved in several industrial projects with non-academic partners through IMPA’s Center for Projects and Innovation (Centro Pi). Recent Centro Pi project themes include:

  • analysis of medical MRI scans;
  • automatic detection of deforestation in the Amazon;
  • seismic inversion and oil exploration in the pre-salt;
  • preventive maintenance;
  • short-term weather forecast;
  • computer vision for document understanding.

Combinatorics

Combinatorics studies the number, size and behavior of discrete structures. Our line of research is divided between extremal and probabilistic combinatorics.

Extreme Combinatorics

Extremal combinatorics studies the largest or smallest size a discrete structure can have. Questions in this area are like: what is the maximum number of edges in a graph with n vertices that does not contain a cycle of size k? What is the maximum size of a subset of the integers {1,…,n} that does not contain an arithmetic progression with k elements? What is the maximum density of a d-dimensional packing of spheres? What is the largest number of vertices in a graph that contains neither a clique of size s nor an independent set of size t?

Probabilistic Combinatorics

Probabilistic combinatorics studies the number and typical behavior of discrete random structures. Of particular interest is the probabilistic method that uses a random structure to demonstrate the existence of an extremal example. Questions in this area are of the type: what is the number of graphs with n vertices that do not contain a clique of size k? What is the distribution of the chromatic number of a random graph chosen uniformly at random? What is the probability that a matrix with 0 or 1 entries chosen uniformly at random is singular?

Methods

The methods used in both areas can be combinatorial, such as inductive arguments and algorithms. On the other hand, more elaborate combinatorial techniques developed over the last 50 years such as dependent random choice, absorption, regularity, containers and the differential equations method are also used. Methods from other areas such as probability, Fourier analysis, dynamical systems, algebraic geometry and high-dimensional geometry are also widely used. Many of the most interesting works in the area mix elementary methods, more advanced combinatorial techniques and methods from other areas.

Visual Computing

The VISGRAF Vision and Computer Graphics Laboratory was created in 1989 to promote and develop research, teaching and project development activities in related areas involving geometric models, images and other digital media.

IMPA’s interest in Computer Graphics dates back some ten years, to the early 1980s, when it acquired a Textronix graphics terminal, which is now part of the laboratory’s historical collection. Visgraf adopts the philosophy that this area is a branch of Computational Applied Mathematics. As such, the group is very interested in the mathematical foundations of Visual Computing and its applications.

The Laboratory’s main areas of research are:

  • Image Analysis and Processing;
  • Image Synthesis and Visualization;
  • Geometric Modeling and Interaction;
  • Animation and Multimedia.

Below is a more detailed summary of the lines of research being carried out by the group.

Modeling and Visualization

  • Hierarchical Mesh Structures;
  • Subdivision surfaces 4-8;
  • Multiscale Shape Synthesis;
  • Dynamic Texture of Implicit Surfaces.

Vision and Image Processing

  • Virtual Arbitrator;
  • Image Quantization;
  • Digital halftone with space-filling curves.

Animation and Multimedia

  • Visorama: Virtual Reality with Panoramas;
  • Motion Capture and Processing;
  • Deformation and Metamorphosis of Graphic Objects;
  • Virtual Scenarios and Image Composition.

Interfaces and Applications

  • VisMed: Visualization and Analysis of Medical Images;
  • 3D photography;
  • Visualization of Geographic Data;
  • Video Databases.

The VISGRAF Laboratory is supported by FINEP, CNPq, FAPERJ, Google, and maintains regular collaboration with PUC-Rio, the École Polytechnique and the Courant Institute of Mathematical Sciences.

 

Fluid Dynamics

Fluid Dynamics is an area of research that began a long time ago, together with Calculus. It brings together techniques from Mathematical Analysis – such as Asymptotic Methods, Approximation Theory, Conservation Law Theory and Reaction-Diffusion Equations – and Dynamic Systems, such as Bifurcation Theory, among others.

Due to its technological relevance and the wide range of interesting mathematical problems it gives rise to, it remains one of the most important areas of Partial Differential Equations. In fact, the first computers were built during the Second World War to decode encrypted information. It soon became apparent that it would have wide-ranging applications, the first of which, on a non-military basis, was computer weather forecasting.

From 1987 onwards, a small research group in Fluid Dynamics with an emphasis on useful applications for the country was established at IMPA, which became the leading research group in this area with a mathematical emphasis.

The main applications are

  • Oil recovery;
  • Numerical Weather Forecasting;
  • Wave Propagation in Heterogeneous Media;
  • Turbulence in liquids.

In more detail, the group has worked on: Flow in Oil Reservoirs, Meteorology, Coastal Wave Propagation and Acoustic Waves in heterogeneous media, Numerical Analysis, Domain Decomposition and Parallel Computing.

The group has a network of Brazilian and foreign collaborators and its activities have received support from various institutions and funding agencies.

Mathematical Economics

Mathematical economics is the application of mathematics to the development of economic models with the aim of building a rigorous and unified economic theory.

The techniques of Functional Analysis, Topology, Differential Topology are widely used in the central economic model. General Equilibrium Theory, Differential Equations and Dynamic Systems provide mathematical economists with the basic tools for analyzing economic processes or dynamics. Probability is fundamental in the study of economic models, where risk and uncertainty are present.

Allied to Mathematical Economics is Econometrics, which studies the properties of data generation processes, techniques for analyzing economic data and methods for estimating and testing economic hypotheses. In this area, the tools developed by Statistics are central.

The main areas of research currently carried out at IMPA are:

  • General equilibrium;
  • Information Economics and Uncertainty;
  • Incomplete Markets;
  • Dynamic Programming and Capital Theory;
  • Capital Theory;
  • Dynamic Programming.

Complex Geometry and Holomorphic Foliations

The theory of complex differential equations began in the 19th century with the work of Briot and Bouquet, Poincaré, Picard, Darboux, Painlevé, Halphen and, at the beginning of the 20th century, Dulac. When an equation is given by polynomials, it naturally defines a foliation by sheets of dimension one in Euclidean space or in one of its compactifications. The main question is to analyze the dynamics of the solutions (the leaves), both locally and globally.

Modern research in the area was taken up by Reeb in France, inspired by Painlevé’s work, and by Petrovsky, Landis and Iliashenko in Russia, motivated by Hilbert’s 16th problem. The 1970s brought intense development in France, with the contributions of Moussu, Mattei, Cerveau, Martinet and Ramis, and in Brazil, with the advances made by the IMPA group. Since then, the group has made a fundamental contribution to establishing important theorems, often with the collaboration of researchers from Brazilian universities.

Phenomena modeled by real polynomial differential equations naturally generate complex differential equations. The interface between the real equation and its complexification leads to a better understanding of the phenomenon being modeled. One of the reasons is that the study of the complexified problem allows the use of tools from Complex Analysis and Algebraic Geometry, revealing non-apparent aspects of the real problem and producing results that can be interpreted in the original context. Conversely, the study of differential equations of the Picard-Fuchs type and those arising from Gauss-Manin connections results in rigorous demonstrations of some fundamental theorems of Algebraic Geometry, such as the Noether-Lefschetz theorem. These equations are satisfied by periods of fibrations and gave rise to Hodge’s theory of algebraic geometry.

The research carried out at IMPA deals with a wide range of problems ranging from classical questions of integrability by means of transcendent functions to more modern questions about the dynamics of foliations and applications in Algebraic Geometry and Hodge Theory.

Some of the lines of this research are:

  • Limit sets of foliations;
  • Cross-sectional structure of complex veneers;
  • Projective leaflets of codimension one;
  • Birational geometry of foliations;
  • Linearization of foliations and normal neighborhoods;
  • Algebraic solutions of algebraic differential equations;
  • Uniformization of the leaves of a complex veneer;
  • Indices and invariants of projective foliations;
  • Hilbert’s 16th problem and zeros of abelian integrals;
  • Picard-Fuchs equations and Picard-Lefschetz theory;
  • Differential equations of modular forms;
  • Calabi-Yau varieties;
  • Hodge cycles and algebraic cycles.

Differential Geometry

Differential Geometry consists of applications of the methods of local and global analysis to Geometry problems.

It has deep interconnections with other domains of mathematics, such as Partial Differential Equations (minimal subvarieties); Topology (Morse Theory and characteristic classes); Complex Analytic Functions (complex varieties); Dynamical Systems (geodesic flow) and Group Theory (homogeneous varieties). The language and models of Differential Geometry have found applications in related fields, such as Relativity and Celestial Mechanics. Given its interdisciplinary nature, Differential Geometry has shown great vitality and has developed in various directions, which present a considerable volume of research today.

The main current lines of research in Differential Geometry are:

  • Minimum and Constant Mean Curvature Subvarieties;
  • Riemannian Varieties;
  • Isometric Immersions.

Symplectic Geometry

Symplectic geometry is a branch of differential geometry with historical roots in the geometric formulation of classical mechanics developed in the nineteenth century, known as the Hamiltonian formalism. Its modern development has been driven by deep interactions with several areas of contemporary mathematics—including topology, dynamical systems, and complex geometry—as well as with mathematical physics.  

Research in symplectic geometry at IMPA is organized around three main themes:

  • Poisson geometry;
  • Symplectic topology;
  • Kahler geometry, Sasakian geometry and special geometries.

Poisson geometry originates in the study of Poisson brackets, introduced in the classical works of Poisson, Jacobi, Lie, and Hamilton. These structures motivate the notion of Poisson manifolds, which generalize symplectic manifolds. Since the 1980s, Poisson geometry has become an active and rapidly evolving field, combining methods from symplectic geometry, foliation theory, and Lie theory. Moreover, Poisson structures arise naturally as semiclassical limits of quantum systems, serving as a bridge between classical differential geometry and noncommutative algebraic structures.  

Symplectic topology emerged from Mikhael Gromov’s groundbreaking work in 1985, which established striking rigidity phenomena in symplectic geometry and introduced the theory of pseudoholomorphic curves in the area. Together with fundamental contributions by Conley, Zehnder, and Witten, Gromov’s ideas paved the way for Andreas Floer’s breakthrough solution of Arnold’s conjecture on the number of fixed points of Hamiltonian diffeomorphisms, as well as for the construction of Floer homology. Since then, a wide range of Floer-type homology theories have been developed, leading to powerful applications in Hamiltonian dynamics and to further rigidity results in symplectic geometry. These developments have given rise to a new area often referred to as symplectic dynamics.

Kahler geometry is a vast theory uniting the methods of symplectic geometry, differential geometry and partial differential equations to study complex manifolds. Lie groups acting on Kahler manifolds are central in symplectic reduction, relating symplectic geometry and geometric invariant theory. In odd dimension, Sasakian geometry plays the same role to contact geometry as Kahler geometry to symplectic. These theories are indispensable in the study of complex manifolds, singularity theory, and mathematical physics. Mirror symmetry relates the symplectic invariants of a Kahler manifold to complex invariants of its mirror dual, giving a way to study the Floer cohomology and Fukaya category of a symplectic manifold using the derived category of its mirror dual. The special geometries (Calabi-Yau, hyperkahler, G2) play the central role in these applications.

The main research directions currently pursued at IMPA include:

  • Poisson manifolds and related geometries: Dirac structures and Courant algebroids, applications to symmetries and reduction;
  • Lie-theoretic aspects of Poisson geometry: Lie groupoids, Lie algebroids and their higher categorical versions;
  • Floer homology, periodic orbits on prescribed energy levels of autonomous Hamiltonian systems, and dynamical phenomena beyond periodic orbits;
  • Computation of various versions of Floer homology and their applications; computation of symplectic capacities; the study of symplectic embeddings;
  • Symplectic aspects of Euler-Lagrange flows and its relation with Floer homology;
  • Variational methods and Euler-Lagrange flows;
  • The relation between convex and symplectic geometries; generalized billiards; integrable systems;
  • Kahler geometry and its applications: geometric invariant theory, symplectic packing, homological mirror symmetry and its application to Fukaya category;
  • Locally conformally Kahler geometry and Sasakian geometry: algebraic geometry and differential geometry on a cone over a contact manifold;
  • Special geometry: hyperkahler, hypercomplex, Calabi-Yau, G2-manifolds; geometric structures on Lie groups, nilmanifolds and solvmanifolds; Monge-Ampere equations on manifolds with special geometries.

Optimization

Activities in this area at IMPA began in the 1970s with the group then called Operations Research. Currently, the group’s research interests focus on continuous optimization and related areas.

Specific research topics include:

  • Iterative methods for large convex optimization or convex feasibility, with applications in image reconstruction from projections (e.g. computed tomography);
  • Computational methods for nonlinear complementarity problems and variational inequalities;
  • Parallel optimization algorithms;
  • Generalizations of the proximal point method for convex optimization and monotone variational inequalities (including recently non-convex and non-monotone cases);
  • New approaches to duality in nonlinear programming;
  • Non-monotone methods for non-linear optimization.

Recently, three new topics have been added: new regularity theories in finite dimension (particularly 2-regularity), extensions of maximal monotone operators, generalizing the epsilon-subdifferential of a convex function, and optimization in Banach spaces.

Probability

Probability theory is fundamentally aimed at modeling phenomena subject to uncertainty. Its use in planning and statistical inference is well known. It has proved to be of great importance in areas such as Electrical Engineering, Information Theory (signal detection, control) and Physics (Statistical, Classical or Quantum Mechanics). In addition, probabilistic models, concepts and methods are now widely used in Chemistry, Social Sciences and Economic Sciences.

The main lines of research at IMPA are:

  • Hydrodynamic Behavior of Particle Systems;
  • Small Random Perturbations of Deterministic Systems;
  • Percolation;
  • Theories of Great Deviations;
  • Markovian Systems with Infinite Interacting Components.

An important objective is to obtain mathematically rigorous models that help us understand sensitive issues related to Statistical Mechanics, such as the hydrodynamic limit. Connections with statistical image reconstruction problems are also examined.

Dynamical Systems and Ergodic Theory

The Theory of Dynamical Systems dates back to the work of Henri Poincaré on differential equations at the end of the 19th century. Since most differential equations cannot be solved by classical methods, usually called integration by quadratures, Poincaré advocated a new approach: instead of seeking explicit formulae for the solutions, describe their qualitative characteristics using topological, geometric, and probabilistic tools, complementing this analysis with a numerical study of the differential equation.

In the 20th century, Birkhoff, Smale, Palis, Anosov, Arnold, Sinai, and many others proved the power of this idea. A crucial step in the development of this theory was the emergence of the notion of the uniformly hyperbolic system, introduced by Smale and used by Anosov in his ergodicity theorem for the geodesic flow on Riemannian manifolds of negative curvature. Moreover, Palis and Smale conjectured that uniformly hyperbolic systems, which are the most chaotic, are also the most stable.

The IMPA group has made leading contributions in topics such as: bifurcation theory; homoclinic tangencies and fractal dimensions; strange attractors; interval transformations; physical measures; partially hyperbolic systems or those with dominated decomposition; interval exchange and Teichmüller flow; spectral theory of Schrödinger cocycles; Lyapunov exponents; and many others.

Ergodic Theory studies the statistical properties of dynamical systems. In mathematical terms, it deals with measures on the configuration space that remain stationary as the phenomenon evolves. How and at what speed does the system evolve from the initial state to equilibrium? In equilibrium, what are the most probable configurations? At what speed does the system return to configurations close to the initial configuration?

This subject originated in the kinetic theory of gases, developed in the 19th century by the physicists Boltzmann, Maxwell, and Gibbs. Gases consist of a huge number of particles (molecules) in constant interaction, making it infeasible to account for the individual behavior of each particle. Alternatively, Boltzmann proposed deducing the experimental properties of gases in nature from a statistical analysis of the entire population (ensemble) of their molecules.

The IMPA research group works on various aspects of this theory, such as: ergodic properties of partially hyperbolic systems; thermodynamic formalism for non-uniformly hyperbolic systems; physical measures of dissipative strange attractors; stochastic stability; mixing properties of interval exchanges, translation flows, and Teichmüller flows; Lyapunov exponents of linear cocycles and diffeomorphisms.

A dynamical system is said to be stable if its behavior does not change qualitatively when its evolution law is slightly modified. For example, if we slightly modify the size, weight, or shape of a pendulum, it remains true that it will oscillate for a while until it stops due to energy dissipation caused by friction.

In nature, there are many examples of stable systems, such as the pendulum with friction, but also others whose behavior is highly sensitive to small variations in the evolution law. For instance, small modifications to an ecological habitat can lead to profound changes in the species that coexist there, including mass extinctions. How can we understand, explain, and predict these facts rigorously? How can we characterize stability or instability from a mathematical viewpoint? These are the fundamental questions of Bifurcation Theory.

The work developed at IMPA in this area since the 1970s uses highly sophisticated tools, including fractal dimensions, to analyze profound and complex changes in dynamical behavior.

Current research lines at IMPA are:

  • Strange attractors, physical measures, stochastic stability;
  • Homoclinic bifurcations and fractal dimensions;
  • Symplectic dynamics;
  • Unidimensional dynamics;
  • Lyapunov exponents and non-uniform hyperbolicity;
  • Partial hyperbolicity and dominated decomposition;
  • Holomorphic dynamics;
  • Rotational theory.