Spring 2024
Spring 2024
This is the webpage of a Learning Seminar at Purdue on Groups, Geometry, Topology, and Dynamics in Spring 2024. We will post abstracts of talks and some lecture notes.
Organizers/Instructors: Lvzhou Chen, Ben McReynolds and Sam Nariman.
Regular Meeting Time: Mondays 10:30-11:45 am.
Location: Math 731.
Module 1: Lattices in Lie groups.
In this lecture series, we will start from the concept of a locally compact group and build towards the concept of an arithmetic lattice in a semisimple Lie group. This will require a fair amount of work to establish the requisite language. The tentative plan for the lectures is as follows:
Lecture 1 (Jan. 22)
Abstract: In this lecture, I (=Ben) will introduce the concept of a locally compact group and some basic structure theory with several examples. I will define what a (left) Haar measure is and Weil's proof of their existence. I will define the concept of a lattice in a locally compact group. I will provide some motivation for studying lattices and why I find them useful. I will give some several explicit examples. I will discuss the classification of lattices in the Euclidean groups and the Bieberbach theorems as well as their almost analogs arising from nilpotent Lie groups.
Lecture 2 (Jan. 29)
Abstract: In this lecture, we start with the concept of a Lie group and the associated Lie algebra. We discuss some basic features of Lie groups (e.g. the exponential map). We then focus on simply connected, connected nilpotent Lie groups. We discuss when lattices exist (i.e. when the nilpotent group is defined over the rationals), Mal'cev rigidity, and connections with abstract group theory. We then discuss what a simple Lie group is and the classification of complex simple Lie groups.
Lecture 3 (Feb. 5)
Abstract: In this lecture, we will start with a review of the real classical simple Lie groups; this requires classical but lesser well known linear algebra concepts and involutions on matrix algebras. We will review the concept of a semi-simple Lie group and some basic structure theory (e.g. Iwasawa decomposition, real rank). We will discuss how the (sectional) curvatures of a natural Riemannian metric on a simple Lie group behave with respect to the Iwasawa decomposition and the curvatures of symmetric spaces.
Lecture canceled on Feb. 12
Lecture 4 (Feb. 19)
Abstract: In this lecture, we will start with the concept of a linear algebraic group and K-forms of algebraic groups. We will review foundational work of Borel and Harish-Chandra of integral points of rationally defined semi-simple algebraic groups. Namely, that the group of integral points is a lattice. We then will use this to construct lattices in the special orthogonal groups SO(p,q). We will finish with a discussion when these lattices are cocompact and the concept of an isotropic/anisotropic bilinear form.
Lecture 5 (Feb. 26)
Abstract: In this lecture, we will start with a review of number fields and restriction of scalars; this is a functor from E-objects to F-objects where E/F is an extension of fields. We then will use restriction of scalars and Borel--Harish-Chandra to produce all of the commensurability classes of arithmetic lattices in SO(p,q) when p+q is odd.
Lecture 6 (March 4)
Abstract: In this lecture, we will start with a review of central simple algebras over a field with an emphasis on number fields; some time will be spent motivating why one might care about such things. We will discuss the Brauer group of a number field and Morita equivalence of central simple algebras. We will discuss the local-to-global principle which is a foundational method for solving problems over number fields. We will then review the major results on central simple algebras over number fields and the determination of the Brauer group of a number field; the main point here being that such things are determined by their local behavior.
Lectures canceled on March 18-April 1
Lecture 7 (April 8)
Abstract: In this lecture, we will describe all of the (irreducible) arithmetic lattices (up to commensurability) in G_{a,b} = (SL(2,R))^a x (SL(2,C))^b. Special attention will be given to the cases a=1,b=0 and a=0,b=1 which are connected to hyperbolic 2- and 3-manifolds. We will discuss a volume formula of Borel for the lattices arising in these constructions. We will end with a finiteness theorem of Borel and some applications to low dimensional topology; we will continue this discussion in Lecture 8 where we will focus on lattices in SL(2,R) and SL(2,C) arising from geometric/topological constructions.
Guest Lecture Series by Steve Hurder (April 15-17)
Scroll down for details.
Lecture 8 (April 22)
Abstract: In this lecture, we will start from the geometric side of building lattices via geometric structures on smooth/topological manifolds. We will review Teichmuller theory on compact surfaces and complex/hyperbolic structures. We will review the geometry of 3-manifolds via the 8 model geometries and the JSJ decomposition of 3-manifolds. We will discuss some basic tools for constructing 3-manifolds like Heegaard splittings, mapping tori of surfaces, and Dehn surgery on knots/links in the 3-sphere (or any closed 3-manifold). We will end with a discussion of non-arithmetic lattices in SO(n,1) due to Gromov and Piatetski-Shapiro.
Lecture 9 (April 29)
Abstract: In this lecture, we will discuss various rigidity results for lattices in semi-simple Lie groups. We will start with Weil's local rigidity theorem which will entail a short discussion of representation and character varieties. We then will discuss Mostow and Mostow-Prasad rigidity. We then will discuss Margulis super-rigidity and his famous arithmeticity theorem. We then will discuss the super-rigidity theorem of Corlette and Gromov-Schoen. We then will discuss Kazhdan's property (T) and Margulis' normal subgroup theorem. Finally, we will end with Serre's conjecture on the congruence subgroup problem.
Guest Lecture Series by Steve Hurder
Please note the unusual time and locations
Lecture 1 (April 15, 1:30-2:45pm, BRNG 1255)
Title: Actions on the circle
Abstract: In this talk, we begin with the action on the circle at infinity of the fundamental group G of a closed Riemann surface S with constant negative curvature. We then consider the deformations of this action by taking deformations of the Riemannian metric on S, and introduce the Mitsumatsu invariant of the deformed actions. We will then show how the Mitsumatsu invariant is related to the extended Godbillon-Vey invariant of the action, and its non-triviality is an obstruction to the local symmetries of the metric.
Lecture 2 (April 16, 10:30-11:45am, BRNG 1254)
Title: Actions on manifolds
Abstract: In this talk, we consider actions of groups on compact manifolds of dimension at least 2. We introduce the notion of an Anosov action, and some consequences of the actions in this case. We give an example of deformations of actions of SL(2,Z) on the 2-torus. We then consider actions on the n-torus for n greater than 2, and show that any deformation of the action of SL(n,Z) must be smoothly rigid. We also discuss more recent results on the classification of Anosov actions of higher rank lattices on the n-torus.
Lecture Recording (Passcode: ?j9R0k+e)
Lecture 3 (April 17, 10:30-11:45am, BRNG B212)
Title: Actions on Cantor sets
Abstract: In this talk, we consider equicontinuous actions on Cantor sets, and discuss some of the results on their classification using the notion of continuous orbit equivalence. Examples of actions of the 3-dimensional Heisenberg group illustrate the complexity of the classification problem for this class of actions.
Lecture Recording (Passcode: 8Z$!m8O0)
Module 2: Homeomorphism and diffeomorphism groups.
Module 3: Rigidity of groups acting on closed manifolds.