Rank Conjectures Across Algebra and Topology
Masterclass
University of Copenhagen, 24-28 June 2024
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The masterclass will explore a family of conjectures spanning multiple disciplines: the Rank Conjectures, each making predictions about the sizes of certain homotopically defined objects.
Update: the masterclass is over, and you can find notes written by each of the speakers here:
Mark Walker: Tackling rank conjectures in algebra using Adams operations
Ergün Yalçın: Free (Z/p)^r Actions on Products of Spheres
Leopold Zoller: The Toral Rank Conjecture
Exercises: Rank Exercises
Halperin’s toral rank conjecture predicts a lower bound on dimension of the rational cohomology of a space with a free torus action. Similar conjectures concerning the actions of elementary abelian p-groups have opened up connections with modular representation theory. Within commutative algebra, the Buchsbaum-Eisenbud-Horrocks 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.
New techniques have recently led to substantial progress on the rank conjectures in algebraic topology, commutative algebra, and modular representation theory. The guest lecturers will share their expertise in these areas, bringing students up to a state of the art understanding of the conjectures, the tools being used to attack them, and the connections between them. The masterclass will highlight open problems and lines of investigation which could potentially set the tone for future work. The lectures will be supplemented by regular problem sessions, bringing the students to the front line of modern research. The masterclass will also function as an opportunity for students to meet other young (and established) researchers in nearby areas, building new academic relationships across disciplines.
Mark Walker:
Title: Tackling rank conjectures in algebra using Adams operations
Abstract: These talks will focus on various conjectures regarding Betti numbers and related invariants of modules, and chain complexes of modules, over local rings. I will focus on using Adams operators, and related constructions, to settle these conjectures in certain cases, and I will also connect these purely algebraic conjectures to topological analogues.
Ergün Yalçın:
Title: Homological methods for finite group actions
Abstract: I will introduce the rank conjecture for free G=(Z/p)^r actions on a finite CW-complex homotopy equivalent to a product of k spheres. I will start with a brief survey of known results due to Smith, Heller, Carlsson, Adem-Browder, Benson-Carlson, and others. Then, I will introduce the basic definitions and homological algebra tools (such as group cohomology, G-CW-complexes, and the Borel construction) for studying this version of the rank conjecture and discuss the proofs of some well-known results from the literature, especially for free actions on (S^n)^k. Then in my last lecture, I will introduce the exponent methods (using hypercohomology) developed by Browder and others, and discuss a theorem that O. Okutan and I proved in 2013 on free actions on products of spheres at high dimensions.
If you want to read ahead, these papers might be useful: "On the non-existence of free actions of elementary abelian groups on products of spheres" by G. Carlsson, "The free rank of symmetry on (S^n)^k" by A. Adem and W. Browder, "Free actions on products of spheres at high dimensions" by O. B. Okutan and E. Yalcin.
Leopold Zoller
Title: Rational homotopy theory, equivariant cohomology, and the toral rank conjecture
Abstract: The Toral Rank Conjecture predicts a certain lower bound on the total dimension of the rational cohomology of a compact space with a free torus action. Despite having seen some progress and partial results during the last 40 years the problem is still wide open. We will discuss how to translate this geometric problem into a purely algebraic question on differential graded algebras. This will involve a very brief introduction to equivariant cohomology and rational homotopy theory. Using this approach we will study several angles from which to attack the conjecture.
If you want to read ahead, "Rational Homotopy Theory" by Felix, Halperin, and Thomas contains all the necessary background and more.
All talks will be in Aud 2, in the HCØ building Universitetsparken 5, 2100 København Ø
Because of the a lot events happening at the same time the canteen will be extremely busy, and we recommend getting to the bio-canteen at 11:50 for an early lunch.
| Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
|
Registration |
Coffee 9:30-10:00 |
Coffee 9:30-10:00 |
Coffee 9:30-10:00 |
Coffee 9:30-10:00 |
|
Walker 1 10:00-11:00 |
Yalçın 2 10:00-11:00 |
Exercises 10:00-11:45 |
Zoller 3 10:00-11:00 |
Zoller 4 10:00-11:00 |
|
Discussion |
Discussion |
Discussion 11:00-11:45 |
Discussion 11:00-11:45 |
|
|
Lunch |
Lunch |
Lunch |
Lunch |
Lunch |
|
Yalçın 1 13:30-14:30 |
Zoller 2 13:30-14:30 |
Yalçın 3 13:30-14:30 |
Exercises 13:30-14:30 |
Yalçın 4 13:30-14:30 |
|
Coffee/cake |
Coffee/cake |
Coffee/cake |
Coffee/cake |
Coffee/cake |
|
Zoller 1 |
Walker 2 |
Walker 3 |
Walker 4 |
Exercises and |
|
Pizza reception
|
Dinner at Starts 17:30 |
The conference/masterclass will take place at the Department of Mathematical Sciences, University of Copenhagen. See detailed instructions on how to reach Copenhagen and the conference venue.
Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB ticket office on your right-hand side as soon as you come out of the arrival area of the airport. DSB has an agreement with 7-Eleven, so many of their shops double as selling points for public transportation.
A journey planner in English is available.
More information on the "find us" webpage.
We kindly ask the participants to arrange their own accommodation.
We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandinavia (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and Ibsen's Hotel.
An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). See information about tickets & prices.
Registration has closed, please write to bpb@math.ku.dk if you're still interested in coming.
Ben Briggs bpb@math.ku.dk
Jesper Grodal jg@math.ku.dk
