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Interactions between homotopy theory and representation theory
Provider: Faculty of Science

Activity no.: 5125-25-00-00There are 100 available seats 
Enrollment deadline: 29/09/2025
PlaceDepartment of Mathematical Sciences
Universitetsparken 5, 2100 København Ø
Date and time10.11.2025, at: 09:00 - 14.11.2025, at: 16:00
Regular seats100
Activity Prices:
  - 1 Participant fee (all participants in add. to course fee)0.00 kr.
  - 2 Course fee PhD student enrolled at UCPH SCIENCE0.00 kr.
  - 3 Course fee PhD student at Danish Universities (except CBS)0.00 kr.
  - 4 Course fee PhD student at Copenhagen Business School3,000.00 kr.
  - 5 Course fee PhD student at foreign university3,000.00 kr.
  - 6 Course fee Master's student at Danish university0.00 kr.
  - 7 Course fee Master's student at foreign university3,000.00 kr.
  - 8 Course fee Employee at university (e.g., postdocs)3,000.00 kr.
  - 9 Course fee Others (e.g., from a private company)8,400.00 kr.
LecturersHenrik Granau Holm
ECTS credits2.50
Contact personHenrik Granau Holm    E-mail address: holm@math.ku.dk
Enrolment Handling/Course OrganiserPhD Administration SCIENCE    E-mail address: phdcourses@science.ku.dk
Enrolment guidelines
This is a specialised course where 50% of the seats are reserved to PhD students enrolled at the Faculty of SCIENCE at UCPH and 50% og the seats are reserved to other applicants.

This course is free of charge for PhD students at Danish universities (except Copenhagen Business School). All other participants must pay the course fee (except if you are a master’s student from a Danish University).

Anyone can apply for the course, but if you are not a PhD student, you will be placed on the waiting list until enrollment deadline. After the enrollment deadline, available seats will be allocated to applicants on the waiting list.


Aim:
The purpose of this week-long masterclass is to explore and deepen the understanding of the rich and evolving interactions between homotopy theory and representation theory. In recent years, these two domains – traditionally seen as distinct – have become increasingly interconnected, with techniques and insights from one leading to major developments in the other. This masterclass provides an opportunity for early-career researchers to engage with cutting-edge research at this interface, guided by leading experts in the field.

Content:
The main program will consist of three lecture series (and some preliminary talks), each focusing on a specific aspect of the interaction between, and the application of, homotopical and representation-theoretic methods.
The plan is as follows:
– Preliminary lectures on stable categories, sheaves, and exchange graphs.
– Gustavo Jasso (University of Cologne) will present a series on his recent work on the derived Auslander-Iyama correspondence.
– Merlin Christ (Institut de Mathématiques de Jussieu - Paris Rive Gauche) will present a series showcasing the use of constructible sheaves on graphs and his work on geometric models.
– Sira Helena Gratz (Aarhus University) will present a series that revolves around her work on cluster algebras and their categorifications.
The masterclass is designed to foster active participation, with ample time allocated for discussions and informal interactions among participants and speakers. It aims not only to provide a solid overview of key aspects of the current landscape, but also to inspire new research directions by illustrating the power of cross-disciplinary approaches.


Learning outcomes
Intended learning outcome for the students who complete the course:

Knowledge
• Knowledge of higher category theory: presentability, constructible sheaves, exit-path categories
• Definition of cluster algebras and geometric models
• Statement and proof of the derived Auslander-Iyama correspondence, as well as usage examples

Skills
• Use of higher category theory in practice
• Performing computations of global sections of sheaves and constructing them from local sections
• Working with n-abelian and n-angulated categories

Competences
• Ability to spot opportunities for use of homotopy theory methods in representation theory
• Awareness of current trends in representation theory
• Understanding of research goals of homotopy theory and representation theory in context


Target Group
PhD students (possibly also advanced master’s students) and postdoctoral researchers from Danish and international universities whose research interests are in homotopy theory, representation theory or neighbouring fields.


Recommended Academic Qualifications
Students should be enrolled in a mathematics PhD program with knowledge of basic concepts at least at the level of a first graduate course in homological algebra, a first graduate course in category theory, a first graduate course in algebraic topology, and a first graduate course in representation theory.


Research Area
Homotopy theory, representation theory.


Teaching and Learning Methods
Preparation (30 hours): The participants are expected to prepare for the course by studying prerequisite material yet to be determined. Each of the three lecture series (see below) will require approximately 10 hours of preparation.
Lectures (12 hours): Three lecture series on complementary subjects are provided by the guest lecturers (4 hours each). Attendance is expected.
Preliminary talks (3 hours): Each of the three lecture series will be preceded by a 1-hour preliminary talk (given by selected participants) providing relevant background material. Attendance is expected.
Discussion sessions (10 hours): In these sessions, the reading material and the content of the lectures will be discussed among the participants and the lecturers. Active participation is expected.
Self-study (15 hours): Students are expected to study the course material during the week of the masterclass by going over lectures and solving exercises. Each of the three lecture series requires 5 hours of self-study.


Type of Assessment
Students will pass the course by attending and actively participating in the lectures and discussion sessions.


Course coordinator
Henrik Granau Holm, Professor, holm@math.ku.dk


Guest Lecturers
Gustavo Jasso (University of Cologne) is a leading expert in representation theory of quivers and algebras as well as in homological and homotopical algebra, and higher category theory.
Merlin Christ (Institut de Mathématiques de Jussieu – Paris Rive Gauche) is a young, very promising researcher in higher category theory, representation theory, and noncommutative algebraic geometry.
Sira Helena Gratz (Aarhus University) is an expert in combinatorial methods in representation theory with particular interest in cluster algebras and their categorifications.

All three guest lecturers are experts in homotopy theory and in representation theory and are therefore ideal to present their unique perspectives on the interaction between, and the application of, methods in these two fields.


Dates
10–14 November 2025


Course location
H.C. Ørsted Instituttet (HCØ)



Seats to PhD students from other Danish universities will be allocated on a first-come, first-served basis and according to the applicable rules.
Applications from other participants will be considered after the deadline for registration.




Cancellation policy
• Cancellations made up to two weeks before the course starts are free of charge.
• Cancellations made less than two weeks before the course starts will be charged a fee of DKK 3.000
• Participants with less than 80% attendance cannot pass the course and will be charged a fee of DKK 5.000
• No-show will result in a fee of DKK 5.000
• Participants who fail to hand in any mandatory exams or assignments cannot pass the course and will be charged a fee of DKK 5.000

Course fee and participant fee
PhD courses offered at the Faculty of SCIENCE have course fees corresponding to different participant types.
In addition to the course fee, there might also be a participant fee.
If the course has a participant fee, this will apply to all participants regardless of participant
type - and in addition to the course fee.

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