The normal subgroup theorem of Margulis expresses that many lattices in semi-simple Lie groups are simple up to finite error. In this talk, we give a short introduction to the normal subgroup theorem , including a brief review of the relevant notions about lattices, amenability, and property (T), as well as a short overview of the proof of the normal subgroup theorem. 1 Margulis's normal subgro...

The Zimmer program refers to a number of questions and conjectures posed by Robert Zimmer in the 1980s concerning smooth actions of lattices in higher-rank Lie groups. We report on some recent progress by the author and collaborators. Lattices in Higher-Rank Lie Groups The primary objects inmy talk are lattices Γ in higher-rank simple Lie groups G. The simplest example of such a lattice is Γ = ...

Lattice Definition. (Lattice) Given n linearly independent vectors b1, , bn ∈ R, the lattice generated by them · · · is defined as L(b1, b2, · · · bn) = { xibi|xi ∈ Z}. We refer to b1, · · · , bn as a basis of the lattice. Equivalently, if we define B as the m × n matrix whose columns are b1, , bn, then the lattice generated by B is · · · L(B) = L(b1, b2, · · · , bn) = {Bx|x ∈ Z}. We say that t...

We prove sharp upper and lower bounds for generalized Calderón's sums associated to frames on LCA groups generated by affine actions of cocompact subgroup translations general measurable families automorphisms. The proof makes use techniques analysis metric spaces, relies a counting estimate lattice points inside balls. will deduce as special cases Calderón-type inequalities expanding automorph...

Abstract. A composite dilation wavelet is a collection of functions generating an orthonormal basis for L2(Rn) under the actions of translations from a full rank lattice and dilations by products of elements of non-commuting groups A and B. A minimally supported frequency composite dilation wavelet has generating functions whose Fourier transforms are characteristic functions of a lattice tilin...

We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with the graph product provided all of its quasiconvex subgroups are separable. W...

Any system that approximates an infinite lattice by a family of finite clusters (with periodic boundary conditions) passes through an intermediate region with enlarged (hidden) symmetry as the system size is increased. The hidden symmetry allows for extra degeneracies and level crossings and has application to exact-diagonalization studies, Monte Carlo simulations, lattice gauge theories, and r...

H is the upper half plane, a complex manifold. It will be helpful to interpret H in multiple ways. A lattice Λ ⊂ C is a free abelian group of rank 2, for which the map Λ ⊗Z R → C is an isomorphism. In other words, Λ is a subgroup of C of the form Zα⊕Zβ, where {α, β} is basis for C/R. Two lattices Λ and Λ′ are homothetic if Λ′ = θΛ for some θ ∈ C∗. This is an equivalence relation, and the equiva...

Let V be a finite dimensional complex linear space and let G be an irreducible finite subgroup of GL(V). For a G-invariant lattice Λ in V of maximal rank, we give a description of structure of the complex torus V /Λ. In particular, we prove that for a wide class of groups, V /Λ is isogenous to a self-product of an elliptic curve, and that in many cases V /Λ is isomorphic to a product of mutuall...

A colored lattice L(c) has a geometrical lattice L. A subgroup lattice L' of L and each of its cosets consist of like-colored points, each coset having a different color. The index of L' in L is given by Delta, the determinant of the matrix (t(jk)) that converts L into L'. This is the order of the factor group {L/L'}, and is also the number n of colors present. The crystal systems-i.e., the com...