# Abstracts

Nir Sharon pdf

Omer Tamuz:
Scenery Reconstruction on Finite Abelian Groups
A binary labeling of a graph is a function from its nodes to {0,1}. Scenery reconstruction is the problem of inferring a labeling given the labels observed by a particle performing a random walk on the graph. We consider the question of when a random walk on a finite abelian group with a given step distribution is reconstructive, or can be used to reconstruct a binary labeling up to a shift. We focus initially on walks on undirected cycles. Matzinger and Lember (2006) give a sufficient condition for reconstructibility on cycles, involving the Fourier transform of the random walk's step distribution. While, as we show, this condition is not in general necessary, our main result is that it is necessary when the length of the cycle is prime and larger than 5, and the step distribution has only rational probabilities. We use this result to show that rational random walks on prime cycles which have non-zero drift (for an appropriate notion of drift) are reconstructive, as is any random walk with a non-symmetric bounded step function, on a large enough cycle.
Joint work with Hilary Finucane and Yariv Yaari

Ahuva Shkop:
Finding something real in Zilber's field.
In 2004, Zilber constructed a class of exponential fields, known as pseudoexponential fields, and proved that there is exactly one pseudoexponential field in every uncountable cardinality up to isomorphism. He conjectured that the pseudoexponential field of size continuum, K, is isomorphic to the classic complex exponential field. Since the complex exponential field contains the real exponential field, one consequence of this conjecture is the existence of a real closed exponential subfield of K. In this talk, I will sketch the construction of real closed exponential subfields of K and discuss some of their properties.

Generalized Clifford algebras
This talk is based on a joint work by Prof. Uzi Vishne and the speaker. Given a field $F$ containing a primitive $p$th root of unity $\rho_p$ and a homogenous form of degree $p$ with $n$ variables $f(u_1,\dots,u_n)$, the Clifford algebra of this form is defined to be $C_f=F[x_1,\dots,x_n : (u_1 x_1+\dots+u_n x_n)^p= f(u_1,\dots,u_n) \forall u_1,\dots,u_n \in F]$. For $p=2$, $f$ is a quadratic form, and it is well-known that its underlying Clifford algebra is a tensor product of $\lfloor \frac{n}{2} \rfloor$ quaternion algebras. For any odd prime $p$ and $n=2$, we know that in $C_f=F[x,y : \dots]$, $y=z_1+\dots+z_{p-1}$ where for all $1 \leq i \leq p-1$, $z_i x=\rho_p^i x z_i$. We prove that the algebra $C_f/$ is an Azumaya algebra whose center is the affine algebra of a hyperelliptic curve of genus $\lfloor \frac{p-1}{2} \rfloor$, and that every simple homomorphic image of $C_f$ is a cyclic algebra of degree $p$. This generalizes the main result of the paper “On the Clifford algebra of a binary cubic form”/D. Haile. If $p=5$ we also prove that every division image of $C_f/$ is either a tensor product of one or two cyclic algebras of degree $5$ and we calculate the center of its ring of quotients explicitly.

Menny Aka
Arithmetic groups with isomorphic finite quotients
Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the the fibers are of unbounded size. In the talk I will concentrate on giving examples and explain how they fit in the scheme of my proof.

Doron Pudar
Uniform Words are Primitive
Let a,b,c,... in S_n be random permutations on n elements, chosen at uniform distribution. What is the distribution of the permutation obtained by a fixed word in the letters a,b,c,..., such as ab,a^2, a^2bc^2b, or aba^(-2)b^(-1)? More concretely, do these new random permutations have uniform distribution? In general, a free word w in F_k is called uniform if for every finite group G, the word map $w: G^k \to G$ induces uniform distribution on G (given uniform distribution on G^k). So which words are uniform? This question is strongly connected to the notion of primitive words in the free group $F_k$. The word w is called primitive is it belongs to some basis, i.e. a generating set of size $k$. It is an easy observation that a primitive word is uniform. It was conjectured that the converse is also true. We prove it for F_2. In a very recent joint work with O. Parzanchevski, we manage to prove the conjecture in full. A key ingredient of the proofs is a new algorithm to detect primitive elements.

Matan Prezma
Homotopy normal maps
Normal maps between discrete groups $N\rightarrow G$ were characterized[FS] as those which induce a compatible topological group structure on the homotopy quotient $EN\times_N G$. Here we deal with topological group maps $N\rightarrow G$ being normal in the same sense as above and hence forming a homotopical analogue to the inclusion of a topological normal subgroup in a reasonable way. We characterize these maps by a compatible simplicial loop space structure on $Bar_\bullet(N,G)$, invariant under homotopy monoidal functors, e.g. Localizations and Completions. In the course of characterizing homotopy normality, we define a notion of a "homotopy action" similar to an $A_{\infty}$ action on a space, but phrased in terms of Segal's 'special $\Delta-$spaces' and seem to be of importance on its own right. As an application of the invariance of normal maps, we give a very short proof to a theorem of Dwyer and Farjoun namely that a localization by a suspended map of a principal fibration of connected spaces is again principal.