---
_id: '8032'
abstract:
- lang: eng
text: "Algorithms in computational 3-manifold topology typically take a triangulation
as an input and return topological information about the underlying 3-manifold.
However, extracting the desired information from a triangulation (e.g., evaluating
an invariant) is often computationally very expensive. In recent years this complexity
barrier has been successfully tackled in some cases by importing ideas from the
theory of parameterized algorithms into the realm of 3-manifolds. Various computationally
hard problems were shown to be efficiently solvable for input triangulations that
are sufficiently “tree-like.”\r\nIn this thesis we focus on the key combinatorial
parameter in the above context: we consider the treewidth of a compact, orientable
3-manifold, i.e., the smallest treewidth of the dual graph of any triangulation
thereof. By building on the work of Scharlemann–Thompson and Scharlemann–Schultens–Saito
on generalized Heegaard splittings, and on the work of Jaco–Rubinstein on layered
triangulations, we establish quantitative relations between the treewidth and
classical topological invariants of a 3-manifold. In particular, among other results,
we show that the treewidth of a closed, orientable, irreducible, non-Haken 3-manifold
is always within a constant factor of its Heegaard genus."
acknowledged_ssus:
- _id: E-Lib
- _id: CampIT
alternative_title:
- IST Austria Thesis
article_processing_charge: No
author:
- first_name: Kristóf
full_name: Huszár, Kristóf
id: 33C26278-F248-11E8-B48F-1D18A9856A87
last_name: Huszár
orcid: 0000-0002-5445-5057
citation:
ama: Huszár K. Combinatorial width parameters for 3-dimensional manifolds. 2020.
doi:10.15479/AT:ISTA:8032
apa: Huszár, K. (2020). *Combinatorial width parameters for 3-dimensional manifolds*.
IST Austria. https://doi.org/10.15479/AT:ISTA:8032
chicago: Huszár, Kristóf. “Combinatorial Width Parameters for 3-Dimensional Manifolds.”
IST Austria, 2020. https://doi.org/10.15479/AT:ISTA:8032.
ieee: K. Huszár, “Combinatorial width parameters for 3-dimensional manifolds,” IST
Austria, 2020.
ista: Huszár K. 2020. Combinatorial width parameters for 3-dimensional manifolds.
IST Austria.
mla: Huszár, Kristóf. *Combinatorial Width Parameters for 3-Dimensional Manifolds*.
IST Austria, 2020, doi:10.15479/AT:ISTA:8032.
short: K. Huszár, Combinatorial Width Parameters for 3-Dimensional Manifolds, IST
Austria, 2020.
date_created: 2020-06-26T10:00:36Z
date_published: 2020-06-26T00:00:00Z
date_updated: 2021-01-12T08:16:39Z
day: '26'
ddc:
- '514'
department:
- _id: UlWa
doi: 10.15479/AT:ISTA:8032
file:
- access_level: open_access
checksum: bd8be6e4f1addc863dfcc0fad29ee9c3
content_type: application/pdf
creator: khuszar
date_created: 2020-06-26T10:03:58Z
date_updated: 2020-07-14T12:48:08Z
file_id: '8034'
file_name: Kristof_Huszar-Thesis.pdf
file_size: 2637562
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content_type: application/x-zip-compressed
creator: khuszar
date_created: 2020-06-26T10:10:06Z
date_updated: 2020-07-14T12:48:08Z
file_id: '8035'
file_name: Kristof_Huszar-Thesis-source.zip
file_size: 7163491
relation: source_file
file_date_updated: 2020-07-14T12:48:08Z
has_accepted_license: '1'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: xviii+120
publication_identifier:
isbn:
- 978-3-99078-006-0
issn:
- 2663-337X
publication_status: published
publisher: IST Austria
related_material:
record:
- id: '7093'
relation: dissertation_contains
status: public
- id: '6556'
relation: dissertation_contains
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
- first_name: Jonathan
full_name: Spreer, Jonathan
last_name: Spreer
title: Combinatorial width parameters for 3-dimensional manifolds
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: dissertation
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
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:
- IST Austria 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*.
IST Austria. https://doi.org/10.15479/AT:ISTA:8156
chicago: Avvakumov, Sergey. “Topological Methods in Geometry and Discrete Mathematics.”
IST Austria, 2020. https://doi.org/10.15479/AT:ISTA:8156.
ieee: S. Avvakumov, “Topological methods in geometry and discrete mathematics,”
IST Austria, 2020.
ista: Avvakumov S. 2020. Topological methods in geometry and discrete mathematics.
IST Austria.
mla: Avvakumov, Sergey. *Topological Methods in Geometry and Discrete Mathematics*.
IST Austria, 2020, doi:10.15479/AT:ISTA:8156.
short: S. Avvakumov, Topological Methods in Geometry and Discrete Mathematics, IST
Austria, 2020.
date_created: 2020-07-23T09:51:29Z
date_published: 2020-07-24T00:00:00Z
date_updated: 2021-01-12T08:17:21Z
day: '24'
ddc:
- '514'
department:
- _id: UlWa
doi: 10.15479/AT:ISTA:8156
file:
- access_level: closed
content_type: application/zip
creator: savvakum
date_created: 2020-07-27T12:44:51Z
date_updated: 2020-07-27T12:44:51Z
file_id: '8178'
file_name: source.zip
file_size: 1061740
relation: source_file
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content_type: application/pdf
creator: savvakum
date_created: 2020-07-27T12:46:53Z
date_updated: 2020-07-27T12:46:53Z
file_id: '8179'
file_name: thesis_pdfa.pdf
file_size: 1336501
relation: main_file
success: 1
file_date_updated: 2020-07-27T12:46:53Z
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language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: '119'
publication_identifier:
unknown:
- 2663-337X
publication_status: published
publisher: IST Austria
related_material:
record:
- id: '6355'
relation: part_of_dissertation
status: public
- id: '75'
relation: part_of_dissertation
status: public
- id: '8182'
relation: part_of_dissertation
status: public
- id: '8183'
relation: part_of_dissertation
status: public
- id: '8184'
relation: part_of_dissertation
status: public
- id: '8185'
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2020'
...
---
_id: '7944'
abstract:
- lang: eng
text: "This thesis considers two examples of reconfiguration problems: flipping
edges in edge-labelled triangulations of planar point sets and swapping labelled
tokens placed on vertices of a graph. In both cases the studied structures – all
the triangulations of a given point set or all token placements on a given graph
– can be thought of as vertices of the so-called reconfiguration graph, in which
two vertices are adjacent if the corresponding structures differ by a single elementary
operation – by a flip of a diagonal in a triangulation or by a swap of tokens
on adjacent vertices, respectively. We study the reconfiguration of one instance
of a structure into another via (shortest) paths in the reconfiguration graph.\r\n\r\nFor
triangulations of point sets in which each edge has a unique label and a flip
transfers the label from the removed edge to the new edge, we prove a polynomial-time
testable condition, called the Orbit Theorem, that characterizes when two triangulations
of the same point set lie in the same connected component of the reconfiguration
graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot.
We additionally provide a polynomial time algorithm that computes a reconfiguring
flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties
of a certain high-dimensional cell complex that has the usual reconfiguration
graph as its 1-skeleton.\r\n\r\nIn the context of token swapping on a tree graph,
we make partial progress on the problem of finding shortest reconfiguration sequences.
We disprove the so-called Happy Leaf Conjecture and demonstrate the importance
of swapping tokens that are already placed at the correct vertices. We also prove
that a generalization of the problem to weighted coloured token swapping is NP-hard
on trees but solvable in polynomial time on paths and stars."
alternative_title:
- IST Austria Thesis
article_processing_charge: No
author:
- first_name: Zuzana
full_name: Masárová, Zuzana
id: 45CFE238-F248-11E8-B48F-1D18A9856A87
last_name: Masárová
orcid: 0000-0002-6660-1322
citation:
ama: Masárová Z. Reconfiguration problems. 2020. doi:10.15479/AT:ISTA:7944
apa: Masárová, Z. (2020). *Reconfiguration problems*. IST Austria. https://doi.org/10.15479/AT:ISTA:7944
chicago: Masárová, Zuzana. “Reconfiguration Problems.” IST Austria, 2020. https://doi.org/10.15479/AT:ISTA:7944.
ieee: Z. Masárová, “Reconfiguration problems,” IST Austria, 2020.
ista: Masárová Z. 2020. Reconfiguration problems. IST Austria.
mla: Masárová, Zuzana. *Reconfiguration Problems*. IST Austria, 2020, doi:10.15479/AT:ISTA:7944.
short: Z. Masárová, Reconfiguration Problems, IST Austria, 2020.
date_created: 2020-06-08T00:49:46Z
date_published: 2020-06-09T00:00:00Z
date_updated: 2021-01-12T08:16:11Z
day: '09'
ddc:
- '516'
- '514'
department:
- _id: HeEd
- _id: UlWa
doi: 10.15479/AT:ISTA:7944
file:
- access_level: open_access
checksum: df688bc5a82b50baee0b99d25fc7b7f0
content_type: application/pdf
creator: zmasarov
date_created: 2020-06-08T00:34:00Z
date_updated: 2020-07-14T12:48:05Z
file_id: '7945'
file_name: THESIS_Zuzka_Masarova.pdf
file_size: 13661779
relation: main_file
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checksum: 45341a35b8f5529c74010b7af43ac188
content_type: application/zip
creator: zmasarov
date_created: 2020-06-08T00:35:30Z
date_updated: 2020-07-14T12:48:05Z
file_id: '7946'
file_name: THESIS_Zuzka_Masarova_SOURCE_FILES.zip
file_size: 32184006
relation: source_file
file_date_updated: 2020-07-14T12:48:05Z
has_accepted_license: '1'
keyword:
- reconfiguration
- reconfiguration graph
- triangulations
- flip
- constrained triangulations
- shellability
- piecewise-linear balls
- token swapping
- trees
- coloured weighted token swapping
language:
- iso: eng
license: https://creativecommons.org/licenses/by-sa/4.0/
month: '06'
oa: 1
oa_version: Published Version
page: '160'
publication_identifier:
eissn:
- 2663-337X
isbn:
- 978-3-99078-005-3
publication_status: published
publisher: IST Austria
related_material:
record:
- id: '5986'
relation: part_of_dissertation
status: public
- id: '7950'
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
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
title: Reconfiguration problems
tmp:
image: /images/cc_by_sa.png
legal_code_url: https://creativecommons.org/licenses/by-sa/4.0/legalcode
name: Creative Commons Attribution-ShareAlike 4.0 International Public License (CC
BY-SA 4.0)
short: CC BY-SA (4.0)
type: dissertation
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2020'
...
---
_id: '6681'
abstract:
- lang: eng
text: "The first part of the thesis considers the computational aspects of the homotopy
groups πd(X) of a topological space X. It is well known that there is no algorithm
to decide whether the fundamental group π1(X) of a given finite simplicial complex
X is trivial. On the other hand, there are several algorithms that, given a finite
simplicial complex X that is simply connected (i.e., with π1(X) trivial), compute
the higher homotopy group πd(X) for any given d ≥ 2.\r\nHowever, these algorithms
come with a caveat: They compute the isomorphism type of πd(X), d ≥ 2 as an abstract
finitely generated abelian group given by generators and relations, but they work
with very implicit representations of the elements of πd(X). We present an algorithm
that, given a simply connected space X, computes πd(X) and represents its elements
as simplicial maps from suitable triangulations of the d-sphere Sd to X. For fixed
d, the algorithm runs in time exponential in size(X), the number of simplices
of X. Moreover, we prove that this is optimal: For every fixed d ≥ 2,\r\nwe construct
a family of simply connected spaces X such that for any simplicial map representing
a generator of πd(X), the size of the triangulation of S d on which the map is
defined, is exponential in size(X).\r\nIn the second part of the thesis, we prove
that the following question is algorithmically undecidable for d < ⌊3(k+1)/2⌋,
k ≥ 5 and (k, d) ̸= (5, 7), which covers essentially everything outside the meta-stable
range: Given a finite simplicial complex K of dimension k, decide whether there
exists a piecewise-linear (i.e., linear on an arbitrarily fine subdivision of
K) embedding f : K ↪→ Rd of K into a d-dimensional Euclidean space."
alternative_title:
- IST Austria Thesis
author:
- first_name: Stephan Y
full_name: Zhechev, Stephan Y
id: 3AA52972-F248-11E8-B48F-1D18A9856A87
last_name: Zhechev
citation:
ama: Zhechev SY. Algorithmic aspects of homotopy theory and embeddability. 2019.
doi:10.15479/AT:ISTA:6681
apa: Zhechev, S. Y. (2019). *Algorithmic aspects of homotopy theory and embeddability*.
IST Austria. https://doi.org/10.15479/AT:ISTA:6681
chicago: Zhechev, Stephan Y. “Algorithmic Aspects of Homotopy Theory and Embeddability.”
IST Austria, 2019. https://doi.org/10.15479/AT:ISTA:6681.
ieee: S. Y. Zhechev, “Algorithmic aspects of homotopy theory and embeddability,”
IST Austria, 2019.
ista: Zhechev SY. 2019. Algorithmic aspects of homotopy theory and embeddability.
IST Austria.
mla: Zhechev, Stephan Y. *Algorithmic Aspects of Homotopy Theory and Embeddability*.
IST Austria, 2019, doi:10.15479/AT:ISTA:6681.
short: S.Y. Zhechev, Algorithmic Aspects of Homotopy Theory and Embeddability, IST
Austria, 2019.
date_created: 2019-07-26T11:14:34Z
date_published: 2019-08-08T00:00:00Z
date_updated: 2021-01-12T08:08:57Z
day: '08'
ddc:
- '514'
department:
- _id: UlWa
doi: 10.15479/AT:ISTA:6681
file:
- access_level: open_access
checksum: 3231e7cbfca3b5687366f84f0a57a0c0
content_type: application/pdf
creator: szhechev
date_created: 2019-08-07T13:02:50Z
date_updated: 2020-07-14T12:47:37Z
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file_name: Stephan_Zhechev_thesis.pdf
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creator: szhechev
date_created: 2019-08-07T13:03:22Z
date_updated: 2020-07-14T12:47:37Z
file_id: '6772'
file_name: Stephan_Zhechev_thesis.tex
file_size: 303988
relation: source_file
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checksum: 86b374d264ca2dd53e712728e253ee75
content_type: application/zip
creator: szhechev
date_created: 2019-08-07T13:03:34Z
date_updated: 2020-07-14T12:47:37Z
file_id: '6773'
file_name: supplementary_material.zip
file_size: 1087004
relation: supplementary_material
file_date_updated: 2020-07-14T12:47:37Z
has_accepted_license: '1'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
page: '104'
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: IST Austria
related_material:
record:
- id: '6774'
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: Algorithmic aspects of homotopy theory and embeddability
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: dissertation
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2019'
...
---
_id: '1123'
abstract:
- lang: eng
text: "Motivated by topological Tverberg-type problems in topological combinatorics
and by classical\r\nresults about embeddings (maps without double points), we
study the question whether a finite\r\nsimplicial complex K can be mapped into
Rd without triple, quadruple, or, more generally, r-fold points (image points
with at least r distinct preimages), for a given multiplicity r ≤ 2. In particular,
we are interested in maps f : K → Rd that have no global r -fold intersection
points, i.e., no r -fold points with preimages in r pairwise disjoint simplices
of K , and we seek necessary and sufficient conditions for the existence of such
maps.\r\n\r\nWe present higher-multiplicity analogues of several classical results
for embeddings, in particular of the completeness of the Van Kampen obstruction
\ for embeddability of k -dimensional\r\ncomplexes into R2k , k ≥ 3. Speciffically,
we show that under suitable restrictions on the dimensions(viz., if dimK = (r
≥ 1)k and d = rk \\ for some k ≥ 3), a well-known deleted product criterion
(DPC ) is not only necessary but also sufficient for the existence of maps without
global r -fold points. Our main technical tool is a higher-multiplicity version
of the classical Whitney trick , by which pairs of isolated r -fold points of
opposite sign can be eliminated by local modiffications of the map, assuming
codimension d – dimK ≥ 3.\r\n\r\nAn important guiding idea for our work was that
suffciency of the DPC, together with an old\r\nresult of Özaydin's on the existence
of equivariant maps, might yield an approach to disproving the remaining open
cases of the the long-standing topological Tverberg conjecture , i.e., to construct
maps from the N -simplex σN to Rd without r-Tverberg points when r not a prime
power and\r\nN = (d + 1)(r – 1). Unfortunately, our proof of the sufficiency
of the DPC requires codimension d – dimK ≥ 3, which is not satisfied for K =
σN .\r\n\r\nIn 2015, Frick [16] found a very elegant way to overcome this \\codimension
3 obstacle" and\r\nto construct the first counterexamples to the topological
Tverberg conjecture for all parameters(d; r ) with d ≥ 3r + 1 and r not a prime
power, by a reduction1 to a suitable lower-dimensional skeleton, for which the
codimension 3 restriction is satisfied and maps without r -Tverberg points exist
by Özaydin's result and sufficiency of the DPC.\r\n\r\nIn this thesis, we present
a different construction (which does not use the constraint method) that yields
counterexamples for d ≥ 3r , r not a prime power. "
acknowledgement: "Foremost, I would like to thank Uli Wagner for introducing me to
the exciting interface between\r\ntopology and combinatorics, and for our subsequent
years of fruitful collaboration.\r\nIn our creative endeavors to eliminate intersection
points, we had the chance to be joined later\r\nby Sergey Avvakumov and Arkadiy
Skopenkov, which led us to new surprises in dimension 12.\r\nMy stay at EPFL and
IST Austria was made very agreeable thanks to all these wonderful\r\npeople: Cyril
Becker, Marek Filakovsky, Peter Franek, Radoslav Fulek, Peter Gazi, Kristof Huszar,\r\nMarek
Krcal, Zuzana Masarova, Arnaud de Mesmay, Filip Moric, Michal Rybar, Martin Tancer,\r\nand
Stephan Zhechev.\r\nFinally, I would like to thank my thesis committee Herbert Edelsbrunner
and Roman Karasev\r\nfor their careful reading of the present manuscript and for
the many improvements they suggested."
alternative_title:
- IST Austria Thesis
article_processing_charge: No
author:
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
citation:
ama: 'Mabillard I. Eliminating higher-multiplicity intersections: an r-fold Whitney
trick for the topological Tverberg conjecture. 2016.'
apa: 'Mabillard, I. (2016). *Eliminating higher-multiplicity intersections: an
r-fold Whitney trick for the topological Tverberg conjecture*. IST Austria.'
chicago: 'Mabillard, Isaac. “Eliminating Higher-Multiplicity Intersections: An r-Fold
Whitney Trick for the Topological Tverberg Conjecture.” IST Austria, 2016.'
ieee: 'I. Mabillard, “Eliminating higher-multiplicity intersections: an r-fold Whitney
trick for the topological Tverberg conjecture,” IST Austria, 2016.'
ista: 'Mabillard I. 2016. Eliminating higher-multiplicity intersections: an r-fold
Whitney trick for the topological Tverberg conjecture. IST Austria.'
mla: 'Mabillard, Isaac. *Eliminating Higher-Multiplicity Intersections: An r-Fold
Whitney Trick for the Topological Tverberg Conjecture*. IST Austria, 2016.'
short: 'I. Mabillard, Eliminating Higher-Multiplicity Intersections: An r-Fold Whitney
Trick for the Topological Tverberg Conjecture, IST Austria, 2016.'
date_created: 2018-12-11T11:50:16Z
date_published: 2016-08-01T00:00:00Z
date_updated: 2021-02-23T00:00:07Z
day: '01'
ddc:
- '500'
department:
- _id: UlWa
file:
- access_level: closed
checksum: 2d140cc924cd1b764544906fc22684ef
content_type: application/pdf
creator: dernst
date_created: 2019-08-13T08:45:27Z
date_updated: 2019-08-13T08:45:27Z
file_id: '6809'
file_name: Thesis_final version_Mabillard_w_signature_page.pdf
file_size: 2227916
relation: main_file
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checksum: 2d140cc924cd1b764544906fc22684ef
content_type: application/pdf
creator: dernst
date_created: 2021-02-22T11:36:34Z
date_updated: 2021-02-22T11:36:34Z
file_id: '9178'
file_name: 2016_Mabillard_Thesis.pdf
file_size: 2227916
relation: main_file
success: 1
file_date_updated: 2021-02-22T11:36:34Z
has_accepted_license: '1'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
page: '55'
publication_status: published
publisher: IST Austria
publist_id: '6237'
related_material:
record:
- id: '2159'
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: 'Eliminating higher-multiplicity intersections: an r-fold Whitney trick for
the topological Tverberg conjecture'
type: dissertation
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2016'
...