Related papers: Counterexamples to the topological Tverberg conjec…

Ordinary maps satisfy topological recursion for a certain spectral curve $(x, y)$. We solve a conjecture from arXiv:1710.07851 that claims that fully simple maps, which are maps with non self-intersecting disjoint boundaries, satisfy…

Let $n\geq 2$ and $\mathbb K $ be a number field of characteristic $0$. Jacobian Conjecture asserts for a polynomial map $\mathcal P$ from $\mathbb K ^n$ to itself, if the determinant of its Jacobian matrix is a nonzero constant in $\mathbb…

Given a closed Riemannian manifold $(N^{n+1},g)$, $n+1 \geq 3$ we prove the compactness of the space of singular, minimal hypersurfaces in $N$ whose volumes are uniformly bounded from above and the $p$-th Jacobi eigenvalue $\lambda_p$'s are…

The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in…

Let $T(d,r) = (r-1)(d+1)+1$ be the parameter in Tverberg's theorem, and call a partition $\mathcal I$ of $\{1,2,\ldots,T(d,r)\}$ into $r$ parts a "Tverberg type". We say that $\mathcal I$ "occurs" in an ordered point sequence $P$ if $P$…

A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in ${\mathbb E}^3$ must be greater than one. In this paper we prove this for $C^{3+\alpha}$-smooth surfaces. The Conjecture is first…

Let $(X,T)$ be a dynamical system where $X$ is a compact metric space and $T:X\rightarrow X$ is continuous and invertible. Assume the Lebesgue covering dimension of $X$ is $d$. We show that for a generic continuous map…

Given a complete simple topological graph $G$, a $k$-face generated by $G$ is the open bounded region enclosed by the edges of a non-self-intersecting $k$-cycle in $G$. Interestingly, there are complete simple topological graphs with the…

This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional…

The structure of the set of positivity-preserving maps between matrix algebras is notoriously difficult to describe. The notable exceptions are the results by St{\o}rmer and Woronowicz from 1960s and 1970s settling the low dimensional…

Let X be a smooth, complete, connected submanifold of dimension n < N in a complex affine space A^N (C), and r is the rank of its Gauss map \gamma, \gamma (x) = T_x (X). The authors prove that if 2 \leq r \leq n - 1, N - n \geq 2, and in…

Our first main result is a uniform bound, in every dimension $k \in \mathbb N$, on the topological Tur\'an numbers of $k$-dimensional simplicial complexes: for each $k \in \mathbb N$, there is a $\lambda_k \ge k^{-2k^2}$ such that for any…

This survey paper deals with upper and lower bounds on the number of $k$-matchings in regular graphs on $N$ vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower…

In this paper, we investigate properties of set-valued mappings that establish connection between the values of this map at two arbitrary points of the domain and the value at their midpoint. Such properties are, for instance, Jensen…

Our main result states that for each finite complex L the category ${\bf TOP}$ of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all…

This work is, in part, a generalization of the article by A.A. Bruen ,T.C Bruen and J.M.McQuillan on Desargues Theorem in arXiv:2007.09175[mathCO]July 17,2020. We prove the extension of Desargues theorem in all dimensions, using 4 different…

Let $X$ be a subset of a Hilbert space. We prove that if $v\colon X\to \mathbb{R}^m$ is such that \begin{equation*} \Big\lVert v(x)-\sum_{i=1}^m t_iv(x_i)\Big\rVert\leq \Big\lVert x-\sum_{i=1}^m t_ix_i\Big\rVert \end{equation*} for all…

Let f: P^1 \to P^1 be a rational map with finite postcritical set P_f. Thurston showed that f induces a holomorphic map \sigma_f of the Teichmueller space T modelled on P_f to itself fixing the basepoint corresponding to the identity map…

The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at…

We extend to binary relational systems the notion of compact and normal structure, introduced by J.P.Penot for metric spaces, and we prove that for the involutive and reflexive ones, every commuting family of relational homomorphisms has a…