The classification of certain linked 33-manifolds in 66-space

We work entirely in the smooth category. An embedding $f:(S^2\times S^1)\sqcup S^3\rightarrow {\mathbb R}^6$ is {\it Brunnian}, if the restriction of $f$ to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$ such that $m\equiv n \pmod{2}$, we explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\times S^1)\sqcup S^3 \rightarrow {\mathbb R}^6$ such that the following theorem holds. Theorem: Any Brunnian embedding $f:(S^2\times S^1)\sqcup S^3\rightarrow {\mathbb R}^6$ is isotopic to $f_{k,m,n}$ for some integers $k,m,n$ such that $m\equiv n \pmod{2}$. Two embeddings $f_{k,m,n}$ and $f_{k',m',n'}$ are isotopic if and only if $k=k'$, $m\equiv m' \pmod{2k}$ and $n\equiv n' \pmod{2k}$. We use Haefliger's classification of embeddings $S^3\sqcup S^3\rightarrow {\mathbb R}^6$ in our proof. The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\rightarrow {\mathbb R}^6$ and $S^3\sqcup S^3\rightarrow {\mathbb R}^6$ is not trivial. Corollary: There exist embeddings $f:(S^2\times S^1)\sqcup S^3\rightarrow {\mathbb R}^6$ and $g,g':S^3\sqcup S^3\rightarrow {\mathbb R}^6$ such that the componentwise embedded connected sum $f\#g$ is isotopic to $f\#g'$ but $g$ is not isotopic to $g'$

How do curved spheres intersect in 3-space?

The following problem was proposed in 2010 by S. Lando. Let $M$ and $N$ be two unions of the same number of disjoint circles in a sphere. Do there always exist two spheres in 3-space such that their intersection is transversal and is a union of disjoint circles that is situated as $M$ in one sphere and as $N$ in the other? Union $M'$ of disjoint circles is {\it situated} in one sphere as union $M$ of disjoint circles in the other sphere if there is a homeomorphism between these two spheres which maps $M'$ to $M$. We prove (by giving an explicit example) that the answer to this problem is "no". We also prove a necessary and sufficient condition on $M$ and $N$ for existing of such intersecting spheres. This result can be restated in terms of graphs. Such restatement allows for a trivial brute-force algorithm checking the condition for any given $M$ and $N$. It is an open question if a faster algorithm exist.Comment: 9 pages, 9 figure

Any cyclic quadrilateral can be inscribed in any closed convex smooth curve

We prove that any cyclic quadrilateral can be inscribed in any closed convex $C^1$-curve. The smoothness condition is not required if the quadrilateral is a rectangle.Comment: 6 pages, 10 figure

Vector and Axial Nucleon Form Factors:A Duality Constrained Parameterization

We present new parameterizations of vector and axial nucleon form factors. We maintain an excellent descriptions of the form factors at low momentum transfers, where the spatial structure of the nucleon is important, and use the Nachtman scaling variable xi to relate elastic and inelastic form factors and impose quark-hadron duality constraints at high momentum transfers where the quark structure dominates. We use the new vector form factors to re-extract updated values of the axial form factor from neutrino experiments on deuterium. We obtain an updated world average value from neutrino-d and pion electroproduction experiments of M_A = 1.014 +- 0.014 GeV/c2. Our parameterizations are useful in modeling neutrino interactions at low energies (e.g. for neutrino oscillations experiments). The predictions for high momentum transfers can be tested in the next generation electron and neutrino scattering experiments.Comment: 5 pages, 3 figures, to be published in EPJ

IST Austria Thesis

We present solutions to several problems originating from geometry and discrete mathematics: existence of equipartitions, maps without Tverberg multiple points, and inscribing quadrilaterals. Equivariant obstruction theory is the natural topological approach to these type of questions. However, for the specific problems we consider it had yielded only partial or no results. We get our results by complementing equivariant obstruction theory with other techniques from topology and geometry

Envy-free division using mapping degree

In this paper we study envy-free division problems. The classical approach to such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions for this map to hit the center of the simplex. The mere continuity of the map is not sufficient for reaching such a conclusion. Classically, one makes additional assumptions on the behavior of the map on the boundary of the simplex (for example, in the Knaster--Kuratowski--Mazurkiewicz and the Gale theorem). We follow Erel Segal-Halevi, Fr\'ed\'eric Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the meaning in economy as the possibility for a player to prefer an empty part in the segment partition problem. We solve the problem positively when $n$, the number of players that divide the segment, is a prime power, and we provide counterexamples for every $n$ which is not a prime power. We also provide counterexamples relevant to a wider class of fair or envy-free division problems when $n$ is odd and not a prime power. In this arxiv version that appears after the official publication we have corrected the statement and the proof of Lemma 3.4.Comment: 16 pages, 3 figure