# Research Teams

##
Algebra

**Algebra, Combinatorics and Cryptology, Prof. Dr. Tran (i.R.)**

**Discrete Mathematics and Algebra, Prof. Dr. G. Törner**

[Research team]

**Commutative Algebra and Combinatorics,**** Prof. Dr. Herzog (i.R.)**

Cohen-Macaulay rings, Maximal Cohen-Macaulay modules, Primary decompositions, Powers of ideals, Gröbner bases, Toric algebras, Stanley Reisner rings, Stanley decompositions, Hibi rings und Hibi ideals, Koszul algebras, Binomial ideals, Edge rings und Edge ideals, Polyominoes.

##
Algebraic Geometry and Number Theory

Essen Seminar for Algebraic Geometry and Arithmetic

**Algebraic Geometry, Prof. Dr. Georg Hein**

Derived categories, moduli spaces of vector bundles, generalized theta divisors.

**Algebraic Geometry, Prof. Dr. Jochen Heinloth**

Algebraic stacks, moduli of bundles, geometric Langlands program. [Research team]

**Algebraic Geometry and ****Algebraic Topology, Prof. Dr. Marc Levine**

Algebraic \(K\)-theory, algebraic geometry, motivic homotopy theory. [Research team]

**Algebraic and Complex Geometry, Prof. Dr. Daniel Greb **

Birational Geometry, Geometric Invariant Theory, Moduli spaces of vector bundles, Kähler Geometry, Hamiltonian group actions and momentum maps. [Research team]

**Arithmetic Geometry, Prof. Dr. M. Bertolini****, Prof. Dr. W. Lempken **

Group theory, representation theory of finite groups, especially: Theory of finite simple groups, character theory of finite groups, permutation codes.

Arithmetic geometry and number theory, especially: Elliptic curves, modular forms, \(L\)-functions, rational points, algebraic cycles. [Research team]

**Arithmetic Geometry, Prof. Dr. J. Kohlhaase**

Representation theory of \(p\)-adic Lie groups, Galois groups of local fields, moduli spaces of formal groups. [Research team]

**Arithmetic Geometry, Prof. Dr. Vytautas Paškūnas**

Representation theoretic methods in number theory, \(p\)-adic Langlands program. [Research team]

**Number Theory and Arithmetic Geometry, in particular explicit and algorithmic methods****, Prof. Dr. Ulrich Görtz, ****Prof. Dr. Dr. h.c. Gerhard Frey (i.R.)**

Modular forms, automorphic representations and Galois representations - Langlands program - Elliptic curves, modular curves, Shimura varieties - Representations of absolute Galois groups of number fields - Linear algebraic groups and their combinatorics - Applications in coding theory and cryptography. [Research team]

##
Analysis

**Applied Functional Analysis, Analysis of Partial Differential Equations, Prof. Dr. P. Wittbold **

Nonlinear evolution equations, nonlinear semigroups, monotone/accretive operators, \(L^1\)-theory, Lebesgue spaces with variable exponents and generalized Orlicz spaces, scalar conservation laws, nonlinear PDEs of mixed type (in particular, elliptic-parabolic equations), integro-partial differential equations (in particular, with fractional derivative in time), nonlinear nonlocal diffusion equations, stochastic nonlinear PDEs, numerical approximation methods (in particular, finite volume methods) for nonlinear PDEs.** **[Research team]

**Differential Equations and Differential Geometry, Prof. Dr. Ulrich Dierkes, Prof. Dr. Gerhard Freiling (i.R.), Prof. Dr. Wolfgang Schreiber (i.R.)**

Partial differential equations - Calculus of Variations - Differential geometry. [Research team]

**Geometric Analysis, Prof. Dr. A. Gastel**

Differential equations and variational problems of Geometric Analysis: existence, regularity, symmetry, nonuniqueness. [Research team]

**Nonlinear Analysis and Modeling**, **Prof. Dr. Patrizio Neff**, **Prof. Dr. Mircea Birsan**

Analysis and numerics of nonlinear PDE systems, elliptic regularity, calculus of variations, modeling in continuum mechanics, elasticity and plasticity theory, microstructures, shells and plates, membranes. [Research team]

**Nonlinear partial differential equations, Prof. Dr. Georg Weiß**

Free boundary problems, free surface flow, singular limits, calculus of variations, regularity questions. [Research team]

**Partial Differential Equations and Calculus of Variations, Prof. Dr. C. Scheven**

Geometric variational problems, geometric flows, degenerate and singular parabolic equations and systems, obstacle problems. [Research team]

##
Didactics of Mathematics

**Prof. Dr. Bärbel Barzel**

[Research team]

**Prof. Dr. Andreas Büchter**

Active hands-on learning in mathematics instruction for understanding and development of mathematical concepts - Mathematics learning and language skills - Spatial abilities and mathematics achievement - Student’s perceptions about mathematical concepts - Curriculum research and development - Mathematics during the initial phase of different courses of studies. [Research team]

**Prof. Dr. Lisa Hefendehl-Hebeker ****(i.R.)**

[Research team]

**Prof. Dr. Gerhard Herden** **(i.R.)**

**Prof. Dr. Hans Niels Jahnke**** (i.R.)**

Genesis of argumention and proof – History of mathematics – History of mathematics in education. [Research team]

**Prof. Dr. Benjamin Rott**

Mathematical problem solving – Heuristic process regulation. [Research team]

**Prof. Dr. Florian Schacht**

Concept formation in mathematics - Digital technologies - Mathematics and language - Epistemological questions and theories in mathematics education - Inferentialism in mathematics education. [Research team]

**Prof. Dr. Petra Scherer**

Research in education and learning processes. [Research team]

**Prof. Dr. Heinz Steinbring**

Basic research in mathematics education – Epistemological analysis of mathematical interaction processes – Development and research of mathematical teaching and learning processes in cooperation with school praxis. [Research team]

##
Engineering Mathematics

**Engineering Mathematics, Prof. Dr. W. Heinrichs**

Spectral and pseudospectral methods - multilevel - Stokes and Navier-Stokes equations - splitting techniques for saddle point problems - stabilization techniques for convection-diffusion problems. [Research team]

##
Numerical Mathematics

**Applied Mathematics and Numerical Analysis, Prof. P. Pozzi, PhD**

Numerical analysis, finite elements methods, geometrical partial differential equations. [Research team]

**Numerical Methods for Partial Differential Equations, Prof. Dr. Johannes Kraus**

Numerical Solution of Partial Differential Equations, Finite Element Methods,Numerical Linear Algebra, Multigrid and Algebraic Multilevel Methods, Block Factorization Preconditioners. [Research team]

**Numerical Mathematics, Prof. Dr. G. Starke**

Mixed Finite Elements, Numerical Solid and Fluid Mechanics. [Research team]

**Numerical Mathematics and Numeric Simulation**, **Prof. Dr. Axel Klawonn**

University of Cologne (co-opted member). [Research team]

##
Optimization

**Inverse Problems, Prof. Dr. Christian Clason**

Parameter identification - nonsmooth optimization - mathematical imaging - medical imaging. [Research team]

**Nonlinear Optimization, Prof. Dr. Arnd Rösch**

Numerical Methods for Nonlinear Optimization Problems Governed by Partial Differential Equations - Optimality Conditions - Discretization and Regularization Concepts. [Research team]

**Optimal Control of Partial Differential Equations, Prof. Dr. Irwin Yousept**

Optimal Control of Partial Differential Equations, Numerical Analysis of Partial Differential Equations, Nonlinear Optimization and Scientific Computing. [Research team]

**Optimization and Algorithmic Discrete Mathematics, Prof. Dr. Rüdiger Schultz**

Stochastic integer programming - two-stage stochastic programming and risk minimization - multi-stage stochastic programming - stochastic shape optimization - test sets for integer programming and computer-algebraic methods - optimization in energy networks under uncertainty - on-line optimization in procedural facilities for more than one output. [Research team]

##
Stochastics

**Applied Stochastics, Prof. Dr. Denis Belomestny, Prof. Dr. Volker Krätschmer,**** ****Prof. Dr. Mikhail Urusov**

Mathematical Statistics, Stochastic Algorithms, Financial and Insurance Mathematics, Quantitative Risk Management. [Research team]

**Stochastic Analysis****, Prof. Martin Hutzenthaler**

*Stochastic Analysis:* Regularity of stochastic (partial) differential equations, Numerical approximations of stochastic (partial) differential equations, Numerical approximations of stochastic backward differential equations, Malliavin Calculus and Stein's method. *Mathematical Biology:* Effect of fluctating selection, Models with persisting altruism. [Research team]

**Probability Theory**, Prof. Dr. A. Winter

Stochastic Analysis, Mathematical Biology, Metric Geometry, Measure-valued Diffusion, Interacting Particle Systems. [Research team]