Tropical varieties, maps and gossip

Frenk, B.J.

DOI:10.6100/IR750815

Published: 01/01/2013

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Citation for published version (APA):Frenk, B. J. (2013). Tropical varieties, maps and gossip Eindhoven: Technische Universiteit Eindhoven DOI:10.6100/IR750815

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TROPICAL VARIETIES, MAPS AND GOSSIP

Bart Frenk

This work is licensed under a Creative Commons Attribution

3.0 Unported License, which can be found via:

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Tropical varieties, maps and gossip / B.J. FrenkTechnische Universiteit Eindhoven 2013

A catalogue record is available from the Eindhoven University of Technology Library.

ISBN: 978-90-386-3343-5

NUR: 918

Cover design: Madelief Brandsma

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Tropical varieties, maps and gossip

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor Promoties in het openbaar te verdedigen

op woensdag 13 maart 2013 om 16.00 uur

door

Bartholomeus Johannes Frenk

geboren te Schiedam

Dit proefschrift is goedgekeurd door de promotor:

prof.dr. A.M. Cohen

Copromotor:dr.ir. J. Draisma

Preface

Tropical mathematics, or tropical geometry are loosely defined terms referring to asubfield of mathematics in which a central role is played by an algebraic structure

called the tropical real numbers. This is the set RYt8u equipped with the operationsof addition and taking the minimum. This structure first found applications in the

algebraic formulation of optimization problems (cf. [16]). In tropical mathematics,however, the emphasis is on geometry over the tropical semifield. For this thesis it is

convenient to make a very rough division of tropical mathematics into two (overlap-

ping) parts.

One part studies images of embedded algebraic varieties under a valuation. Such

images, called tropicalizations or tropical varieties, are (underlying sets of) polyhedralcomplexes that have a characterization in terms of the algebra of the tropical real

numbers and the ideal of the variety. The founding idea is that the properties of

tropicalizations reflect properties of the algebraic variety in question. One can then

use combinatorial techniques to derive algebraic geometric results, and vice versa.

A celebrated example is G. Mikhalkins computation of Gromov-Witten invariants by

counting tropical curves in [45].The other part studies the algebra and geometry of the tropical real numbers with-

out reference to a valued field. For example, it includes functional analysis over the

tropical real numbers (cf. [43]), and studies abstractions of tropical varieties. A rea-son to study such abstractions is that they are applicable to the aforementioned part

of tropical geometry, while the techniques involved do not depend on the characteris-

tic of the field. As argued in Chapter 4 the concept of a matroid falls naturally within

this part of tropical geometry.

Chapters 1 and 2 fit within the first part. The first chapter deals with reformulating

and proving the main properties of the tropicalization of an algebraic variety. None of

these results are new, but as far as we know have not been collected in a published

source. The second chapter explores the relation between unirationality of algebraic

varieties and tropical unirationality of its tropicalization.Chapter 3, 4 and 5 fit within the second part. Chapter 4 is the largest chapter of

this thesis. Its aim is to construct a category whose objects are tropical linear spaces.

i

ii

These spaces are abstractions of tropicalizations of vector spaces and are closely re-

lated to the valuated matroids of [23]. The reason for constructing such a categoryis that in general maps between tropical varieties are problematic. For example, in

the context of tropical linear spaces the naive notion of a linear map does not map

subspaces to subspaces. Chapter 4 attempts to remedy that situation.

Chapter 3 puts the tropical semifield into the broader context of idempotent semi-

fields. Its main purpose is to understand which algebraic properties of idempotent

semifields are essential for Chapter 4. The last chapter, Chapter 5 is about the monoid

generated by distance matrices under tropical matrix multiplication. This monoid is

related to the gossip problem and referred to as the gossip monoid. We make use of

techniques related to realizing a finite metric by a graph to compute polyhedral fan

structures on the gossip monoids of square matrices of size at most 4.

Acknowledgements

To me, sometimes an appropriate metaphor for being a Ph.D. candidate in mathe-

matics is tropical. It is that of deep, dark jungle. An environment without clear paths,

in which it is often difficult to distinguish the friendly from the fearful, at least at

first sight. A place easy to get lost in. That I came out in one piece is greatly due

to the efforts of my guide and copromotor, dr. ir. Jan Draisma. Of the many things

I learned from him the most important was probably to face unknown mathematics

with courage, no matter how fearsome it might appear. He also wrote the original

Tropical Algebraic Groups NWO project proposal that was the starting point of this

thesis.

During exploration one meets the people along the way that helped the project

progress in some way or another. They appear here in no particular order.

I would like to thank Prof. Dr. Eva Maria Feichtner for the invitation to do part of

my thesis work at the University of Bremen, where I spent three months. Life there

was made easier, both professionally and personally, by the people of the DiscreteStructures in Algebraic Geometry group and the capoeristas of Ginga Brasil Bremen. Iextend my thanks.

My promotor, prof. dr. Arjeh Cohen, was kind enough to accept me as his official

Ph.D. student. The three aforementioned people were also part of my reading com-

mittee, along with Prof. Dr. Thorsten Theobald and Prof. dr. Andries Brouwer, all of

which I would like to thank for their helpful corrections and comments on the first

draft of my thesis. It goes without saying that any errors and omissions left in the

thesis are entirely mine. The reading committee members are also part of my defence

committee, along with Prof. Dr. Michael Joswig en Prof. dr. Peter Butkovic. I would

like to thank all of them for their time and effort.

iii

My colleagues and fellow Ph.D. candidates at the Eindhoven University of Technol-

ogy. In particular, Jan-Willem Knopper, for his patience in dealing with the computer-

related problems I sometimes had, and Maxim Hendriks, that guided me gently

through the myriad of choices one has to make when deciding how to print ones

thesis.

Of course, not all of my time in the last four years was spent exploring this

metaphorical jungle.

The capoeira-lessons of Mestre Tayson in Eindhoven were always a very good way

to take my mind of the mathematics. I would like to thank him and his students, my

friends, in no particular order: Danni, Damian, Tsveti, Mike, Roelof, Dirk, Imre, Qorin,

Matilde.

I would like to thank my parents for their continuous support. I consider myself

lucky that my twin sister Myrthe worked on a Ph.D. project at the University of Maas-

tricht while I worked on mine in Eindhoven. This was particularly helpful during the

periods of frustration that are bound to come up when doing research. Talking to her

about such matters