In this paper we introduce the concept of a $G$-pseudomanifold, which is an equivariant version of the stratified spaces defined by Goresky and Mac Pherson for compact Lie group actions.

In this paper we study the Reiter $P_{2}$ condition for commutative hypergroups and give necessary and sufficient conditions for $x \in \mathrm{supp}\, \pi$, where $\pi$ is the Plancherel measure. Finally we apply general results to characterize $\mathrm{supp} \, \pi$ in the case of polynomial hypergroups.

The flow category of a Morse-Bott-Smale function $f_{A}:G_{n}(\mathbb{C}^{\infty}) \rightarrow \mathbb{R}$ is shown to be related to the flow category of the action functional on the universal cover of $\mathcal{L}G_{n,n+k}(\mathbb{C})$ via a group action. The Floer homotopy type and the associated cohomology ring of $f_{A}:G_{n}(\mathbb{C}) \rightarrow \mathbb{R}$ are computed. When $n = 1$ this cohomology ring is the Floer cohomology of $G_{1,1+k}(\mathbb{C})$.

Tsirelson's space $T$ is known to be distortable but it is open as to whether or not $T$ is arbitrarily distortable. For $n \in \mathbb{N}$ the norm $||\cdot||_{n}$ of the Tsirelson space $T(S_{n},2^{-n})$ is equivalent to the standard norm on $T$. We prove there exists $K \lt \infty$ so that for all $n$, ||\cdot||_{n} does not $K$ distort any subspace $Y$ of $T$.

A necessary and sufficient condition for the existence and the uniqueness of a conformal metric on a $2$-sphere of constant curvature 1 and with three conical singularities of prescribed order is given.

A modified Rankin-Selberg integral is defined for the automorphic functions not of rapid decay on the congruence subgroup of $SL(2,Z)$. Analytic continuation and functional equations are obtained for this Rankin-Selberg integral.

In answer to a question by Becker, Rubel, and Henson, we show that countable subsets of $\mathbb{C}$ can be used as complete invariants for Riemann surfaces considered up to conformal equivalence, and that this equivalence relation is itself Borel in a natural Borel structure on the space of all such surfaces. We further proceed to precisely calculate the classification difficulty of this equivalence relation in terms of the modem theory of Borel equivalence relations.

On the other hand we show that the analog of Becker, Rubel, and Henson's question has a negative solution in (complex) dimension $n \geq 2$.

For a symmetric $\alpha$-stable process $X$ on $\mathbf{R}^{n}$ with $0 \lt \alpha \lt 2$, $n \geq 2$ and a domain $D \subset \mathbf{R}^{n}$, let $L^{D}$ be the infinitesimal generator of the subprocess of $X$ killed upon leaving $D$. For a Kato class function $q$, it is shown that $L^{d}+q$ is intrinsic ultracontractive on a Hölder domain $D$ of order 0. Then this is used to establish the conditional gauge theorem for $X$ on bounded Lipschitz domains in $\mathbf{R}^{n}$. It is also shown that the conditional lifetimes for symmetric stable process in a Hölder domain of order 0 are uniformly bounded.

By an integral-elementary function we mean any real function that can be obtained from the constants, sin $x$, $e^{x}$, $\log x$, and arcsin $x$ (defined on ($-1,1$)) using the basic algebraic operations, composition and integration. The rank of an integral-elementary function $f$ is the depth of the formula defining $f$. The integral-elementary functions of rank $\leq n$ are real-analytic and satisfy a common algebraic differential equation $P_{n} (f,f',\ldots,f^(k))=0$ with integer coefficients.

We prove that every continuous function $f:\mathbf{R} \rightarrow \mathbf{R}$ can be approximated uniformly by integral-elementary functions of bounded rank. Consequently, there exists an algebraic differential equation with integer coefficients such that its everywhere analytic solutions approximate every continuous function uniformly. This solves a problem posed by L. A. Rubel.

Using the same basic functions as above, but allowing only the basic algebraic operations and compositions, we obtain the class of elementary functions. We show that every differentiable function with a derivative not exceeding an iterated exponential can be uniformly approximated by elementary functions of bounded rank. If we include the function arcsin $x$ defined on $[-1,1]$, then the resulting class of naive-elementary functions will approximate every continuous function uniformly.

We also show that every sequence can be uniformly approximated by elementary functions, and that every integer sequence can be represented .in the form $f(n)$, where $f$ is naive-elementary.

We study smooth Spin(4) actions on closed, orientable $8$-dimensional manifolds, where Spin(4) is isomorphic to the group $\mathrm{SU}(2) \times \mathrm{SU}(2)$. We examine the isotropy structures that can arise, and give an equivariant classification in the case where the set of exceptional orbits, stabilized by finite-cyclic goups, is empty.

We show that a non-Sasakian contact metric manifold whose characteristic vector field belongs to the $(k,\mu)$-nullity distribution is completely determined locally by its dimension and the values for $k$ and $\mu$. We present explicit examples for all possible dimensions and all possible $(k,\mu)$.