Discrete Geometry, Discrete Differential Geometry and Visualization
Discrete geometry combines combinatorics and convex geometry. There are many connections to adjacent fields, including linear and combinatorial optimization, functional analysis, number theory, topology, algebraic and tropical geometry. Key topics include the characterization, classification and enumeration of convex polytopes. For instance, toric varieties can be described most naturally in terms of normal fans of lattice polytopes. This bridge between discrete and algebraic geometry is crucial for modern methods solving systems of polynomial equations.
Discrete differential geometry starts with the observation that proper choices of discretization for many smooth manifolds are not merely approximations but share special structural properties of the smooth limits.
For instance, for surfaces defined by integrable equations, the analogous discrete integrable systems shed great light on the essence of integrability. One can follow Whitney to develop discrete exterior calculus by requiring exact sequences at the discrete level. The related theory of discrete harmonic maps and discrete minimal surfaces can also be explained in terms of finite element methods and, therefore, directly links with numerical applications. Various techniques related to circle packings and circle patterns seem to give good notions of discrete conformal and holomorphic maps.
Berlin research groups
Computer Graphicstext will follow |
TU Berlin (Computer Science) |
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Geometry and integrable systemsOur fields of interest are: Geometry, Visualization, Mathematical Physics, in particular, Discrete Differential Geometry, Differential Geometry, Integrable Systems, Riemann Surfaces. |
TU Berlin |
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Discrete and convex geometryThe fields of discrete and convex geometry deal with polytopes and more generally convex bodies in Euclidean space. Their interplay with different fields, like algebraic geometry and optimization, is rich and full of surprising connections. The study of triangulations of polytopes for example has many open directions and is relevant in questions in toric geometry. Invariants like covering radius and width of convex bodies are fundamental in discrete geometry and have important applications in mixed integer programming. |
FU Berlin |
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| Georg Loho | Discrete and Geometric MethodsTropical geometry studies polyhedral shadows of classical (algebraic) geometry. Discrete optimization is concerned with optimization problems over discrete structures. An important class of neural networks represents exactly piecewise-linear functions. These are, among others, application areas of combinatorial and polyhedral techniques. This motivates the fundamental study of discrete and geometric methods together with their applications. |
FU Berlin |
| Christian Haase | Discrete Methods in Algebraic GeometryThere is a rich interplay of algebraic geometry, polyhedral geometry and combinatorics. Deep theorems from algebraic geometry and commutative algebra are applied to combinatorial problems. Conversely, algebraic problems are reduced to considerations in discrete geometry. For instance, the theory of toric varieties is an established and very active field of research. Methods from tropical geometry and Newton-Okounkov theory have widened the scope of these methods. Applications come from matroids, auction theory, integer linear optimization and even the study of the expressivity of neural networks. |
FU Berlin |
Convex Geometry, Geometry of Numbers, and the Theory of Integer Programming Convex geometry studies metric properties or extremal problems of convex sets, usually in high dimensions. A typical question might be what is the maximal volume slice of the cube [-1,1]^n - any guess? Answers to such harmless-looking questions are of interest for problems in other areas as well, e.g., in Geometry of Numbers or in Integer Programming. In Geometry of Numbers, the interplay of convex sets with the discrete structure of lattice points is studied. For instance, an optimal upper bound on the max-norm of a non-trivial integral solution of a homogenous system of linear equations is obtained from the above slicing problem. A central aspect in the Theory of Integer Programming is the structure of integral solutions of linear optimization problems. Here, for instance, bounds in the above slicing problem can be used in order to obtain transference bounds connecting proximity and sparsity of solutions of integer programs. |
TU Berlin |
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Polyhedral, Tropical and Algorithmic GeometryMichael Joswig is interested in all aspects of polyhedral, tropical and algorithmic geometry, and this includes topics such as optimization, combinatorial topology and applications to biology. He is also active in designing and applying mathematical software (polymake, OSCAR). |
TU Berlin |
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Mathematical Geometry Processingtext will follow |
FU Berlin |
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Discrete and differential geometryMediating between discrete and differential geometry, our research explores connections between projective and non-euclidean geometries and other areas within and outside of mathematics, e.g., applications of hyperbolic geometry in computer science and number theory. |
TU Berlin |
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Dynamical Systems of geometric originKey words: discrete differential geometry, structure preserving discretization of differential equations, variational and symplectic integrators, integrable discretization, integrable models of statistical mechanics |
TU Berlin |
Links to other areas
There are natural, strong collaborative ties to the research groups in differential geometry and mathematical physics (RTA 1), and to those in discrete mathematics (RTA 4).
In particular, these include: Borot, Felsner, Rote, Sullivan
Core Courses
The BMS Core Courses in Area 5 and details about their content can be found under "Course Program":
Area 5 - Core Courses
Advanced Courses
Topics for typical advanced courses include: Convex Geometry; Mathematical Visualization; Discrete Riemann Surfaces; Tropical Geometry; Lattice Polytopes; Special Discretizations of Surfaces
The current advanced courses are listed under "Course Program":
Area 5 - Advanced Courses
