Loughborough University

Local-Global Conjectures for Spetses

Loughborough UniversityLoughborough, United Kingdom

Course overview

A real reflection group is a finite group of matrices with real entries generated by reflections. Such groups occur ubiquitously as Weyl groups in the classifications of simple complex Lie groups and algebraic groups, and in the theories of braid groups, knots, and Hecke algebras. A complex reflection group is a finite group of matrices with complex entries generated by pseudo-reflections - matrices that fix a hyperplane.

Certain data in the representation theory of a rational reductive group, for example its unipotent representation degrees, can be given generically in terms of the Weyl group. A “spets” is a more general object, associated with a complex reflection group, to which similar data can be assigned. The theory of spetses dates back to the 1990s, but exciting recent developments in algebraic topology have broadened its remit to include generalised theories of unipotent blocks, fusion systems, and characters.

A wealth of questions then present themselves, most pertinently whether the local-global conjectures - a series of far-reaching and important open problems in modular representation theory - can be extended and proved in this generalised setting. This will be a key focus of the project. 94% of Loughborough’s research impact is rated world-leading or internationally excellent.

REF 2021

Key highlights

  • Research on complex reflection groups and their associated spetses
  • Focus on extending local-global conjectures in modular representation theory
  • Explores unipotent blocks, fusion systems and characters in generalised settings
  • Draws on recent developments in algebraic topology
  • 94% of research impact rated world-leading or internationally excellent (REF 2021)

Entry requirements

  • Degree: 2:1 — 2:1

Scholarships

  • Alumni Bursary

    10% of tuition fees which will be credited to the student’s tuition fee account.

    Up to 10% of fees

  • Alumni Discount

    UK students: £1,500 tuition fee discount for one academic year. International students: £3,000 tuition fee discount for one academic year.

  • Care Leaver's Scholarship

  • Excellence Scholarships

    UK students: £3,000 off tuition fees; International students: £6,000 off tuition fees which will be credited to the student’s tuition fee account.

  • International Sports Student Postgraduate Bursary

    Our International Sports Student Postgraduate Bursary offers a £12,000 tuition fee discount (set against the tuition fees of your chosen postgraduate taught programme ONLY).

    Up to £12,000

  • London 10th Anniversary Award

    £5,000 off tuition fees.

    Up to £5,000

  • Loughborough Business School Scholarship

    £1,500 (UK students) and £3,000 (international students) off tuition fees which will be credited to the student’s tuition fee account.

  • MBA Progress with Purpose Scholarship

    50% of tuition fees which will be credited to the student’s tuition fee account.

    Up to 50% of fees

  • Qatari Ghanim Al-Muftah Excellence Scholarship

Careers this course leads to

  • Postdoctoral Research Associate
  • University Lecturer in Mathematics
  • Research Mathematician
  • Data Scientist
  • Cryptographer
  • Quantitative Analyst

Cost of study

UK/Home: £5,238 | International: £23,100

Admission

Applications are made through AHZ at no cost. Share your transcripts, English test scores and personal statement, and your AHZ Counsellor will review your profile, submit your application to Loughborough University, and guide you through interview, offer, CAS and visa — right up to enrolment.

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