TY - CONF
AB - We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call homotopic curve shortening (HCS), starts with a closed curve (which might self-intersect) in the presence of a set P⊂ ℝ² of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent. We find experimentally that, if the initial curve is held fixed and P is chosen to be either a very fine regular grid or a uniformly random point set, then HCS behaves at the limit like the affine curve-shortening flow (ACSF). This connection between HCS and ACSF generalizes the link between "grid peeling" and the ACSF observed by Eppstein et al. (2017), which applied only to convex curves, and which was studied only for regular grids. We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature. Furthermore, the number of self-intersections of a curve, or intersections between two curves (appropriately defined), does not increase. Finally, if the initial curve is simple, then the number of inflection points (appropriately defined) does not increase.
AU - Avvakumov, Sergey
AU - Nivasch, Gabriel
ID - 7991
SN - 18688969
T2 - 36th International Symposium on Computational Geometry
TI - Homotopic curve shortening and the affine curve-shortening flow
VL - 164
ER -
TY - THES
AB - 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.
AU - Avvakumov, Sergey
ID - 8156
TI - Topological methods in geometry and discrete mathematics
ER -
TY - JOUR
AU - Avvakumov, Sergey
AU - Wagner, Uli
AU - Mabillard, Isaac
AU - Skopenkov, A. B.
ID - 9308
IS - 6
JF - Russian Mathematical Surveys
SN - 0036-0279
TI - Eliminating higher-multiplicity intersections, III. Codimension 2
VL - 75
ER -
TY - GEN
AB - Suppose that $n\neq p^k$ and $n\neq 2p^k$ for all $k$ and all primes $p$. We prove that for any Hausdorff compactum $X$ with a free action of the symmetric group $\mathfrak S_n$ there exists an $\mathfrak S_n$-equivariant map $X \to
{\mathbb R}^n$ whose image avoids the diagonal $\{(x,x\dots,x)\in {\mathbb R}^n|x\in {\mathbb R}\}$.
Previously, the special cases of this statement for certain $X$ were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We
take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of $\mathfrak S_n$-equivariant maps from the boundary
$\partial\Delta^{n-1}$ of $(n-1)$-simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser's conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem.
AU - Avvakumov, Sergey
AU - Kudrya, Sergey
ID - 8182
T2 - arXiv
TI - Vanishing of all equivariant obstructions and the mapping degree
ER -
TY - GEN
AB - Denote by ∆N the N-dimensional simplex. A map f : ∆N → Rd is an almost r-embedding if fσ1∩. . .∩fσr = ∅ whenever σ1, . . . , σr are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and d ≥ 2r + 1, then there is an almost r-embedding ∆(d+1)(r−1) → Rd. This was improved by Blagojevi´c–Frick–Ziegler using a simple construction of higher-dimensional counterexamples by taking k-fold join power of lower-dimensional ones. We improve this further (for d large compared to r): If r is not a prime power and N := (d+ 1)r−r l
d + 2 r + 1 m−2, then there is an almost r-embedding ∆N → Rd. For the r-fold van Kampen–Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard–Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the Ozaydin theorem on existence of equivariant maps.
AU - Avvakumov, Sergey
AU - Karasev, R.
AU - Skopenkov, A.
ID - 8184
T2 - arXiv
TI - Stronger counterexamples to the topological Tverberg conjecture
ER -
TY - GEN
AB - In this paper we study envy-free division problems. The classical approach to some of such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions when this map hits the center of the simplex. The mere continuity is not sufficient for such a conclusion, the usual assumption (for example, in the Knaster--Kuratowski--Mazurkiewicz and the Gale theorem) is a certain boundary condition.
We follow Erel Segal-Halevi, Fr\'ed\'eric Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the economic meaning of 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 partition problems when $n$ is odd and not a prime power.
AU - Avvakumov, Sergey
AU - Karasev, Roman
ID - 8185
T2 - arXiv
TI - Envy-free division using mapping degree
ER -
TY - GEN
AU - Pokusaeva, Victoria
AU - Usmanova, Dinara R.
AU - Putintseva, Ekaterina V.
AU - Espinar, Lorena
AU - Sarkisyan, Karen
AU - Mishin, Alexander S.
AU - Bogatyreva, Natalya S.
AU - Ivankov, Dmitry
AU - Akopyan, Arseniy
AU - Avvakumov, Sergey
AU - Povolotskaya, Inna S.
AU - Filion, Guillaume J.
AU - Carey, Lucas B.
AU - Kondrashov, Fyodor
ID - 9789
TI - Multiple alignment of His3 orthologues
ER -
TY - GEN
AU - Pokusaeva, Victoria
AU - Usmanova, Dinara R.
AU - Putintseva, Ekaterina V.
AU - Espinar, Lorena
AU - Sarkisyan, Karen
AU - Mishin, Alexander S.
AU - Bogatyreva, Natalya S.
AU - Ivankov, Dmitry
AU - Akopyan, Arseniy
AU - Avvakumov, Sergey
AU - Povolotskaya, Inna S.
AU - Filion, Guillaume J.
AU - Carey, Lucas B.
AU - Kondrashov, Fyodor
ID - 9790
TI - A statistical summary of segment libraries and sequencing results
ER -
TY - JOUR
AB - Characterizing the fitness landscape, a representation of fitness for a large set of genotypes, is key to understanding how genetic information is interpreted to create functional organisms. Here we determined the evolutionarily-relevant segment of the fitness landscape of His3, a gene coding for an enzyme in the histidine synthesis pathway, focusing on combinations of amino acid states found at orthologous sites of extant species. Just 15% of amino acids found in yeast His3 orthologues were always neutral while the impact on fitness of the remaining 85% depended on the genetic background. Furthermore, at 67% of sites, amino acid replacements were under sign epistasis, having both strongly positive and negative effect in different genetic backgrounds. 46% of sites were under reciprocal sign epistasis. The fitness impact of amino acid replacements was influenced by only a few genetic backgrounds but involved interaction of multiple sites, shaping a rugged fitness landscape in which many of the shortest paths between highly fit genotypes are inaccessible.
AU - Pokusaeva, Victoria
AU - Usmanova, Dinara R.
AU - Putintseva, Ekaterina V.
AU - Espinar, Lorena
AU - Sarkisyan, Karen
AU - Mishin, Alexander S.
AU - Bogatyreva, Natalya S.
AU - Ivankov, Dmitry
AU - Akopyan, Arseniy
AU - Avvakumov, Sergey
AU - Povolotskaya, Inna S.
AU - Filion, Guillaume J.
AU - Carey, Lucas B.
AU - Kondrashov, Fyodor
ID - 6419
IS - 4
JF - PLoS Genetics
TI - An experimental assay of the interactions of amino acids from orthologous sequences shaping a complex fitness landscape
VL - 15
ER -
TY - GEN
AB - We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer m≥2; this result was previously known for prime powers m=pk. We also give a higher-dimensional generalization.
AU - Akopyan, Arseniy
AU - Avvakumov, Sergey
AU - Karasev, Roman
ID - 75
TI - Convex fair partitions into arbitrary number of pieces
ER -
TY - JOUR
AB - We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle.
AU - Akopyan, Arseniy
AU - Avvakumov, Sergey
ID - 6355
JF - Forum of Mathematics, Sigma
SN - 2050-5094
TI - Any cyclic quadrilateral can be inscribed in any closed convex smooth curve
VL - 6
ER -
TY - JOUR
AB - We classify smooth Brunnian (i.e., unknotted on both components) embeddings (S2 × S1) ⊔ S3 → ℝ6. Any Brunnian embedding (S2 × S1) ⊔ S3 → ℝ6 is isotopic to an explicitly constructed embedding fk,m,n for some integers k, m, n such that m ≡ n (mod 2). Two embeddings fk,m,n and fk′ ,m′,n′ are isotopic if and only if k = k′, m ≡ m′ (mod 2k) and n ≡ n′ (mod 2k). We use Haefliger’s classification of embeddings S3 ⊔ S3 → ℝ6 in our proof. The relation between the embeddings (S2 × S1) ⊔ S3 → ℝ6 and S3 ⊔ S3 → ℝ6 is not trivial, however. For example, we show that there exist embeddings f: (S2 ×S1) ⊔ S3 → ℝ6 and g, g′ : S3 ⊔ S3 → ℝ6 such that the componentwise embedded connected sum f # g is isotopic to f # g′ but g is not isotopic to g′.
AU - Avvakumov, Serhii
ID - 1522
IS - 1
JF - Moscow Mathematical Journal
TI - The classification of certain linked 3-manifolds in 6-space
VL - 16
ER -
TY - GEN
AB - We study conditions under which a finite simplicial complex $K$ can be mapped to $\mathbb R^d$ without higher-multiplicity intersections. An almost $r$-embedding is a map $f: K\to \mathbb R^d$ such that the images of any $r$
pairwise disjoint simplices of $K$ do not have a common point. We show that if $r$ is not a prime power and $d\geq 2r+1$, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost $r$-embedding of
the $(d+1)(r-1)$-simplex in $\mathbb R^d$. This improves on previous constructions of counterexamples (for $d\geq 3r$) based on a series of papers by M. \"Ozaydin, M. Gromov, P. Blagojevi\'c, F. Frick, G. Ziegler, and the second and fourth present authors. The counterexamples are obtained by proving the following algebraic criterion in codimension 2: If $r\ge3$ and if $K$ is a finite $2(r-1)$-complex then there exists an almost $r$-embedding $K\to \mathbb R^{2r}$ if and only if there exists a general position PL map $f:K\to \mathbb R^{2r}$ such that the algebraic intersection number of the $f$-images of any $r$ pairwise disjoint simplices of $K$ is zero. This result can be restated in terms of cohomological obstructions or equivariant maps, and extends an analogous codimension 3 criterion by the second and fourth authors. As another application we classify ornaments $f:S^3 \sqcup S^3\sqcup S^3\to \mathbb R^5$ up to ornament
concordance. It follows from work of M. Freedman, V. Krushkal and P. Teichner that the analogous criterion for $r=2$ is false. We prove a lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample.
AU - Avvakumov, Sergey
AU - Mabillard, Isaac
AU - Skopenkov, A.
AU - Wagner, Uli
ID - 8183
T2 - arXiv
TI - Eliminating higher-multiplicity intersections, III. Codimension 2
ER -