09:30 to 10:20 
Toshiyuki Kobayashi (The University of Tokyo, Japan) 
Minicourse Global Analysis of Locally Symmetric Spaces with Indefinite Metric Lecture 3
The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as pseudoRiemannian geometry, familiar to us as the spacetime of relativity theory, and more generally in pseudoRiemannian geometry of general signature, surprising little wa known about global properties of the geometry even if we impose a locally homogeneous structure. This theme has been developed rapidly in the last three decades.
In the series of lectures, I plan to discuss two topics by the general theory and some typical examples.
1. Global geometry: Properness criterion and its quantitative estimate for the action of discrete groups of isometries on reductive homogeneous spaces, existence problem of compact manifolds modeled on homogeneous spaces, and their deformation theory.
2. Spectral analysis: Construction of periodic L2eigenfunctions for the Laplacian with indefinite signature, stability question of eigenvalues under deformation of geometric structure, and spectral decomposition on the locally homogeneous space of indefinite metric.



10:35 to 11:25 
Marek Kaluba & Piotr Nowak (Karlsruhe Institute for Technology, Germany & IMPAN, Poland) 
Property (T) for automorphism groups of free groups Lecture 1 Talk 1 (Kaluba) We will discuss the notion of positivity in rings, understood as being a sum of squares. This problem is classically relevant for polynomials, and in the context of property (T) it appears in the group ring and its augmentation ideal. We will discuss a computerassisted approach, via semidefinite programming, of verifying whether an element of the group ring can be expressed as a sum of squares. An important feature of this method is that despite using numerical information, in certain situations it provides a rigorous proof of positivity. Together with a characterization of property (T) via positivity in the group ring this provides an approach to proving Kazhdan’s property (T) for some groups.
Talk 2 (Nowak) We will use the methods described in the first talk to prove property (T) for automorphism groups of free groups. First we will consider a singular case of Aut(F5) but the main focus will be the infinite case. We will describe an inductive procedure that allows to deduce property (T) for families of groups like SLn(Z) and Aut(Fn) from positivity of a single group ring element. Using this inductive approach we will prove property (T) for Aut(Fn) for all n ≥ 6, and reprove property (T) for SLn(Z), n ≥ 3. We will also discuss several consequences and applications, in particular we will show new estimates for Kazhdan constants.



12:00 to 12:50 
Alireza Salehi Golsefidy (UCSD, USA) 
Random walks on group extensions Lindenstrauss and Varju asked the following questions: for every prime p, let Sp be a symmetric generating set of Gp := SL(2, Fp)×SL(2, Fp). Suppose the family of Cayley graphs {Cay(SL(2, Fp), pri(Sp))} is a family of expanders, where pri is the projection to the ith component. Is it true that the family of Cayley graphs {Cay(Gp, Sp)} is a family of expanders? We answer this question and go beyond that by describing randomwalks in various group extensions. (This is a joint work with Srivatsa Srinivas.)



15:00 to 15:50 
Alex Lubotzky & Bharatram Rangarajan (Weizmann Institute, Israel & Hebrew University of Jerusalem, Isreal) 
Minicourse Uniform Stability of Higherrank Arithmetic Groups Lecture 3 Lattices in highrank semisimple groups enjoy a number of rigidity properties like superrigidity, quasiisometric rigidity, firstorder rigidity, and more. In these lectures, we will add another one: uniform ( a.k.a. Ulam) stability. Namely, it will be shown that (most) such lattices D satisfy: every finitedimensional unitary ”almostrepresentation” of D ( almost w.r.t. to a submultiplicative norm on the complex matrics) is a small deformation of a true unitary representation. This extends a result of Kazhdan (1982) for amenable groups and or BurgerOzawaThom (2013) for SL(n,Z), n > 2. The main technical tool is a new cohomology theory (”asymptotic cohomology”) that is related to bounded cohomology in a similar way to the connection of the last one with ordinary cohomology. The vanishing of H2 w.r.t. to a suitable module implies the above stability. The talks are based on a joint work of the speakers with L. Glebsky and N. Monod. See arXiv:2301.00476.



16:30 to 17:20 
Marek Kaluba & Piotr Nowak (Karlsruhe Institute for Technology, Germany & IMPAN, Poland) 
Property (T) for automorphism groups of free groups Lecture 2 

