We consider amalgamated free product II_{1}factors$M$=$M$_{1*$B$}$M$_{2*$B$}… and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra$Q$⊂$M$can be unitarily conjugated into one of the$M$_{$i$}’s. We apply this to the case where the$M$_{$i$}’s are w-rigid II_{1}factors, with$B$equal to either$C$, to a Cartan subalgebra$A$in$M$_{$i$}, or to a regular hyperfinite II_{1}subfactor$R$in$M$_{$i$}, to obtain the following type of unique decomposition results, àla Bass–Serre: If$M$= ($N$_{1 * C}N_{2*$C$}…)^{$t$}, for some$t$> 0 and some other similar inclusions of algebras$C$⊂$N$_{$i$}then, after a permutation of indices, ($B$⊂$M$_{$i$}) is inner conjugate to ($C$⊂$N$_{$i$})^{$t$}, for all$i$. Taking$B$=$C$and $ M_{i} = {\left( {L{\left( {Z^{2} \rtimes F_{2} } \right)}} \right)}^{{t_{i} }} $ , with {$t$_{$i$}}_{$i$⩾1}=$S$a given countable subgroup of$R$_{+}^{*}, we obtain continuously many non-stably isomorphic factors$M$with fundamental group $ {\user1{\mathcal{F}}}{\left( M \right)} $ equal to$S$. For$B$=$A$, we obtain a new class of factors$M$with unique Cartan subalgebra decomposition, with a large subclass satisfying $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ and Out(M) abelian and calculable. Taking$B$=$R$, we get examples of factors with $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ , Out($M$) =$K$, for any given separable compact abelian group$K$.

No abstract uploaded!

Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. Erd\"{o}s raised the problem of determining $f(n)$. Erd\"{o}s conjectured that there exists a positive constant $c$ such that $ex(n,C_{2k})\geq cn^{1+1/k}$. Haj\'{o}s conjecture that every simple even graph on $n$ vertices can be decomposed into at most $n/2$ cycles. We present the problems, conjectures related to these problems and we summarize the know results. We do not think Haj\'{o}s conjecture is true.

This survey paper is based on the lecture notes for the mini course in the summer school at Yau Mathematics Science Center, Tsinghua University, 2014. We describe and characterize all random subsets $K$ of simply connected domain which satisfy the ``conformal restriction" property. There are two different types of random sets: the chordal case and the radial case. In the chordal case, the random set $K$ in the upper half-plane $\HH$ connects two fixed boundary points, say 0 and $\infty$, and given that $K$ stays in a simply connected open subset $H$ of $\HH$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is any conformal map from $H$ onto $\HH$ fixing 0 and $\infty$. In the radial case, the random set $K$ in the upper half-plane $\HH$ connects one fixed boundary points, say 0, and one fixed interior point, say $i$, and given that $K$ stays in a simply connected open subset $H$ of $\HH$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is the conformal map from $H$ onto $\HH$ fixing 0 and $i$. It turns out that the random set with conformal restriction property are closely related to the intersection exponents of Brownian motion. The construction of these random sets relies on Schramm Loewner Evolution with parameter $\kappa=8/3$ and Poisson point processes of Brownian excursions and Brownian loops.

For a real analytic periodic function 𝜙: ℝ→ℝ, an integer 𝑏≥2 and 𝜆∈(1/𝑏,1), we prove the following dichotomy for the Weierstrass-type function 𝑊(𝑥)=∑_{𝑛≥0} 𝜆^𝑛 𝜙(𝑏^𝑛 𝑥): Either W(x) is real analytic, or the Hausdorff dimension of its graph is equal to 2+log𝑏𝜆. Furthermore, given b and 𝜙, the former alternative only happens for finitely many 𝜆 unless 𝜙 is constant.