Ori Parzanchevski

Isoperimetric inequalities in simplicial complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov’s notion of geometric overlap. Using the work of Gundert and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes.

The isospectral fruits of representation theory: quantum graphs and drums

We present a method which enables one to construct isospectral objects, such as quantum graphs and drums. One aspect of the method is based on representation theory arguments which are shown and proved. The complementary part concerns techniques of assembly which are both stated generally and demonstrated. For this purpose, quantum graphs are grist to the mill. We develop the intuition that stands behind the construction as well as the practical skills of producing isospectral objects. We discuss the theoretical implications which include Sunadas theorem of isospectrality (Sunada 1985 Ann. Math. 121 169) arising as a particular case of this method. A gallery of new isospectral examples is presented, and some known examples are shown to result from our theory.

Linear Representations and Isospectrality with Boundary Conditions

We present a method for constructing families of isospectral systems, using linear representations of finite groups. We focus on quantum graphs, for which we give a complete treatment. However, the method presented can be applied to other systems such as manifolds and two-dimensional drums. This is demonstrated by reproducing some known isospectral drums, and new examples are obtained as well. In particular, Sunada’s method (Ann. Math. 121, 169–186, 1985) is a special case of the one presented.

Measure Preserving Words are Primitive

A word w 2 Fk, the free group on k generators, is called primitive if it belongs to some basis of Fk. Associated with w and a finite group G is the word map w : G×...×G ! G defined on the direct product of k copies of G. We call w measure preserving if given uniform measure on G×...×G, the image of this word map induces uniform measure on G (for every finite group G). It is easy to see that every primitive word is measure preserving, and several authors have conjectured that the two properties are, in fact, equivalent. Here we prove this conjecture. The main ingredients of the proof include random coverings of Stallings graphs, algebraic extensions of free groups and Mobius inversions. Our methods yield the stronger result that a subgroup of Fk is measure preserving iff it is a free factor. As an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of Fk form a closed set in this topology.

Mixing in High-Dimensional Expanders

We establish a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to as mixing, or pseudorandomness). Recently, an analogue of this Lemma was proved for simplicial complexes of arbitrary dimension, provided that the skeleton of the complex is complete. More precisely, it was shown that a concentrated spectrum of the simplicial Hodge Laplacian implies a similar type of pseudo-randomness as in graphs. In this paper we remove the assumption of a complete skeleton, showing that simultaneous concentration of the Laplace spectra in all dimensions implies pseudo-randomness in any complex. We discuss various applications and present some open questions.

On the Fourier expansion of word maps

Frobenius observed that the number of times an element of a finite group is obtained as a commutator is given by a specific combination of the irreducible characters of the group. More generally, for any word w the number of times an element is obtained by substitution in w is a class function. Thus, it has a presentation as a combination of irreducible characters, called its Fourier expansion. In this paper we present formulas regarding the Fourier expansion of words in which some letters appear twice. These formulas give simple proofs for classical results, as well as new ones.

Super-Golden-Gates for PU(2)

Abstract To each of the symmetry groups of the Platonic solids we adjoin a carefully designed involution yielding topological generators of PU (2) which have optimal covering properties as well as efficient navigation. These are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups. The generators give super efficient 1-qubit quantum gates and are natural building blocks for the design of universal quantum gates.

On

We study finite G-sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: if M is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then M has isospectral non-isometric covers.

G

This paper studies the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.