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Metric and analytic methods
Kaluža, Vojtěch ; Tancer, Martin (advisor) ; Kleiner, Bruce (referee) ; Fulek, Radoslav (referee)
The thesis deals with two separate problems. In the first part we show that the regular n×n grid of points in Z2 cannot be recovered from an arbitrary n2 -element subset of Z2 using only mappings with prescribed maximum stretch independent of n. This provides a negative answer to a question of Uriel Feige from 2002. The present approach builds on the work of Burago and Kleiner and McMullen from 1998 on bilipschitz non-realisable densities and bilipschitz non-equivalence of separated nets in the plane. We describe a procedure that takes a positive, measurable function and encodes it into a sequence of discrete sets. Then we show that applying this procedure to a typical positive, continuous function on the unit square yields a counter-example to Feige's question. Along the way we provide a new proof of a result on bilipschitz decomposition for Lipschitz regular mappings, which was originally proved by Bonk and Kleiner in 2002. In the second part we provide a constructive proof for the strong Hanani- Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi from 2009, the presented approach does not rely on characterisation of embeddability into the projective plane via forbidden minors. 1
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Lipschitz mappings of discrete sets
Kaluža, Vojtěch ; Matoušek, Jiří (advisor) ; Šámal, Robert (referee)
In this thesis we consider Feige's question of whether there always exists a constantly Lipschitz bijection of an n2 -element set S ⊂ Z2 onto a regular lattice of n × n points in Z2 . We propose a solution of this problem in case the points of the set S form a long rectangle or they are arranged in the shape of a square without a part of its interior points. The main part is a summary of Burago's and Kleiner's article [2] and the article by McMullen [12] dealing with the problem of existence of separated nets in R2 that are not bi-Lipschitz equivalent to the integer lattice. This problem looks similar to Feige's problem. According to these articles we construct a separated net that is not bi-Lipschitz equivalent to the integer lattice, using a positive bounded measurable function that is not the Jacobian of a bi-Lipschitz homeomorphism almost everywhere. We present McMullen's construction of such a function and we complete his proof of its correctness. 1
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Metric and analytic methods
Kaluža, Vojtěch ; Tancer, Martin (advisor) ; Kleiner, Bruce (referee) ; Fulek, Radoslav (referee)
The thesis deals with two separate problems. In the first part we show that the regular n×n grid of points in Z2 cannot be recovered from an arbitrary n2 -element subset of Z2 using only mappings with prescribed maximum stretch independent of n. This provides a negative answer to a question of Uriel Feige from 2002. The present approach builds on the work of Burago and Kleiner and McMullen from 1998 on bilipschitz non-realisable densities and bilipschitz non-equivalence of separated nets in the plane. We describe a procedure that takes a positive, measurable function and encodes it into a sequence of discrete sets. Then we show that applying this procedure to a typical positive, continuous function on the unit square yields a counter-example to Feige's question. Along the way we provide a new proof of a result on bilipschitz decomposition for Lipschitz regular mappings, which was originally proved by Bonk and Kleiner in 2002. In the second part we provide a constructive proof for the strong Hanani- Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi from 2009, the presented approach does not rely on characterisation of embeddability into the projective plane via forbidden minors. 1
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Lipschitz mappings of discrete sets
Kaluža, Vojtěch ; Matoušek, Jiří (advisor) ; Šámal, Robert (referee)
In this thesis we consider Feige's question of whether there always exists a constantly Lipschitz bijection of an n2 -element set S ⊂ Z2 onto a regular lattice of n × n points in Z2 . We propose a solution of this problem in case the points of the set S form a long rectangle or they are arranged in the shape of a square without a part of its interior points. The main part is a summary of Burago's and Kleiner's article [2] and the article by McMullen [12] dealing with the problem of existence of separated nets in R2 that are not bi-Lipschitz equivalent to the integer lattice. This problem looks similar to Feige's problem. According to these articles we construct a separated net that is not bi-Lipschitz equivalent to the integer lattice, using a positive bounded measurable function that is not the Jacobian of a bi-Lipschitz homeomorphism almost everywhere. We present McMullen's construction of such a function and we complete his proof of its correctness. 1
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