---
_id: '10220'
abstract:
- lang: eng
text: "We study conditions under which a finite simplicial complex K can be mapped
to ℝd without higher-multiplicity intersections. An almost r-embedding is a map
f: K → ℝd 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 ≥ 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 ℝd. This improves on
previous constructions of counterexamples (for d ≥ 3r) based on a series of papers
by M. Özaydin, M. Gromov, P. Blagojević, F. Frick, G. Ziegler, and the second
and fourth present authors.\r\n\r\nThe counterexamples are obtained by proving
the following algebraic criterion in codimension 2: If r ≥ 3 and if K is a finite
2(r − 1)-complex, then there exists an almost r-embedding K → ℝ2r if and only
if there exists a general position PL map f: K → ℝ2r 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 a cohomological obstruction and extends an
analogous codimension 3 criterion by the second and fourth authors. As another
application, we classify ornaments f: S3 ⊔ S3 ⊔ S3 → ℝ5 up to ornament concordance.\r\n\r\nIt
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."
acknowledgement: Research supported by the Swiss National Science Foundation (Project
SNSF-PP00P2-138948), by the Austrian Science Fund (FWF Project P31312-N35), by the
Russian Foundation for Basic Research (Grants No. 15-01-06302 and 19-01-00169),
by a Simons-IUM Fellowship, and by the D. Zimin Dynasty Foundation Grant. We would
like to thank E. Alkin, A. Klyachko, V. Krushkal, S. Melikhov, M. Tancer, P. Teichner
and anonymous referees for helpful comments and discussions.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: Arkadiy B.
full_name: Skopenkov, Arkadiy B.
last_name: Skopenkov
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. Eliminating higher-multiplicity
intersections. III. Codimension 2. Israel Journal of Mathematics. 2021;245:501–534.
doi:10.1007/s11856-021-2216-z
apa: Avvakumov, S., Mabillard, I., Skopenkov, A. B., & Wagner, U. (2021). Eliminating
higher-multiplicity intersections. III. Codimension 2. Israel Journal of Mathematics.
Springer Nature. https://doi.org/10.1007/s11856-021-2216-z
chicago: Avvakumov, Sergey, Isaac Mabillard, Arkadiy B. Skopenkov, and Uli Wagner.
“Eliminating Higher-Multiplicity Intersections. III. Codimension 2.” Israel
Journal of Mathematics. Springer Nature, 2021. https://doi.org/10.1007/s11856-021-2216-z.
ieee: S. Avvakumov, I. Mabillard, A. B. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity
intersections. III. Codimension 2,” Israel Journal of Mathematics, vol.
245. Springer Nature, pp. 501–534, 2021.
ista: Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. 2021. Eliminating higher-multiplicity
intersections. III. Codimension 2. Israel Journal of Mathematics. 245, 501–534.
mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections. III.
Codimension 2.” Israel Journal of Mathematics, vol. 245, Springer Nature,
2021, pp. 501–534, doi:10.1007/s11856-021-2216-z.
short: S. Avvakumov, I. Mabillard, A.B. Skopenkov, U. Wagner, Israel Journal of
Mathematics 245 (2021) 501–534.
date_created: 2021-11-07T23:01:24Z
date_published: 2021-10-30T00:00:00Z
date_updated: 2023-08-14T11:43:55Z
day: '30'
department:
- _id: UlWa
doi: 10.1007/s11856-021-2216-z
external_id:
arxiv:
- '1511.03501'
isi:
- '000712942100013'
intvolume: ' 245'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1511.03501
month: '10'
oa: 1
oa_version: Preprint
page: '501–534 '
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P31312
name: Algorithms for Embeddings and Homotopy Theory
publication: Israel Journal of Mathematics
publication_identifier:
eissn:
- 1565-8511
issn:
- 0021-2172
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
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relation: earlier_version
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relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Eliminating higher-multiplicity intersections. III. Codimension 2
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 245
year: '2021'
...
---
_id: '11446'
abstract:
- lang: eng
text: Suppose that n is not a prime power and not twice a prime power. We prove
that for any Hausdorff compactum X with a free action of the symmetric group Sn,
there exists an Sn-equivariant map X→Rn whose image avoids the diagonal {(x,x,…,x)∈Rn∣x∈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 Sn-equivariant
maps from the boundary ∂Δn−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.
acknowledgement: S. Avvakumov has received funding from the European Research Council
under the European Union’s Seventh Framework Programme ERC Grant agreement ERC StG
716424–CASe. S. Kudrya was supported by the Austrian Academic Exchange Service (OeAD),
ICM-2019-13577.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Sergey
full_name: Kudrya, Sergey
id: ecf01965-d252-11ea-95a5-8ada5f6c6a67
last_name: Kudrya
citation:
ama: Avvakumov S, Kudrya S. Vanishing of all equivariant obstructions and the mapping
degree. Discrete & Computational Geometry. 2021;66(3):1202-1216. doi:10.1007/s00454-021-00299-z
apa: Avvakumov, S., & Kudrya, S. (2021). Vanishing of all equivariant obstructions
and the mapping degree. Discrete & Computational Geometry. Springer
Nature. https://doi.org/10.1007/s00454-021-00299-z
chicago: Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions
and the Mapping Degree.” Discrete & Computational Geometry. Springer
Nature, 2021. https://doi.org/10.1007/s00454-021-00299-z.
ieee: S. Avvakumov and S. Kudrya, “Vanishing of all equivariant obstructions and
the mapping degree,” Discrete & Computational Geometry, vol. 66, no.
3. Springer Nature, pp. 1202–1216, 2021.
ista: Avvakumov S, Kudrya S. 2021. Vanishing of all equivariant obstructions and
the mapping degree. Discrete & Computational Geometry. 66(3), 1202–1216.
mla: Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions
and the Mapping Degree.” Discrete & Computational Geometry, vol. 66,
no. 3, Springer Nature, 2021, pp. 1202–16, doi:10.1007/s00454-021-00299-z.
short: S. Avvakumov, S. Kudrya, Discrete & Computational Geometry 66 (2021)
1202–1216.
date_created: 2022-06-17T08:45:15Z
date_published: 2021-10-01T00:00:00Z
date_updated: 2023-02-23T13:26:41Z
day: '01'
doi: 10.1007/s00454-021-00299-z
extern: '1'
external_id:
arxiv:
- '1910.12628'
intvolume: ' 66'
issue: '3'
keyword:
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Theoretical Computer Science
language:
- iso: eng
month: '10'
oa_version: Preprint
page: 1202-1216
publication: Discrete & Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '8182'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Vanishing of all equivariant obstructions and the mapping degree
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 66
year: '2021'
...
---
_id: '9308'
acknowledgement: This research was carried out with the support of the Russian Foundation
for Basic Research(grant no. 19-01-00169)
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: A. B.
full_name: Skopenkov, A. B.
last_name: Skopenkov
citation:
ama: Avvakumov S, Wagner U, Mabillard I, Skopenkov AB. Eliminating higher-multiplicity
intersections, III. Codimension 2. Russian Mathematical Surveys. 2020;75(6):1156-1158.
doi:10.1070/RM9943
apa: Avvakumov, S., Wagner, U., Mabillard, I., & Skopenkov, A. B. (2020). Eliminating
higher-multiplicity intersections, III. Codimension 2. Russian Mathematical
Surveys. IOP Publishing. https://doi.org/10.1070/RM9943
chicago: Avvakumov, Sergey, Uli Wagner, Isaac Mabillard, and A. B. Skopenkov. “Eliminating
Higher-Multiplicity Intersections, III. Codimension 2.” Russian Mathematical
Surveys. IOP Publishing, 2020. https://doi.org/10.1070/RM9943.
ieee: S. Avvakumov, U. Wagner, I. Mabillard, and A. B. Skopenkov, “Eliminating higher-multiplicity
intersections, III. Codimension 2,” Russian Mathematical Surveys, vol.
75, no. 6. IOP Publishing, pp. 1156–1158, 2020.
ista: Avvakumov S, Wagner U, Mabillard I, Skopenkov AB. 2020. Eliminating higher-multiplicity
intersections, III. Codimension 2. Russian Mathematical Surveys. 75(6), 1156–1158.
mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections, III.
Codimension 2.” Russian Mathematical Surveys, vol. 75, no. 6, IOP Publishing,
2020, pp. 1156–58, doi:10.1070/RM9943.
short: S. Avvakumov, U. Wagner, I. Mabillard, A.B. Skopenkov, Russian Mathematical
Surveys 75 (2020) 1156–1158.
date_created: 2021-04-04T22:01:22Z
date_published: 2020-12-01T00:00:00Z
date_updated: 2023-08-14T11:43:54Z
day: '01'
department:
- _id: UlWa
doi: 10.1070/RM9943
external_id:
arxiv:
- '1511.03501'
isi:
- '000625983100001'
intvolume: ' 75'
isi: 1
issue: '6'
language:
- iso: eng
main_file_link:
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url: https://arxiv.org/abs/1511.03501
month: '12'
oa: 1
oa_version: Preprint
page: 1156-1158
publication: Russian Mathematical Surveys
publication_identifier:
issn:
- 0036-0279
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
related_material:
record:
- id: '8183'
relation: earlier_version
status: public
- id: '10220'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Eliminating higher-multiplicity intersections, III. Codimension 2
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 75
year: '2020'
...
---
_id: '7991'
abstract:
- lang: eng
text: '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.'
alternative_title:
- LIPIcs
article_number: 12:1 - 12:15
article_processing_charge: No
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Gabriel
full_name: Nivasch, Gabriel
last_name: Nivasch
citation:
ama: 'Avvakumov S, Nivasch G. Homotopic curve shortening and the affine curve-shortening
flow. In: 36th International Symposium on Computational Geometry. Vol 164.
Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:10.4230/LIPIcs.SoCG.2020.12'
apa: 'Avvakumov, S., & Nivasch, G. (2020). Homotopic curve shortening and the
affine curve-shortening flow. In 36th International Symposium on Computational
Geometry (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum
für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2020.12'
chicago: Avvakumov, Sergey, and Gabriel Nivasch. “Homotopic Curve Shortening and
the Affine Curve-Shortening Flow.” In 36th International Symposium on Computational
Geometry, Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.
https://doi.org/10.4230/LIPIcs.SoCG.2020.12.
ieee: S. Avvakumov and G. Nivasch, “Homotopic curve shortening and the affine curve-shortening
flow,” in 36th International Symposium on Computational Geometry, Zürich,
Switzerland, 2020, vol. 164.
ista: 'Avvakumov S, Nivasch G. 2020. Homotopic curve shortening and the affine curve-shortening
flow. 36th International Symposium on Computational Geometry. SoCG: Symposium
on Computational Geometry, LIPIcs, vol. 164, 12:1-12:15.'
mla: Avvakumov, Sergey, and Gabriel Nivasch. “Homotopic Curve Shortening and the
Affine Curve-Shortening Flow.” 36th International Symposium on Computational
Geometry, vol. 164, 12:1-12:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2020, doi:10.4230/LIPIcs.SoCG.2020.12.
short: S. Avvakumov, G. Nivasch, in:, 36th International Symposium on Computational
Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.
conference:
end_date: 2020-06-26
location: Zürich, Switzerland
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2020-06-22
date_created: 2020-06-22T09:14:19Z
date_published: 2020-06-01T00:00:00Z
date_updated: 2021-01-12T08:16:23Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.4230/LIPIcs.SoCG.2020.12
external_id:
arxiv:
- '1909.00263'
file:
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date_updated: 2020-07-14T12:48:06Z
file_id: '8007'
file_name: 2020_LIPIcsSoCG_Avvakumov.pdf
file_size: 575896
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month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P31312
name: Algorithms for Embeddings and Homotopy Theory
publication: 36th International Symposium on Computational Geometry
publication_identifier:
isbn:
- '9783959771436'
issn:
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publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: Homotopic curve shortening and the affine curve-shortening flow
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/3.0/legalcode
name: Creative Commons Attribution 3.0 Unported (CC BY 3.0)
short: CC BY (3.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 164
year: '2020'
...
---
_id: '8156'
abstract:
- lang: eng
text: '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.'
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
citation:
ama: Avvakumov S. Topological methods in geometry and discrete mathematics. 2020.
doi:10.15479/AT:ISTA:8156
apa: Avvakumov, S. (2020). Topological methods in geometry and discrete mathematics.
Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:8156
chicago: Avvakumov, Sergey. “Topological Methods in Geometry and Discrete Mathematics.”
Institute of Science and Technology Austria, 2020. https://doi.org/10.15479/AT:ISTA:8156.
ieee: S. Avvakumov, “Topological methods in geometry and discrete mathematics,”
Institute of Science and Technology Austria, 2020.
ista: Avvakumov S. 2020. Topological methods in geometry and discrete mathematics.
Institute of Science and Technology Austria.
mla: Avvakumov, Sergey. Topological Methods in Geometry and Discrete Mathematics.
Institute of Science and Technology Austria, 2020, doi:10.15479/AT:ISTA:8156.
short: S. Avvakumov, Topological Methods in Geometry and Discrete Mathematics, Institute
of Science and Technology Austria, 2020.
date_created: 2020-07-23T09:51:29Z
date_published: 2020-07-24T00:00:00Z
date_updated: 2023-12-18T10:51:01Z
day: '24'
ddc:
- '514'
degree_awarded: PhD
department:
- _id: UlWa
doi: 10.15479/AT:ISTA:8156
file:
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publication_identifier:
issn:
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publisher: Institute of Science and Technology Austria
related_material:
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relation: part_of_dissertation
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status: public
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- id: '75'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
title: Topological methods in geometry and discrete mathematics
type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2020'
...
---
_id: '8182'
abstract:
- lang: eng
text: "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 $X \\to\r\n{\\mathbb
R}^n$ whose image avoids the diagonal $\\{(x,x\\dots,x)\\in {\\mathbb R}^n|x\\in
{\\mathbb R}\\}$.\r\n 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\r\ntake 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\r\n$\\partial\\Delta^{n-1}$
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."
article_number: '1910.12628'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Sergey
full_name: Kudrya, Sergey
id: ecf01965-d252-11ea-95a5-8ada5f6c6a67
last_name: Kudrya
citation:
ama: Avvakumov S, Kudrya S. Vanishing of all equivariant obstructions and the mapping
degree. arXiv.
apa: Avvakumov, S., & Kudrya, S. (n.d.). Vanishing of all equivariant obstructions
and the mapping degree. arXiv. arXiv.
chicago: Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions
and the Mapping Degree.” ArXiv. arXiv, n.d.
ieee: S. Avvakumov and S. Kudrya, “Vanishing of all equivariant obstructions and
the mapping degree,” arXiv. arXiv.
ista: Avvakumov S, Kudrya S. Vanishing of all equivariant obstructions and the mapping
degree. arXiv, 1910.12628.
mla: Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions
and the Mapping Degree.” ArXiv, 1910.12628, arXiv.
short: S. Avvakumov, S. Kudrya, ArXiv (n.d.).
date_created: 2020-07-30T10:45:08Z
date_published: 2019-10-28T00:00:00Z
date_updated: 2023-09-07T13:12:17Z
day: '28'
department:
- _id: UlWa
external_id:
arxiv:
- '1910.12628'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1910.12628
month: '10'
oa: 1
oa_version: Preprint
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P31312
name: Algorithms for Embeddings and Homotopy Theory
publication: arXiv
publication_status: submitted
publisher: arXiv
related_material:
record:
- id: '11446'
relation: later_version
status: public
- id: '8156'
relation: dissertation_contains
status: public
status: public
title: Vanishing of all equivariant obstructions and the mapping degree
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2019'
...
---
_id: '8184'
abstract:
- lang: eng
text: "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\r\nd + 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. "
acknowledgement: We would like to thank F. Frick for helpful discussions
article_number: '1908.08731'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: R.
full_name: Karasev, R.
last_name: Karasev
- first_name: A.
full_name: Skopenkov, A.
last_name: Skopenkov
citation:
ama: Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological
Tverberg conjecture. arXiv.
apa: Avvakumov, S., Karasev, R., & Skopenkov, A. (n.d.). Stronger counterexamples
to the topological Tverberg conjecture. arXiv. arXiv.
chicago: Avvakumov, Sergey, R. Karasev, and A. Skopenkov. “Stronger Counterexamples
to the Topological Tverberg Conjecture.” ArXiv. arXiv, n.d.
ieee: S. Avvakumov, R. Karasev, and A. Skopenkov, “Stronger counterexamples to the
topological Tverberg conjecture,” arXiv. arXiv.
ista: Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological
Tverberg conjecture. arXiv, 1908.08731.
mla: Avvakumov, Sergey, et al. “Stronger Counterexamples to the Topological Tverberg
Conjecture.” ArXiv, 1908.08731, arXiv.
short: S. Avvakumov, R. Karasev, A. Skopenkov, ArXiv (n.d.).
date_created: 2020-07-30T10:45:34Z
date_published: 2019-08-23T00:00:00Z
date_updated: 2023-09-08T11:20:02Z
day: '23'
department:
- _id: UlWa
external_id:
arxiv:
- '1908.08731'
isi:
- '000986519600004'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1908.08731
month: '08'
oa: 1
oa_version: Preprint
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P31312
name: Algorithms for Embeddings and Homotopy Theory
publication: arXiv
publication_status: submitted
publisher: arXiv
related_material:
record:
- id: '8156'
relation: dissertation_contains
status: public
status: public
title: Stronger counterexamples to the topological Tverberg conjecture
type: preprint
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2019'
...
---
_id: '8185'
abstract:
- lang: eng
text: "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.\r\n 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\r\npartition 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."
article_number: '1907.11183'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Roman
full_name: Karasev, Roman
last_name: Karasev
citation:
ama: Avvakumov S, Karasev R. Envy-free division using mapping degree. arXiv.
doi:10.48550/arXiv.1907.11183
apa: Avvakumov, S., & Karasev, R. (n.d.). Envy-free division using mapping degree.
arXiv. https://doi.org/10.48550/arXiv.1907.11183
chicago: Avvakumov, Sergey, and Roman Karasev. “Envy-Free Division Using Mapping
Degree.” ArXiv, n.d. https://doi.org/10.48550/arXiv.1907.11183.
ieee: S. Avvakumov and R. Karasev, “Envy-free division using mapping degree,” arXiv.
.
ista: Avvakumov S, Karasev R. Envy-free division using mapping degree. arXiv, 1907.11183.
mla: Avvakumov, Sergey, and Roman Karasev. “Envy-Free Division Using Mapping Degree.”
ArXiv, 1907.11183, doi:10.48550/arXiv.1907.11183.
short: S. Avvakumov, R. Karasev, ArXiv (n.d.).
date_created: 2020-07-30T10:45:51Z
date_published: 2019-07-25T00:00:00Z
date_updated: 2023-09-07T13:12:17Z
day: '25'
department:
- _id: UlWa
doi: 10.48550/arXiv.1907.11183
external_id:
arxiv:
- '1907.11183'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1907.11183
month: '07'
oa: 1
oa_version: Preprint
project:
- _id: 26611F5C-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P31312
name: Algorithms for Embeddings and Homotopy Theory
publication: arXiv
publication_status: submitted
related_material:
link:
- relation: later_version
url: https://doi.org/10.1112/mtk.12059
record:
- id: '8156'
relation: dissertation_contains
status: public
status: public
title: Envy-free division using mapping degree
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2019'
...
---
_id: '6419'
abstract:
- lang: eng
text: 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.
article_number: e1008079
article_processing_charge: No
author:
- first_name: Victoria
full_name: Pokusaeva, Victoria
id: 3184041C-F248-11E8-B48F-1D18A9856A87
last_name: Pokusaeva
orcid: 0000-0001-7660-444X
- first_name: Dinara R.
full_name: Usmanova, Dinara R.
last_name: Usmanova
- first_name: Ekaterina V.
full_name: Putintseva, Ekaterina V.
last_name: Putintseva
- first_name: Lorena
full_name: Espinar, Lorena
last_name: Espinar
- first_name: Karen
full_name: Sarkisyan, Karen
id: 39A7BF80-F248-11E8-B48F-1D18A9856A87
last_name: Sarkisyan
orcid: 0000-0002-5375-6341
- first_name: Alexander S.
full_name: Mishin, Alexander S.
last_name: Mishin
- first_name: Natalya S.
full_name: Bogatyreva, Natalya S.
last_name: Bogatyreva
- first_name: Dmitry
full_name: Ivankov, Dmitry
id: 49FF1036-F248-11E8-B48F-1D18A9856A87
last_name: Ivankov
- first_name: Arseniy
full_name: Akopyan, Arseniy
id: 430D2C90-F248-11E8-B48F-1D18A9856A87
last_name: Akopyan
orcid: 0000-0002-2548-617X
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Inna S.
full_name: Povolotskaya, Inna S.
last_name: Povolotskaya
- first_name: Guillaume J.
full_name: Filion, Guillaume J.
last_name: Filion
- first_name: Lucas B.
full_name: Carey, Lucas B.
last_name: Carey
- first_name: Fyodor
full_name: Kondrashov, Fyodor
id: 44FDEF62-F248-11E8-B48F-1D18A9856A87
last_name: Kondrashov
orcid: 0000-0001-8243-4694
citation:
ama: Pokusaeva V, Usmanova DR, Putintseva EV, et al. An experimental assay of the
interactions of amino acids from orthologous sequences shaping a complex fitness
landscape. PLoS Genetics. 2019;15(4). doi:10.1371/journal.pgen.1008079
apa: Pokusaeva, V., Usmanova, D. R., Putintseva, E. V., Espinar, L., Sarkisyan,
K., Mishin, A. S., … Kondrashov, F. (2019). An experimental assay of the interactions
of amino acids from orthologous sequences shaping a complex fitness landscape.
PLoS Genetics. Public Library of Science. https://doi.org/10.1371/journal.pgen.1008079
chicago: Pokusaeva, Victoria, Dinara R. Usmanova, Ekaterina V. Putintseva, Lorena
Espinar, Karen Sarkisyan, Alexander S. Mishin, Natalya S. Bogatyreva, et al. “An
Experimental Assay of the Interactions of Amino Acids from Orthologous Sequences
Shaping a Complex Fitness Landscape.” PLoS Genetics. Public Library of
Science, 2019. https://doi.org/10.1371/journal.pgen.1008079.
ieee: V. Pokusaeva et al., “An experimental assay of the interactions of
amino acids from orthologous sequences shaping a complex fitness landscape,” PLoS
Genetics, vol. 15, no. 4. Public Library of Science, 2019.
ista: Pokusaeva V, Usmanova DR, Putintseva EV, Espinar L, Sarkisyan K, Mishin AS,
Bogatyreva NS, Ivankov D, Akopyan A, Avvakumov S, Povolotskaya IS, Filion GJ,
Carey LB, Kondrashov F. 2019. An experimental assay of the interactions of amino
acids from orthologous sequences shaping a complex fitness landscape. PLoS Genetics.
15(4), e1008079.
mla: Pokusaeva, Victoria, et al. “An Experimental Assay of the Interactions of Amino
Acids from Orthologous Sequences Shaping a Complex Fitness Landscape.” PLoS
Genetics, vol. 15, no. 4, e1008079, Public Library of Science, 2019, doi:10.1371/journal.pgen.1008079.
short: V. Pokusaeva, D.R. Usmanova, E.V. Putintseva, L. Espinar, K. Sarkisyan, A.S.
Mishin, N.S. Bogatyreva, D. Ivankov, A. Akopyan, S. Avvakumov, I.S. Povolotskaya,
G.J. Filion, L.B. Carey, F. Kondrashov, PLoS Genetics 15 (2019).
date_created: 2019-05-13T07:58:38Z
date_published: 2019-04-10T00:00:00Z
date_updated: 2023-08-25T10:30:37Z
day: '10'
ddc:
- '570'
department:
- _id: FyKo
doi: 10.1371/journal.pgen.1008079
ec_funded: 1
external_id:
isi:
- '000466866000029'
file:
- access_level: open_access
checksum: cf3889c8a8a16053dacf9c3776cbe217
content_type: application/pdf
creator: dernst
date_created: 2019-05-14T08:26:08Z
date_updated: 2020-07-14T12:47:30Z
file_id: '6445'
file_name: 2019_PLOSGenetics_Pokusaeva.pdf
file_size: 3726017
relation: main_file
file_date_updated: 2020-07-14T12:47:30Z
has_accepted_license: '1'
intvolume: ' 15'
isi: 1
issue: '4'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '665385'
name: International IST Doctoral Program
publication: PLoS Genetics
publication_identifier:
eissn:
- '15537404'
publication_status: published
publisher: Public Library of Science
quality_controlled: '1'
related_material:
record:
- id: '9789'
relation: research_data
status: public
- id: '9790'
relation: research_data
status: public
- id: '9797'
relation: research_data
status: public
scopus_import: '1'
status: public
title: An experimental assay of the interactions of amino acids from orthologous sequences
shaping a complex fitness landscape
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 15
year: '2019'
...
---
_id: '9789'
article_processing_charge: No
author:
- first_name: Victoria
full_name: Pokusaeva, Victoria
id: 3184041C-F248-11E8-B48F-1D18A9856A87
last_name: Pokusaeva
orcid: 0000-0001-7660-444X
- first_name: Dinara R.
full_name: Usmanova, Dinara R.
last_name: Usmanova
- first_name: Ekaterina V.
full_name: Putintseva, Ekaterina V.
last_name: Putintseva
- first_name: Lorena
full_name: Espinar, Lorena
last_name: Espinar
- first_name: Karen
full_name: Sarkisyan, Karen
id: 39A7BF80-F248-11E8-B48F-1D18A9856A87
last_name: Sarkisyan
orcid: 0000-0002-5375-6341
- first_name: Alexander S.
full_name: Mishin, Alexander S.
last_name: Mishin
- first_name: Natalya S.
full_name: Bogatyreva, Natalya S.
last_name: Bogatyreva
- first_name: Dmitry
full_name: Ivankov, Dmitry
id: 49FF1036-F248-11E8-B48F-1D18A9856A87
last_name: Ivankov
- first_name: Arseniy
full_name: Akopyan, Arseniy
id: 430D2C90-F248-11E8-B48F-1D18A9856A87
last_name: Akopyan
orcid: 0000-0002-2548-617X
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Inna S.
full_name: Povolotskaya, Inna S.
last_name: Povolotskaya
- first_name: Guillaume J.
full_name: Filion, Guillaume J.
last_name: Filion
- first_name: Lucas B.
full_name: Carey, Lucas B.
last_name: Carey
- first_name: Fyodor
full_name: Kondrashov, Fyodor
id: 44FDEF62-F248-11E8-B48F-1D18A9856A87
last_name: Kondrashov
orcid: 0000-0001-8243-4694
citation:
ama: Pokusaeva V, Usmanova DR, Putintseva EV, et al. Multiple alignment of His3
orthologues. 2019. doi:10.1371/journal.pgen.1008079.s010
apa: Pokusaeva, V., Usmanova, D. R., Putintseva, E. V., Espinar, L., Sarkisyan,
K., Mishin, A. S., … Kondrashov, F. (2019). Multiple alignment of His3 orthologues.
Public Library of Science. https://doi.org/10.1371/journal.pgen.1008079.s010
chicago: Pokusaeva, Victoria, Dinara R. Usmanova, Ekaterina V. Putintseva, Lorena
Espinar, Karen Sarkisyan, Alexander S. Mishin, Natalya S. Bogatyreva, et al. “Multiple
Alignment of His3 Orthologues.” Public Library of Science, 2019. https://doi.org/10.1371/journal.pgen.1008079.s010.
ieee: V. Pokusaeva et al., “Multiple alignment of His3 orthologues.” Public
Library of Science, 2019.
ista: Pokusaeva V, Usmanova DR, Putintseva EV, Espinar L, Sarkisyan K, Mishin AS,
Bogatyreva NS, Ivankov D, Akopyan A, Avvakumov S, Povolotskaya IS, Filion GJ,
Carey LB, Kondrashov F. 2019. Multiple alignment of His3 orthologues, Public Library
of Science, 10.1371/journal.pgen.1008079.s010.
mla: Pokusaeva, Victoria, et al. Multiple Alignment of His3 Orthologues.
Public Library of Science, 2019, doi:10.1371/journal.pgen.1008079.s010.
short: V. Pokusaeva, D.R. Usmanova, E.V. Putintseva, L. Espinar, K. Sarkisyan, A.S.
Mishin, N.S. Bogatyreva, D. Ivankov, A. Akopyan, S. Avvakumov, I.S. Povolotskaya,
G.J. Filion, L.B. Carey, F. Kondrashov, (2019).
date_created: 2021-08-06T08:38:50Z
date_published: 2019-04-10T00:00:00Z
date_updated: 2023-08-25T10:30:36Z
day: '10'
department:
- _id: FyKo
doi: 10.1371/journal.pgen.1008079.s010
month: '04'
oa_version: Published Version
publisher: Public Library of Science
related_material:
record:
- id: '6419'
relation: used_in_publication
status: public
status: public
title: Multiple alignment of His3 orthologues
type: research_data_reference
user_id: 6785fbc1-c503-11eb-8a32-93094b40e1cf
year: '2019'
...
---
_id: '9790'
article_processing_charge: No
author:
- first_name: Victoria
full_name: Pokusaeva, Victoria
id: 3184041C-F248-11E8-B48F-1D18A9856A87
last_name: Pokusaeva
orcid: 0000-0001-7660-444X
- first_name: Dinara R.
full_name: Usmanova, Dinara R.
last_name: Usmanova
- first_name: Ekaterina V.
full_name: Putintseva, Ekaterina V.
last_name: Putintseva
- first_name: Lorena
full_name: Espinar, Lorena
last_name: Espinar
- first_name: Karen
full_name: Sarkisyan, Karen
id: 39A7BF80-F248-11E8-B48F-1D18A9856A87
last_name: Sarkisyan
orcid: 0000-0002-5375-6341
- first_name: Alexander S.
full_name: Mishin, Alexander S.
last_name: Mishin
- first_name: Natalya S.
full_name: Bogatyreva, Natalya S.
last_name: Bogatyreva
- first_name: Dmitry
full_name: Ivankov, Dmitry
id: 49FF1036-F248-11E8-B48F-1D18A9856A87
last_name: Ivankov
- first_name: Arseniy
full_name: Akopyan, Arseniy
id: 430D2C90-F248-11E8-B48F-1D18A9856A87
last_name: Akopyan
orcid: 0000-0002-2548-617X
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Inna S.
full_name: Povolotskaya, Inna S.
last_name: Povolotskaya
- first_name: Guillaume J.
full_name: Filion, Guillaume J.
last_name: Filion
- first_name: Lucas B.
full_name: Carey, Lucas B.
last_name: Carey
- first_name: Fyodor
full_name: Kondrashov, Fyodor
id: 44FDEF62-F248-11E8-B48F-1D18A9856A87
last_name: Kondrashov
orcid: 0000-0001-8243-4694
citation:
ama: Pokusaeva V, Usmanova DR, Putintseva EV, et al. A statistical summary of segment
libraries and sequencing results. 2019. doi:10.1371/journal.pgen.1008079.s011
apa: Pokusaeva, V., Usmanova, D. R., Putintseva, E. V., Espinar, L., Sarkisyan,
K., Mishin, A. S., … Kondrashov, F. (2019). A statistical summary of segment libraries
and sequencing results. Public Library of Science. https://doi.org/10.1371/journal.pgen.1008079.s011
chicago: Pokusaeva, Victoria, Dinara R. Usmanova, Ekaterina V. Putintseva, Lorena
Espinar, Karen Sarkisyan, Alexander S. Mishin, Natalya S. Bogatyreva, et al. “A
Statistical Summary of Segment Libraries and Sequencing Results.” Public Library
of Science, 2019. https://doi.org/10.1371/journal.pgen.1008079.s011.
ieee: V. Pokusaeva et al., “A statistical summary of segment libraries and
sequencing results.” Public Library of Science, 2019.
ista: Pokusaeva V, Usmanova DR, Putintseva EV, Espinar L, Sarkisyan K, Mishin AS,
Bogatyreva NS, Ivankov D, Akopyan A, Avvakumov S, Povolotskaya IS, Filion GJ,
Carey LB, Kondrashov F. 2019. A statistical summary of segment libraries and sequencing
results, Public Library of Science, 10.1371/journal.pgen.1008079.s011.
mla: Pokusaeva, Victoria, et al. A Statistical Summary of Segment Libraries and
Sequencing Results. Public Library of Science, 2019, doi:10.1371/journal.pgen.1008079.s011.
short: V. Pokusaeva, D.R. Usmanova, E.V. Putintseva, L. Espinar, K. Sarkisyan, A.S.
Mishin, N.S. Bogatyreva, D. Ivankov, A. Akopyan, S. Avvakumov, I.S. Povolotskaya,
G.J. Filion, L.B. Carey, F. Kondrashov, (2019).
date_created: 2021-08-06T08:50:15Z
date_published: 2019-04-10T00:00:00Z
date_updated: 2023-08-25T10:30:36Z
day: '10'
department:
- _id: FyKo
doi: 10.1371/journal.pgen.1008079.s011
month: '04'
oa_version: Published Version
publisher: Public Library of Science
related_material:
record:
- id: '6419'
relation: used_in_publication
status: public
status: public
title: A statistical summary of segment libraries and sequencing results
type: research_data_reference
user_id: 6785fbc1-c503-11eb-8a32-93094b40e1cf
year: '2019'
...
---
_id: '75'
abstract:
- lang: eng
text: 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.
article_number: '1804.03057'
article_processing_charge: No
arxiv: 1
author:
- first_name: Arseniy
full_name: Akopyan, Arseniy
id: 430D2C90-F248-11E8-B48F-1D18A9856A87
last_name: Akopyan
orcid: 0000-0002-2548-617X
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Roman
full_name: Karasev, Roman
last_name: Karasev
citation:
ama: Akopyan A, Avvakumov S, Karasev R. Convex fair partitions into arbitrary number
of pieces. 2018. doi:10.48550/arXiv.1804.03057
apa: Akopyan, A., Avvakumov, S., & Karasev, R. (2018). Convex fair partitions
into arbitrary number of pieces. arXiv. https://doi.org/10.48550/arXiv.1804.03057
chicago: Akopyan, Arseniy, Sergey Avvakumov, and Roman Karasev. “Convex Fair Partitions
into Arbitrary Number of Pieces.” arXiv, 2018. https://doi.org/10.48550/arXiv.1804.03057.
ieee: A. Akopyan, S. Avvakumov, and R. Karasev, “Convex fair partitions into arbitrary
number of pieces.” arXiv, 2018.
ista: Akopyan A, Avvakumov S, Karasev R. 2018. Convex fair partitions into arbitrary
number of pieces. 1804.03057.
mla: Akopyan, Arseniy, et al. Convex Fair Partitions into Arbitrary Number of
Pieces. 1804.03057, arXiv, 2018, doi:10.48550/arXiv.1804.03057.
short: A. Akopyan, S. Avvakumov, R. Karasev, (2018).
date_created: 2018-12-11T11:44:30Z
date_published: 2018-09-13T00:00:00Z
date_updated: 2023-12-18T10:51:02Z
day: '13'
department:
- _id: HeEd
- _id: JaMa
doi: 10.48550/arXiv.1804.03057
ec_funded: 1
external_id:
arxiv:
- '1804.03057'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1804.03057
month: '09'
oa: 1
oa_version: Preprint
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
publication_status: published
publisher: arXiv
related_material:
record:
- id: '8156'
relation: dissertation_contains
status: public
status: public
title: Convex fair partitions into arbitrary number of pieces
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2018'
...
---
_id: '6355'
abstract:
- lang: eng
text: 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.
article_number: e7
article_processing_charge: No
arxiv: 1
author:
- first_name: Arseniy
full_name: Akopyan, Arseniy
id: 430D2C90-F248-11E8-B48F-1D18A9856A87
last_name: Akopyan
orcid: 0000-0002-2548-617X
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
citation:
ama: Akopyan A, Avvakumov S. Any cyclic quadrilateral can be inscribed in any closed
convex smooth curve. Forum of Mathematics, Sigma. 2018;6. doi:10.1017/fms.2018.7
apa: Akopyan, A., & Avvakumov, S. (2018). Any cyclic quadrilateral can be inscribed
in any closed convex smooth curve. Forum of Mathematics, Sigma. Cambridge
University Press. https://doi.org/10.1017/fms.2018.7
chicago: Akopyan, Arseniy, and Sergey Avvakumov. “Any Cyclic Quadrilateral Can Be
Inscribed in Any Closed Convex Smooth Curve.” Forum of Mathematics, Sigma.
Cambridge University Press, 2018. https://doi.org/10.1017/fms.2018.7.
ieee: A. Akopyan and S. Avvakumov, “Any cyclic quadrilateral can be inscribed in
any closed convex smooth curve,” Forum of Mathematics, Sigma, vol. 6. Cambridge
University Press, 2018.
ista: Akopyan A, Avvakumov S. 2018. Any cyclic quadrilateral can be inscribed in
any closed convex smooth curve. Forum of Mathematics, Sigma. 6, e7.
mla: Akopyan, Arseniy, and Sergey Avvakumov. “Any Cyclic Quadrilateral Can Be Inscribed
in Any Closed Convex Smooth Curve.” Forum of Mathematics, Sigma, vol. 6,
e7, Cambridge University Press, 2018, doi:10.1017/fms.2018.7.
short: A. Akopyan, S. Avvakumov, Forum of Mathematics, Sigma 6 (2018).
date_created: 2019-04-30T06:09:57Z
date_published: 2018-05-31T00:00:00Z
date_updated: 2023-09-19T14:50:12Z
day: '31'
ddc:
- '510'
department:
- _id: UlWa
- _id: HeEd
- _id: JaMa
doi: 10.1017/fms.2018.7
ec_funded: 1
external_id:
arxiv:
- '1712.10205'
isi:
- '000433915500001'
file:
- access_level: open_access
checksum: 5a71b24ba712a3eb2e46165a38fbc30a
content_type: application/pdf
creator: dernst
date_created: 2019-04-30T06:14:58Z
date_updated: 2020-07-14T12:47:28Z
file_id: '6356'
file_name: 2018_ForumMahtematics_Akopyan.pdf
file_size: 249246
relation: main_file
file_date_updated: 2020-07-14T12:47:28Z
has_accepted_license: '1'
intvolume: ' 6'
isi: 1
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
publication: Forum of Mathematics, Sigma
publication_identifier:
issn:
- 2050-5094
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
related_material:
record:
- id: '8156'
relation: dissertation_contains
status: public
status: public
title: Any cyclic quadrilateral can be inscribed in any closed convex smooth curve
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 6
year: '2018'
...
---
_id: '1522'
abstract:
- lang: eng
text: '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′.'
acknowledgement: "I thank A. Skopenkov for telling me about the problem and for his
useful remarks. I also thank A. Sossinsky,\r\nA. Zhubr, M. Skopenkov, P. Akhmetiev,
and an anonymous referee for their feedback. Author was partially\r\nsupported
by Dobrushin fellowship, 2013, and by RFBR grant 15-01-06302."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Serhii
full_name: Avvakumov, Serhii
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
citation:
ama: Avvakumov S. The classification of certain linked 3-manifolds in 6-space. Moscow
Mathematical Journal. 2016;16(1):1-25. doi:10.17323/1609-4514-2016-16-1-1-25
apa: Avvakumov, S. (2016). The classification of certain linked 3-manifolds in 6-space.
Moscow Mathematical Journal. Independent University of Moscow. https://doi.org/10.17323/1609-4514-2016-16-1-1-25
chicago: Avvakumov, Sergey. “The Classification of Certain Linked 3-Manifolds in
6-Space.” Moscow Mathematical Journal. Independent University of Moscow,
2016. https://doi.org/10.17323/1609-4514-2016-16-1-1-25.
ieee: S. Avvakumov, “The classification of certain linked 3-manifolds in 6-space,”
Moscow Mathematical Journal, vol. 16, no. 1. Independent University of
Moscow, pp. 1–25, 2016.
ista: Avvakumov S. 2016. The classification of certain linked 3-manifolds in 6-space.
Moscow Mathematical Journal. 16(1), 1–25.
mla: Avvakumov, Sergey. “The Classification of Certain Linked 3-Manifolds in 6-Space.”
Moscow Mathematical Journal, vol. 16, no. 1, Independent University of
Moscow, 2016, pp. 1–25, doi:10.17323/1609-4514-2016-16-1-1-25.
short: S. Avvakumov, Moscow Mathematical Journal 16 (2016) 1–25.
date_created: 2018-12-11T11:52:30Z
date_published: 2016-01-01T00:00:00Z
date_updated: 2022-02-25T10:15:57Z
day: '01'
department:
- _id: UlWa
doi: 10.17323/1609-4514-2016-16-1-1-25
external_id:
arxiv:
- '1408.3918'
intvolume: ' 16'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1408.3918
month: '01'
oa: 1
oa_version: Preprint
page: 1 - 25
publication: Moscow Mathematical Journal
publication_identifier:
eissn:
- 1609-4514
publication_status: published
publisher: Independent University of Moscow
publist_id: '5652'
quality_controlled: '1'
scopus_import: '1'
status: public
title: The classification of certain linked 3-manifolds in 6-space
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 16
year: '2016'
...
---
_id: '8183'
abstract:
- lang: eng
text: "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 $r$\r\npairwise 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\r\nthe $(d+1)(r-1)$-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\r\nconcordance. 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."
acknowledgement: We would like to thank A. Klyachko, V. Krushkal, S. Melikhov, M.
Tancer, P. Teichner and anonymous referees for helpful discussions.
article_number: '1511.03501'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: A.
full_name: Skopenkov, A.
last_name: Skopenkov
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity
intersections, III. Codimension 2. arXiv.
apa: Avvakumov, S., Mabillard, I., Skopenkov, A., & Wagner, U. (n.d.). Eliminating
higher-multiplicity intersections, III. Codimension 2. arXiv.
chicago: Avvakumov, Sergey, Isaac Mabillard, A. Skopenkov, and Uli Wagner. “Eliminating
Higher-Multiplicity Intersections, III. Codimension 2.” ArXiv, n.d.
ieee: S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity
intersections, III. Codimension 2,” arXiv. .
ista: Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity
intersections, III. Codimension 2. arXiv, 1511.03501.
mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections, III.
Codimension 2.” ArXiv, 1511.03501.
short: S. Avvakumov, I. Mabillard, A. Skopenkov, U. Wagner, ArXiv (n.d.).
date_created: 2020-07-30T10:45:19Z
date_published: 2015-11-15T00:00:00Z
date_updated: 2023-09-07T13:12:17Z
day: '15'
department:
- _id: UlWa
external_id:
arxiv:
- '1511.03501'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1511.03501
month: '11'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: submitted
related_material:
record:
- id: '9308'
relation: later_version
status: public
- id: '10220'
relation: later_version
status: public
- id: '8156'
relation: dissertation_contains
status: public
status: public
title: Eliminating higher-multiplicity intersections, III. Codimension 2
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2015'
...