About
Dr Koeck is an algebraist with research interests in Algebraic Geometry, Algebraic K-Theory, Algebraic Number Theory and Homological Algebra. A common theme in Koeck's research form group actions on spaces in Algeba, Geometry and Number Theory. More specifically he has worked in (equivariant) Riemann-Roch theory, operations on higher K-theory, Galois module theory, Belyi theory and simplicial homological algebra.
You can update this in . Select 鈥楨dit profile鈥. Under the heading and then 鈥楥urriculum and research description鈥, select 鈥楢dd profile information鈥. In the dropdown menu, select - 鈥楢bout鈥.
Write about yourself in the third person. Aim for 100 to 150 words covering the main points about who you are and what you currently do. Clear, simple language is best. You can include specialist or technical terms.
You鈥檒l be able to add details about your research, publications, career and academic history to other sections of your staff profile.
Research
Research interests
- Riemann-Roch theory
- (Operations on) Higher Algebraic K-Theory
- Equivariant Riemann-Roch theory
- Geometric Galois Module Theory
Current research
Dr Koeck's research falls into the broadly understood area of Riemann-Roch theory. Riemann-Roch theory originates from the fundamental problem of computing the dimension of so-called Riemann-Roch spaces such as the vector space of global sections of a vector bundle or, more classically, the vector space of global meromorphic functions on a compact Riemann surface that satisfy certain pole and zero order conditions. The classical and celebrated Riemann-Roch theorem from the 19th century which solves the latter problem has seen vast generalisations over the past century which are central to Algebraic K-Theory, Algebraic Geometry, Arithmetic Geometry and Differential Geometry. Dr Koeck's research is particularly focussing on the following topics:
- (Operations on) Higher Algebraic K-Theory: Exterior power operations furnish the Grothendieck group and higher K-groups of a ring or a scheme with additional structure which is at the heart of Grothendieck鈥檚 Riemann-Roch Theory. While higher K-groups have originally been defined and studied using homotopy theory, there are now purely algebraic definitions of both higher K-groups and exterior power operations available that invite to be studied from the purely algebraic point of view.
- Equivariant Riemann-Roch theory: In the equivariant context, Riemann-Roch spaces are studied as group representations when a group acts on the vector bundle compatible with a given action of the group on the underlying space. Many versions of equivariant Riemann-Roch theorems have been proved over the past three decades or so. Current research in this area concerns refinements of the equivariant Adams-Riemann-Roch theorem and equivariant Riemann-Roch theorems for inertial K-theory or for orbifold varieties.
- Geometric Galois Module Theory: Relating global invariants to invariants that are created from local data is a fundamental and widely studied topic in number theory. In Geometric Galois Theory, an equivariant version of this local-global principle is studied in the situation when a (finite) group acts on a variety over a finite field. More precisely, a goal is to relate epsilon constants appearing in functional equations of Artin L-functions to global invariants that are extracted from the equivariant structure of certain Riemann-Roch spaces. Particularly explicit and deep theorems exist when the space is a curve and the action of the group is tame. Interesting and sought-after generalisations concern wild actions on curves and actions on surfaces.
You can update the information for this section in .
Research groups
Any research groups you belong to will automatically appear on your profile. Speak to your line manager if these are incorrect. Please do not raise a ticket in Ask HR.
Research interests
Add up to 5 research interests. The first 3 will appear in your staff profile next to your name. The full list will appear on your research page. Keep these brief and focus on the keywords people may use when searching for your work. Use a different line for each one.
In , select 鈥楨dit profile鈥. Under the heading 'Curriculum and research description', select 'Add profile information'. In the dropdown menu, select 'Research interests: use separate lines'.
Current research
Update this in . Select 鈥楨dit profile鈥 and then 鈥楥urriculum and research description - Current research鈥.
Describe your current research in 100 to 200 words. Write in the third person. Include broad key terms to help people discover your work, for example, 鈥渟ustainability鈥 or 鈥渇ashion textiles鈥.
Research projects
Research Council funded projects will automatically appear here. The active project name is taken from the finance system.
Publications
Public outputs that list you as an author will appear here, once they鈥檙e validated by the ePrints Team. If you鈥檙e missing any outputs that you鈥檝e added to Pure, they may be waiting for validation.
Supervision
Current PhD Students
Contact your Faculty Operating Service team to update PhD students you supervise and any you鈥檝e previously supervised. Making this information available will help potential PhD applicants to find you.
Teaching
Bernhard has taught on a number of modules, including:
Mathematics A
Further Number Theory
You can update your teaching description in . Select 鈥楨dit profile鈥. Under the heading and then 鈥楥urriculum and research description鈥 , select 鈥楢dd profile information鈥. In the dropdown menu, select 鈥 鈥楾eaching Interests鈥. Describe your teaching interests and your current responsibilities. Aim for 200 words maximum.
Courses and modules
Contact the Curriculum and Quality Assurance (CQA) team for your faculty to update this section.
External roles and responsibilities
These are the public-facing activities you鈥檇 like people to know about.
This section will only display on your public profile if you鈥檝e added content.
You can update your external roles and responsibilities in . Select 鈥+ Add content鈥 and then 鈥楢ctivity鈥, your 鈥楶ersonal鈥 tab and then 鈥楢ctivities鈥. Choose which activities you want to show on your public profile.
You can hide activities from your public profile. Set the visibility as 'Backend' to only show this information within Pure, or 'Confidential' to make it visible only to you.
Biography
PhD Regensburg, Habilitation Karlsruhe, Heisenberg Fellow
You can update your biography section in . Select your 鈥楶ersonal鈥 tab then 鈥楨dit profile鈥. Under the heading, and 鈥楥urriculum and research description鈥, select 鈥楢dd profile information鈥. In the dropdown menu, select - 鈥楤iography鈥. Aim for no more than 400 words.
This section will only appear if you enter the information into .
Prizes
You can update this section in . Select 鈥+Add content鈥 and then 鈥楶rize鈥. using the 鈥楶rizes鈥 section.
You can choose to hide prizes from your public profile. Set the visibility as 鈥楤ackend鈥 to only show this information within Pure, or 鈥楥onfidential鈥 to make it visible only to you.