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Rank Conjectures Across Algebra and Topology
Provider: Faculty of Science

Activity no.: 5595-24-07-31
Enrollment deadline: 17/05/2024
PlaceDepartment of Mathematical Sciences
Universitetsparken 5, 2100 København Ø
Date and time17.06.2024, at: 09:00 - 21.06.2024, at: 16:00
[antalgange]5
Regular seats30
ECTS credits2.00
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserJesper Grodal    E-mail address: jg@math.ku.dk
Written languageEnglish
Teaching languageEnglish
Exam requirementsStudents will pass the course by participating actively in the lectures and exercise sessions, and solving selected problems from the exercise sessions that demonstrate mastery of the core concepts and techniques.
Course workload
Course workload categoryHours
Lectures15.00
Exercises10.00
Evaluation1.00
Preparation / Self-Study29.00
Sum55.00
Aim and content
This week-long masterclass will explore a family of conjectures spanning multiple disciplines: the so-called Rank Conjectures, each making predictions about the sizes of certain homotopically defined objects. Guest lecturers will share their expertise in algebraic topology, commutative algebra, and modular representation theory, bringing students up to a state of the art understanding of the Rank Conjectures in all of these areas, as well as the connections between them.

Torus actions are fundamental and ubiquitous in equivariant topology, and Halperin’s toral rank conjecture has been a source of much activity in this field for decades. Similar rank conjectures concerning the actions of elementary abelian p-groups have opened up important connections with modular representation theory. Within commutative algebra, the Buchsbaum-Eisenbud-Horrocks conjecture has been equally significant; this conjecture predicts a lower bound on the Betti numbers of modules over regular local rings, and has turned out to hold a strikingly close relationship with the toral rank conjecture.

In recent years researchers have begun bridging these areas more closely, developing new methods and making significant progress on longstanding problems. This has to a flurry of interest that the masterclass will exploit. Copenhagen is also a world focal point for modern methods in homotopy theory, and it is expected that this environment will provide a unique opportunity to share local expertise, providing young researchers with new and underutilised tools that they can use to attack old problems.

There will be a lecture series by Mark Walker presenting the state of the art on the rank conjectures in commutative algebra, culminating in a presentation of his recent breakthrough results. Other guest lecturers will present courses on the various rank conjectures, emphasising their avatars in different disciplines.

The lectures will be supplemented by regular problem sessions with instructive examples to illustrate the material from the lectures, and to ultimately bring the students to the front line of modern research. The final problem session will collect open problems and lines of investigations from participants which could potentially set the tone for future work.

The masterclass will also function as an opportunity for PhD students to meet other young researchers in nearby areas, giving them an opportunity to learn techniques from outside their main field of study while building new academic relationships across disciplines.

Learning outcome
Knowledge:
· A clear picture of the state of the art and open problems in the areas surrounding the rank conjectures, including an understanding the connections between the various rank conjectures in algebra and topology
· A complete understanding of the arguments used in Mark Walker’s paper “Total Betti numbers of modules of finite projective dimension”.

Skills:
· Mastery of topological and algebraic techniques used by the guest lecturers, especially using K-theoretic methods to establish bounds in homological algebra

Competences:
· Ability to use and apply standard methods in rational homotopy theory
· Ability to work with spaces with finite group actions, and a practical understanding of the connections with modular representation theory
· Ability to work with differential graded algebras and differential graded modules in commutative algebra, in order to construct counterexamples and prove results

Literature
“Total Betti numbers of modules of finite projective dimension,” by Walker.

“The Total Rank Conjecture in Characteristic Two” by VandeBogert and Walker.

“Counter-Examples to a generalised Toral Rank Conjecture” by Amann.

Other works to be determined by the guest lecturers.

Target group
PhD students at UCPH, other Danish and international universities who are interested in algebraic topology, commutative algebra, or modular representation theory.

Teaching and learning methods
Students are expected to prepare for the course by studying prerequisite material to be determined (approximately 15 hours).

3 lecture series on complementary subjects provided by the guest lecturers (5 hours each, 15 hours total). Attendance is expected.

10 hours of exercise sessions on questions related to the lecture material. Active participation is expected.

Students are expected to the study the course material during the week of the masterclass, by going over lectures and working on some of the exercises alone (approximately 14 hours).

Students will meet with an organiser or guest lecturer for 1 hour to discuss exercises and course material, to determine that the student has met the criteria to pass.

Lecturers
Mark Walker (University of Nebraska, Lincoln)
Tackling rank conjectures in algebra using Adams operations

Ergün Yalçin (Bilkent University)
Homological methods for finite group actions

Leopold Zoller (University of Munich)
Rational homotopy theory, equivariant cohomology, and the toral rank conjecture

Remarks
Please register at the conference webpage:
https://www.math.ku.dk/english/calendar/events/rank-conjectures/

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