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.
Adam Chapman
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/
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.
Ohad Noy Feldheim
Rigidity of 3-colorings of the d-dimensional discrete torus
We prove that a uniformly chosen proper coloring of
Z_{2n}^d with 3 colors has a very rigid structure when the
dimension d is sufficiently high. The coloring takes one color on
almost all of either the even or the odd sub-lattice. In
particular, one color appears on nearly half of the lattice sites.
This model is the zero temperature case of the 3-states anti-
ferromagnetic Potts model, which has been studied extensively in
statistical mechanics. The proof involves results about graph
homomorphisms and various combinatorial methods, and follows a
topological intuition. Joint work with Ron Peled.
Naomy Feldheim
Zeroes of Gaussian Analytic Functions with
Translation-Invariant Distribution
Following Wiener, we study zeroes of Gaussian analytic functions
in a strip in the complex plane, with translation-invariant distribution.
We show that the zeroes
obey a "law of large numbers" provided that
the spectral measure has no atoms.
Then we state a counterpart of this theorem for a natural class of Gaussian
analytic functions which have a symmetry with respect to the real axis.
To that end, we shall present a new formula for the average number
of zeroes which might be of independent interest.