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Research Topics

1. ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS

Problems in differential equations, completely integrable systems, spectral theory and geometric aspects of nonlinear functions. The subjects are strongly related to each other.

Projects:

i. Properties of Solutions of Second-Order Elliptic PDE
ii. Global Geometry of Nonlinear Differential Operators
iii. Topics in Nonlinear Analysis and Spectral Theory
iv. Regularity in Kinetic Equations
v. Error Prediction for Spectral Methods
vi. Modeling for Population Analysis
vii. PDE and Analysis

2. COMBINATORICS

Discrete structures with emphasis on theories of coverage for dimers.

Projects:

i. Combinatorics of Domino Tilings
ii. Logic and Combinatorics

3. COMPUTER GRAPHICS AND GEOMETRY PROCESSING

Both in academics and industry, computing is a fundamental tool that raises mathematic challenge in the acting and optimization of multidimensional geometric data. These studies require methods from several mathematic fields that include: topology, geometry, combinatorics, functional analysis, algebra and numeric methods.

Projects:

i. Approximation of invariant curves
ii. Computational Methods in the Characterization of Porous Media in Aquifers.
iii. Computational topology and mesh structure
iv. Visualization, animation and interface
v. Large Scale Data Visualization
vi. Non-photorealistic rendering

4. MATHEMATICAL PHYSICS

The Mathematical Physics occupies the space between Theoretical Physics and Pure Mathematics. Mathematically based physical theories, building models with the standard of rigor required of any mathematical area, and creates new mathematical structures.

Projects:

i. Fundamentals of physics

5. ALGEBRAIC GEOMETRY

Algebraic Geometry studies properties of spaces locally defined by polynomial equations. Particularly important are properties invariant by birational transformations, that is, invariant by isomorphims over dense open sets, and not necessarily over the whole variety: birational geometry gives rise to nice classifications of curves, surfaces, and higher dimensional varieties. Directly linked with birational geometry is the study of moduli spaces, that is, spaces (variety, schemes, stacks) that parametrize isomorphism (or birational) classes of objects, that can be varieties, vector bundles, sheaves.

Projects:

i. Hilbert schemes of points..
ii.Moduli Spaces of Sheaves.

6. DIFFERENTIAL GEOMETRY

Varieties endowed with different structures such as Riemannian metric or foliation, minimal surfaces or constant mean curvature, compact leaves and leaves curvature. The methods used are Geometric, Analytical and Topological. .

Projects:
i. Lagrangian Dynamics, global geometry and topology of varieties.
ii. Foliations whose leaves have Thurston geometries
iii. Affine Geometry
iv. Minimal Surfaces and Constant Mean Curvature
v. Symplectic Geometry and Group Action
vi. Differential Geometry and Lie Groups

7. PROBABILITY AND STOCHASTIC PROCESSES

The Theory of Stochastic Processes examines the evolution (temporal or special) of systems with random behavior. Its techniques allow extracting the collective behavior of systems composed by a large number of components.

Projects:

i. Stochastic Methods in Finance and Actuarial
ii. Nonlinear Fluctuations of Particle Systems
iii. Phase Transition in partial differential equations
iv. Stochastic Modeling in Actuarial Sciences and Financial Markets

8. DYNAMICAL SYSTEMS

The study of asymptotic behavior of orbits of endomorphism, diffeomorphism and flows, with emphasis on intrinsic properties. The focus is on the stability problems and ways in which this feature disappears.

Projects:

i. Ergodic properties of non-uniformly hyperbolic dynamical systems. .
ii. Bifurcation and cycles
iii. Geodesic flows in manifolds without conjugate points.
iv. Lagrangian flows.
v. Geometric Theory of Control
vi. Robust Transitivity and weak hyperbolicity

9. TOPOLOGY

The study of topology in foliation theory, group actions and geometry.

Projects:

i. The topology of the space of Locally Convex Curves on the two-sphere
ii. Algebraic sets and Invariant Foliations
iii. Asymptotic linking of Rk actions
iv. Stability of Compact Actions

Research Topics - PDF

  Mathematics Department / PUC-Rio
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