This summer school on high-dimensional expanders is the first event organized in the framework of the EOS-project "High-dimensional expanders and Kac–Moody–Steinberg groups". It consists of 7 short series of lectures by experts in the field and is meant to be accessible to a wide audience of both young and established researchers.

Topics include:

• Ramanujan graphs and expander graphs
• Bruhat–Tits buildings
• Kazhdan's property (T)
• Hecke algebras
• Lattices in p-adic groups
• Ramanujan complexes and high-dimensional expanders
• Applications

Alain Valette: Ramanujan graphs

The mini-course will start with a recap on algebraic graph theory (the adjacency matrix of a graph and the combinatorial significance of its eigenvalues). This leads first to expander graphs, then to Ramanujan graphs which are the best expanders from the spectral point of view. The name Ramanujan graphs was coined in 1986 by Lubotzky-Phillips-Sarnak as a sales pitch for the original construction of such graphs, that involved deep number theory: we will sketch that construction, that provides only graphs whose degree is of the form "1 + prime power". We will end with the probabilistic proof of Marcus-Spielman-Srivastava who showed in 2013 the existence of bipartite Ramanujan graphs of arbitrary degree (so that the name Ramanujan maybe is not so well-chosen, after all).

[Exercises]   [Recordings]

Petra Schwer: Bruhat–Tits buildings

This mini-course covers constructions of Bruhat-Tits buildings as well es some of their geometric and combinatorial properties. Buildings are geometric objects associated with reductive groups over non-archimedian local fields potentially with a discrete valuation. We will see explicit constructions of Bruhat-Tits buildings, axiomatic descriptions and the connection to BN-pairs. The most important geometric features of a building are its apartments and retractions onto apartments, which will also be introduced in the course.

Furthermore it will be explained how the geometry of retractions and the combinatorics of folded galleries provides a unified framework to study orbits in affine flag varieties. When \( X \) is a Bruhat-Tits building for a group over a local field, we can then relate labeled folded galleries and shadows to double coset intersections in affine flag varieties. This will allow us to hint at connections with the Hecke algebras discussed in Anne-Marie Aubert's course.

[Notes part 1]   [Notes part 2]   [Notes part 3]   [Recordings]

Indira Chatterji : Kazhdan's property (T)

Kazhdan’s property (T) for a group \( G \) is the isolation of the trivial representation in the unitary dual of \( G \). The first lecture will be devoted to a general introduction to property (T), the second lecture will explore consequences of property (T), and in particular that finite quotients give a family of expanders. The last lecture will be devoted to explaining a proof of property (T) for \( \mathrm{SL}(n,\mathbb{R}) \) due to Hee Oh and using estimates on the decay of the coefficients for an irreducible unitary representation.

[Exercises]   [Recordings]

François Thilmany: Lattices in p-adic Lie groups

In these lectures, we will explore the construction and basic properties of lattices in p-adic Lie groups. The provisional plan is the following:

• In lecture 1, we start with the general theory of lattice subgroups in locally compact groups. We define ‘lattice’, and give examples of lattices in some familiar groups. This will lead us to a famous theorem of Borel and Harish Chandra, describing a general procedure to construct arithmetic lattices in Lie groups. We briefly discuss the quotient of the group by a lattice, and state a criterion to determine when it is compact. To the extent permitted by time, we will survey the fundamental rigidity and arithmeticity theorems of Margulis.
• Lecture 2 will be dedicated to the specific properties of lattices in simple p-adic groups. In particular, we show that such lattices are always cocompact, virtually torsion-free, and finitely presented. To this end, the action on the Bruhat-Tits building of the p-adic group will be a useful tool. We will also discuss strong approximation for arithmetic lattices.
• The contents of lecture 3 will be shaped by the needs of the other courses. We briefly discuss the representation theory of arithmetic p-adic lattices and the implications for the spectral properties of their action on the building. The remaining time may be spent reviewing some concepts or discussing some of the exercices.

[Exercises]

Anne-Marie Aubert: Local representation theory and Hecke algebras

The mini-course will start with basic facts on the representation theory of locally compact groups. Next, we will describe the decomposition into irreducible unitary representations of the quasi-regular representation of a p-adic group \( G \) on \( L^2(G/L) \), where \( L \) is a lattice in \( G \).

We will introduce various Hecke algebras attached to \( G \), in particular the spherical Hecke algebra and the Iwahori-spherical Hecke algebra. We will study the representations of these algebras, and will explicit their relation with the above mentioned decomposition.

We will also explain the use of these Hecke algebras in the understanding of complex representations of a p-adic group \( G \), notably in the classification of the irreducible representations of \( G \), and how they can be related to the Bruhat-Tits building of \( G \).

[Exercises]   [Slides part 1]   [Slides part 2]   [Slides part 3]   [Recordings]

Shai Evra : Ramanujan complexes and high-dimensional expanders

In these lectures, we will introduce the notions of Ramanujan complexes and high dimensional (HD) expanders, and present some of their applications. The plan for the talks is as follows:

1. Ramanujan complexes and Ramanujan conjectures — We will present several definitions of Ramanujan complexes, and for a finite quotient of a Bruhat-Tits building, describe a p-adic and an automorphic representation-theoretic characterizations of being Ramanujan.
2. Spectral HD-expanders — After briefly recalling the notion of a spectral expander graph, we will review its HD generalizations by Garland and others and describe some applications, including a partite mixing lemma, high chromatic number and the geometric overlapping property of Gromov.
3. Cohomological HD-expanders — Expander graphs are defined via the Cheeger constant, and thanks to the discrete Cheeger inequality this notion is qualitatively equivalent to spectral expansion. We will review the HD generalizations of the Cheeger constant, now known as \( \mathbb{F}_2 \)-coboundary expansion, originally due to Linial-Meshulam and independently to Gromov. We will give a counterexample to the Cheeger inequality in higher dimensions, and present an application to Gromov's problem on the topological overlapping property.

[Recordings]

Ori Parzanchevski : Optimal expansion in groups and complexes

The main theme of these talks is the relations between explicit constructions of Ramanujan complexes and results on optimal expansion in finite groups and compact Lie groups. The plan of the talks is:

1. Explicit constructions — We will explain how specially chosen arithmetic lattices give rise to explicit constructions of Ramanujan (and non-Ramanujan) complexes, as clique complexes of finite Cayley graphs.
2. Optimal expansion — We will survey the study of random walks on Ramanujan complexes, and show that the total-variation mixing time of certain walks is optimal in an appropriate sense, and that these walks exhibit Diaconis' cutoff phenomenon. An interesting notion which will arise in this study is that of Ramanujan digraphs.
3. Expansion in groups — From the perspective of group theory, the Lubotzky-Phillips-Sarnak (LPS) construction gives an optimal expansion result for (certain Cayley graphs of) the finite group \( \mathrm{PGL}_2(q) \). It is natural to seek similar connections between \( \mathrm{PGL}_d(q) \) and Ramanujan complexes, and we shall explain such a connection, which is more involved than the graph case. In addition to Ramanujan graphs, the work of LPS produced optimal topological generators for the Lie group \( \mathrm{PU}(2) \), which have recently found an application to quantum computations on a single qubit (so-called “Golden Gates”). Explicit constructions of Ramanujan complexes can again be connected to optimal generators of \( \mathrm{PU}(d) \), which correspond to computations on more qubits, but again the story becomes more complicated, and much is still unknown.

[Exercises]   [Recordings]

This schedule is still subject to changes.

The list of registered participants (and a group picture) can be found here.

• The summer school takes place from Monday, May 22 (around 10am) until Friday, May 26 (around 1pm).
• There will be no subscription fee.
• We can offer (limited) financial support for young (i.e., at most postdoc level) participants, upon approval.
• The deadline to apply for the summer school is Friday May 5, and April 21 if you apply for financial support.
• The summer school will take place at Ghent University, Campus "Ledeganck", Auditorium 6.
• We will have a reception on Monday evening and a conference dinner on Wednesday evening.
• We plan to have a social event on Wednesday afternoon; more details will follow later.
• We ask all participants (except the speakers) to book their own accommodation. Here is a list of suggested hotels, which we kept short so as to maximize the likelihood that many participants will share the same hotel/hostel.
• Warning: some participants have received an email from travellerpoint.org related to accommodation. This is a scam. Do not respond to these messages.
• Questions? Send us an email: "hdx (at) ugent.be".

Registration is closed.

Ghent University
Université Catholique de Louvain
FWO-FNRS EOS programme