Related papers: Knaster's problem for $(Z_2)^k$-symmetric subsets …

In this paper we exhibit a similarity between Euclidean Ramsey problems and Knaster-type problems. By borrowing ideas from Ramsey theory we prove weak Knaster properties of non-equatorial triangles in spheres, and of simplices in Euclidean…

In this paper some new cases of Knaster's problem on continuous maps from spheres are established. In particular, we consider an almost orbit of a $p$-torus $X$ on the sphere, a continuous map $f$ from the sphere to the real line or real…

We introduce notions of finiteness obstruction, Euler characteristic, L^2-Euler characteristic, and M\"obius inversion for wide classes of categories. The finiteness obstruction of a category Gamma of type (FP) is a class in the projective…

We consider the problem $F=f(\nu)$ for strictly convex, closed hypersurfaces in $S^{n+1}$ and solve it for curvature functions $F$ the inverses of which are of class $(K)$.

We construct a series of patch type solutions for incompressible Euler equation on $\mathbb S^2$, which constitutes the regularization for steady or traveling point vortex systems. We first prove the existence of $k$-fold symmetric patch…

In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new…

We describe the algebraic K-theory of the $K(1)$-local sphere and the category of type 2 finite spectra in terms of K-theory of discrete rings and topological cyclic homology. We find an infinite family of 2-torsion classes in the $K_0$ of…

Generalizing a result (the case $k = 1$) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k + 1$ belongs to the generalized Walkup class ${\cal K}_k(2k + 1)$, i.e., all its vertex links are $k$-stacked…

In this paper, we construct a family of global solutions to the incompressible Euler equation on a standard 2-sphere. These solutions are odd-symmetric with respect to the equatorial plane and rotate with a constant angular speed around the…

We provide an angular parametrization of the special unitary group $\textrm{SU}(2^{n})$ generalizing Euler angles for $\textrm{SU}(2)$ by successively applying the KAK decomposition. We then determine constraint equations for the parametric…

We prove a general result that relates certain pushouts of $E_k$-algebras to relative tensors over $E_{k+1}$-algebras. Specializations include a number of established results on classifying spaces, resolutions of modules, and (co)homology…

In a classical, quartic field theory with $SU(N) \times Z_2$ symmetry, a class of kink solutions can be found analytically for one special choice of parameters. We construct these solutions and determine their energies. In the limit $N\to…

The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by Wyser and Yong representing the $K$-theory classes of the closures of these…

We unify problems about the equivariant geometry of symmetric quiver representation varieties, in the finite type setting, with the corresponding problems for symmetric varieties $GL(n)/K$ where $K$ is an orthogonal or symplectic group. In…

Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not…

We derive formulas and algorithms for Kitaev's invariants in the periodic table for topological insulators and superconductors for finite disordered systems on lattices with boundaries. We find that K-theory arises as an obstruction to…

We prove some existence results for the Webster scalar curvature problem on the Heisenberg group and on the unit sphere of ${\mathbb C}^{n+1}$, under the assumption of some natural symmetries of the prescribed curvatures. We use variational…

The K\"unneth Theorem for equivariant (complex) K-theory K^*_G, in the form developed by Hodgkin and others, fails dramatically when G is a finite group, and even when G is cyclic of order 2. We remedy this situation in this very simplest…

First, we prove a special case of Knaster's problem, implying that each symmetric convex body in R^3 admits an inscribed cube. We deduce it from a theorem in equivariant topology, which says that there is no S_4-equivariant map from SO(3)…

In the paper the foundation of the $k$-orbit theory is developed. The theory opens a new simple way to the investigation of groups and multidimensional symmetries. The relations between combinatorial symmetry properties of a $k$-orbit and…