Let ${{\Theta }^{[j]}}$ be an analogue of the Ramanujan theta operator for Siegel modular forms. For a given prime $p$ , we give the weights of elements of mod $p$ kernel of ${{\Theta }^{[j]}}$ , where the mod $p$ kernel of ${{\Theta }^{[j]}}$ is the set of all Siegel modular forms $F$ such that ${{\Theta }^{[j]}}(F)$ is congruent to zero modulo $p$ . In order to construct examples of the mod $p$ kernel of ${{\Theta }^{[j]}}$ from any Siegel modular forms, we introduce new operators ${{A}^{(j)}}(M)$ and show the modularity of $F|{{A}^{\left( j \right)}}\left( M \right)$ when $F$ is a Siegel modular form. Finally, we give some examples of the mod $p$ kernel of ${{\Theta }^{[j]}}$ and the filtrations of some of them.

Let $W$ be a compact simply connected triangulated manifold with boundary and let $K\,\subset \,W$ be a subpolyhedron. We construct an algebraic model of the rational homotopy type of $W\text{ }\!\!\backslash\!\!\text{ K}$ out of a model of the map of pairs $\left( K,\,K\cap \partial W \right)\,\to \,\left( W,\,\partial W \right)$ under some high codimension hypothesis.

We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicit models of these configuration spaces for a large class of compact manifolds.

We address the classification problem for graph ${{C}^{*}}$-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition $\left( K \right)$, so that the graph ${{C}^{*}}$-algebras may come with uncountably many ideals.

We find that in this generality, stable isomorphism of graph ${{C}^{*}}$-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the C*-algebras having isomorphic $K$-theories. This proves in turn that under this condition, the graph ${{C}^{*}}$-algebras are in fact classifiable by $K$-theory, providing, in particular, complete classification when the ${{C}^{*}}$- algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs.

Our results are applied to discuss the classification problem for the quantumlens spaces defined by Hong and Szymański, and to complete the classification of graph ${{C}^{*}}$-algebras associated with all simple graphs with four vertices or less.

Culler and Vogtmann defined a simplicial space $O\left( g \right)$, called outer space, to study the outer automorphism group of the free group ${{F}_{g}}$. Using representation theoretic methods, we give an embedding of $O\left( g \right)$ into the analytification of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$, the $S{{L}_{2}}\left( \mathbb{C} \right)$ character variety of ${{F}_{g}}$, reproving a result of Morgan and Shalen. Then we show that every point $v$ contained in a maximal cell of $O\left( g \right)$ defines a flat degeneration of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ to a toric variety $X\left( {{P}_{\Gamma }} \right)$. We relate $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ and $X\left( v \right)$ topologically by showing that there is a surjective, continuous, proper map ${{\Xi }_{v}}\,:\,x\left( {{F}_{g}}\,,\,S{{L}_{2}}\,\left( \mathbb{C} \right) \right)\,\to \,X\left( v \right)$. We then show that this map is a symplectomorphism on a dense open subset of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ with respect to natural symplectic structures on $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ and $X\left( v \right)$. In this way, we construct an integrable Hamiltonian system in $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ for each point in a maximal cell of $O\left( g \right)$, and we show that each $v$ defines a topological decomposition of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ derived from the decomposition of $X\left( {{P}_{\Gamma }} \right)$ by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell in $O\left( g \right)$ all arise as divisorial valuations built from an associated projective compactification of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$.

We introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital ${{\text{C}}^{*}}$-algebras $P\,\subset \,A$ with index finite, and show that an action $\alpha$ from a finite group $G$ on a simple unital ${{\text{C}}^{*}}$- algebra $A$ has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation $E:\,A\,\to \,{{A}^{G}}\,$ has the tracial Rokhlin property. Let $\mathcal{C}$ be a class of infinite dimensional stably finite separable unital ${{\text{C}}^{*}}$-algebras that is closed under the following conditions:

(1) If $A\,\in \,\mathcal{C}$ and $B\,\cong \,A$, then $B\,\in \,\mathcal{C}$.

(2) If $A\,\in \,\mathcal{C}$ and $n\,\in \,\mathbb{N}$, then ${{M}_{n}}\left( A \right)\,\in \,\mathcal{C}$.

(3) If $A\,\in \,\mathcal{C}$ and $p\,\in \,A$ is a nonzero projection, then $pAp\,\in \,\mathcal{C}$.

Suppose that any ${{\text{C}}^{*}}$-algebra in $\mathcal{C}$ is weakly semiprojective. We prove that if $A$ is a local tracial ${{\text{C}}^{*}}$-algebra in the sense of Fan and Fang and a conditional expectation $E:\,A\,\to \,P$ is of index-finite type with the tracial Rokhlin property, then $P$ is a unital local tracial $\mathcal{C}$-algebra.

The main result is that if $A$ is simple, separable, unital nuclear, Jiang–Su absorbing and $E:\,A\,\to \,P$ has the tracial Rokhlin property, then $P$ is Jiang–Su absorbing. As an application, when an action α from a finite group $G$ on a simple unital ${{\text{C}}^{*}}$-algebra $A$ has the tracial Rokhlin property, then for any subgroup $H$ of $G$ the fixed point algebra ${{A}^{H}}$ and the crossed product algebra $A{{\rtimes }_{{{\alpha }_{|H}}}}$$H$ is Jiang–Su absorbing. We also show that the strict comparison property for a Cuntz semigroup $W\left( A \right)$ is hereditary to $W\left( P \right)$ if $A$ is simple, separable, exact, unital, and $E:\,A\,\to \,P$ has the tracial Rokhlin property.

We consider three point positive masses moving on ${{S}^{2}}$ and ${{H}^{2}}$. An Eulerian-relative equilibrium is a relative equilibrium where the three masses are on the same geodesic. In this paper we analyze the spectral stability of these kind of orbits where the mass at the middle is arbitrary and the masses at the ends are equal and located at the same distance from the central mass. For the case of ${{S}^{2}}$, we found a positive measure set in the set of parameters where the relative equilibria are spectrally stable, and we give a complete classification of the spectral stability of these solutions, in the sense that, except on an algebraic curve in the space of parameters, we can determine if the corresponding relative equilibrium is spectrally stable or unstable. On ${{H}^{2}}$, in the elliptic case, we prove that generically all Eulerian-relative equilibria are unstable; in the particular degenerate case when the two equal masses are negligible, we get that the corresponding solutions are spectrally stable. For the hyperbolic case we consider the system where the mass in the middle is negligible; in this case the Eulerian-relative equilibria are unstable.

For a Shimura variety of Hodge type with hyperspecial level structure at a prime $p$, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when $p\,>\,2$. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn, and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elements $w$ in the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding to $w$ is smooth of dimension $l\left( w \right)$ (i.e., the length of $w$) if it is non-empty. We also determine the closure of each stratum.