Variations of Hochschild and cyclic cohomology for DG-categories and applications to algebra, geometry and topology

Differential graded categories or DG-categories are at the heart of current research and important applications in algebra, algebraic geometry and algebraic topology. The main goal of the proposed PhD project is to extend, generalise and apply the fundamental results of Neumann-Szymik from their 2017 paper in Selecta Mathematica from absolute Hochschild cohomology to cyclic and relative Hochschild cohomology of DG-categories. In their paper, the authors interpret the so called characteristic homomorphism as an edge morphism for a spectral sequence, converging to the DG-Hochschild cohomology, and then study its connections to various phenomena in algebra, geometry and topology. The proposed project aims to generalise this work in a more flexible setting in order to apply the framework to various classification problems in algebra, geometry, topology and mathematical physics. Cyclic cohomology was introduced in the 1980s by Fields medalist Alain Connes to obtain algebraic invariants for phenomena in noncommutative geometry, which originated in quantum physics.

DG-categories are a natural framework where these invariants live and their general nature allow for a vast application of such invariants in different areas. As categorical methods are also becoming important tools in information science and data analysis, these invariants would also be of benefit for possible applications in these fields, where algebraic invariants are needed to distinguish different types of data and information. The non-commutative nature of these invariants make them usable for the theoretical study of quantum phenomena manifested already in the non-commutative algebraic nature of Heisenberg’s classical uncertainty relations. Hochschild cohomology also provides a rich algebraic invariant for the topology of strings as manifested through the algebraic-categorical nature of string diagrams.

The PhD student will embark on a journey using methods from homological algebra to construct and calculate new algebraic invariants to obtain insights into several fundamental non-commutative phenomena in algebra, geometry and topology, which also manifest themselves in quantum physics, data analysis and computer science.

Entry requirements
Applicants are required to hold/or expect to obtain a UK Bachelor Degree 2:1 or better in a relevant subject. The University of Leicester English language requirements apply where applicable.

How to apply
The online application and supporting documents are due by Monday 21st January 2019.

Any applications submitted after the deadline will not be accepted for the studentship scheme.

References should arrive no later than Monday 28th January 2019.

Applicants are advised to apply well in advance of the deadline, so that we can let you know if anything is missing from your application.

Required Materials:

1. Online application form
2. Two academic references
3. Transcripts
4. Degree certificate/s (if awarded)
5. Curriculum Vitae
6. CSE Studentship Form
7. English language qualification

Applications which are not complete by the deadline will not be considered for the studentship scheme. It is the responsibility of the applicant to ensure the application form and documents are received by the relevant deadlines.

All applications must be submitted online, along with the supporting documents as per the instructions on the website.

Please ensure that all email addresses, for yourself and your referees, are correct on the application form.

Project / Funding Enquiries
Application enquiries to [Email Address Removed]
Closing date for applications – 21st January 2019