Real Analysis, Quantitative Topology, and Geometric Complexity

Contents 1 Mappings and distortion 2 The mathematics of good behavior much of the time, and the BMO frame of mind 3 Finite polyhedra and combinatorial parameterization problems 4 Quantitative topology, and calculus on singular spaces 5 Uniform rectifiability Appendices A Fourier transform calculations B Mappings with branching C More on existence and behavior of homeomorphisms D Doing pretty well with spaces which may not have nice coordinates E Some simple facts related to homologyComment: 161 pages, Latex2

Purity of monoids and characteristic-free splittings in semigroup rings

Inspired by methods in prime characteristic in commutative algebra, we introduce and study combinatorial invariants of seminormal monoids. We relate such numbers with the singularities and homological invariants of the semigroup ring associated to the monoid. Our results are characteristic independent

Calculus of Variations

Since its invention, the calculus of variations has been a central field of mathematics and physics, providing tools and techniques to study problems in geometry, physics and partial differential equations. On the one hand, steady progress is made on long-standing questions concerning minimal surfaces, curvature flows and related objects. On the other hand, new questions emerge, driven by applications to diverse areas of mathematics and science. The July 2012 Oberwolfach workshop on the Calculus of Variations witnessed the solutions of famous conjectures and the emerging of exciting new lines of research

Results and questions on matchings in groups and vector subspaces of fields

A matching from a finite subset $A$ of an abelian group to another subset $B$ is a bijection $f:A\rightarrow B$ with the property that $a+f(a)$ never lies in $A$. A matching is called acyclic if it is uniquely determined by its multiplicity function. Motivated by a question of E. K. Wakeford on canonical forms for symmetric tensors, the study of matchings and acyclic matchings in abelian groups was initiated by C. K. Fan and J. Losonczy in [16, 26], and was later generalized to the context of vector subspaces in a field extension [13, 1]. We discuss the acyclic matching and weak acyclic matching properties and we provide results on the existence of acyclic matchings in finite cyclic groups. As for field extensions, we completely classify field extensions with the linear acyclic matching property. The analogy between matchings in abelian groups and in field extensions is highlighted throughout the paper and numerous open questions are presented for further inquiry.Comment: 17 pages, minor corrections, subsection 2.2 is shortened. To appear in Journal of Algebr

Mittojen hienorakenne

The main goal of this dissertation is to study the local distribution and irregularities of measures in mostly Euclidean setting. The research belongs to the field of Geometric Measure Theory. The thesis consists of an overview and three refereed research articles. The first article concerns the relationship between Hausdorff- and packing dimensions of measures and the local distribution of measures. There are many ways to quantify local distribution and here we consider local homogeneity, conical densities and porosity. Historically, there have already been many results for these notions of local distribution, but our contribution is to generalize and simplify many of the earlier results, and most importantly, provide a unified framework where such results could be proved. This framework is based on local entropy averages, a recently introduced way to calculate dimensions of measures inspired by dynamical systems. In the second and third articles we consider another notion that describes the local irregularities of measures: tangent measures. Tangent measures were rigorously defined and studied by D. Preiss in 1987 and they provided a powerful tool in the study of rectifiability. In this thesis we consider the possible relationship between tangent measures and the original measure. Our motivation is to strengthen the heuristics that it is not in general possible to deduce information from just the tangent measures of the underlying measure without further assumptions from the measure. In the second paper we construct a highly singular measure, a non-doubling measure, for which every tangent measure is equivalent to Lebesgue measure. The existence of such a measure provides a natural extension to a previous result by Preiss and it also provides a direct counterexample to the characterisation of porosity with tangent measures for general measures, which was previously unknown. In the third paper we prove that for a typical measure in the Euclidean space, in the sense of Baire category, the set of tangent measures consists of all non-zero measures at almost every point with respect to the underlying measure. This result was already proved by T. O'Neil in this PhD thesis from 1994, but we provide another self-contained proof using different techniques. Moreover, we record previously unknown corollaries and sharpen the result by T. O'Neil. Furthermore, we are able to use similar ideas in the setting of micromeasures, which are a symbolic way to define tangent measures in trees, and prove an analogous result in this setting.Mitat ovat modernin matematiikan yksi keskeisimmistä työkaluista. Niiden avulla voidaan luonnollisesti kuvailla massan jakautumista joukoissa ja ne ovat mahdollistaneet matemaattisen analyysin ja todennäköisyysteorian muovautumisen nykyiseen muotoonsa. Näin mitat ovat myös keskeisessä roolissa monissa sovelluksissa, kuten fysiikassa, biologiassa, tilastotieteissä ja taloustieteissä. Väitöskirjassa tarkastellaan erityisesti mittojen geometrisia ominaisuuksia. Päätulokset koskevat mittojen hienorakennetta ja eri tapoja liittää mittojen paikallinen käyttäytyminen niiden globaaliin rakenteeseen. Väitöskirja koostuu yleistajuisesta johdannosta ja kolmesta vertaisarvioidusta tieteellisestä artikkelista. Ensimmäinen artikkeli käsittelee mittojen dimensioiden ja massan jakautumisen yhteyksiä. Massan jakautumista voidaan kuvailla monilla eri tavoilla ja tässä väitöskirjassa käytämme jakautumista kuvailemaan lokaalia homogeenisuutta, kartiotiheyksiä ja huokoisuutta. Jo pitkään ennen väitöskirjan laatimista on ollut tiedossa monia tuloksia liittyen dimensioiden ja massan jakautumisen yhteyksiin mutta tässä väitöskirjassa yleistämme ja yksinkertaistamme monia aikaisempia tuloksia. Lisäksi ehkä tärkein panoksemme on rakentaa uusi yhtenäinen työympäristö, jota käytimme kaikkien tuloksiemme todistamiseen. Tämä työympäristö perustuu niin sanottuihin lokaaleihin entropiakeskiarvoihin, jotka ovat saaneet inspiraationsa dynaamisten systeemien teoriasta. Toisessa ja kolmannessa artikkelissa käsittelemme mittojen tangenttimittoja. Tangenttimitat ovat eräänlaisia mittojen derivaattamittoja ja niiden avulla voi kuvailla mittojen hienorakennetta. Tangenttimittojen tutkimus alkoi D. Preissin vuonna 1987 julkaistusta tutkimuksesta, jossa hän määritteli ja tutki systemaattisesti tangenttimittoja. Preissin tekniikat olivat merkittäviä, sillä ne soveltuivat luonnollisesti niin sanottujen suoristuvien joukkojen teoriaan. Tämän väitöskirjan motivaatio on tutkia miten tangenttimitat liittyvät alkuperäisen mitan ominaisuuksiin. Erityisesti päämääränä on vahvistaa heuristiikkaa siitä, ettei mitan globaalista rakenteesta voi yleisesti päätellä mitään vain mitan tangenttimittojen avulla ellei mitan rakenteesta oleta jo alunperin jotain. Väitöskirjan toisessa artikkelissa rakennamme hyvin epäsäännöllisen mitan, niin sanotun epätuplaavan mitan, jonka kaikki tangenttimitat ovat hyvin säännöllisiä: ne ovat ekvivalentteja Lebesguen mitan kanssa. Tämä tulos laajentaa luonnollisesti Preissin aikaisempaa vastaavaa tulosta ja myös osoittaa, ettei mittojen huokoisuutta voi karakterisoida tangettimittojen avulla yleisesti. Kolmannessa artikkelissa osoitamme, että tyypillisellä mitalla tangenttimitat ovat kaikki sallitut mitat melkein kaikissa pisteissä mitan suhteen. Tuloksen oli kuitenkin osoittanut T. O'Neil väitöskirjassaan vuodelta 1994 mutta tässä väitöskirjassa esitämme sille uuden itsenäisen todistuksen. Näytämme myös ennestään tuntemattomia seurauksia ja pystymme sanomaan jotain tuloksen tarkkuudesta. Lisäksi tutkimme symbolisia tangenttimittoja mitoille puissa - niin sanottuja mikromittoja - ja osoitamme analogisen ominaisuuden tyypillisten mittojen mikromitoille

Anisotropic energies in geometric measure theory

In this thesis we focus on different problems in the Calculus of Variations and Geometric Measure Theory, with the common peculiarity of dealing with anisotropic energies. We can group them in two big topics: 1. The anisotropic Plateau problem: Recently in [37], De Lellis, Maggi and Ghiraldin have proposed a direct approach to the isotropic Plateau problem in codimension one, based on the “elementary” theory of Radon measures and on a deep result of Preiss concerning rectifiable measures. In the joint works [44],[38],[43] we extend the results of [37] espectively to any codimension, to the anisotropic setting in codimension one and to the anisotropic setting in any codimension. For the latter result, we exploit the anisotropic counterpart of Allard’s rectifiability Theorem, [2], which we prove in [42]. It asserts that every d-varifold in Rn with locally bounded anisotropic first variation is d-rectifiable when restricted to the set of points in Rn with positive lower d-dimensional density. In particular we identify a necessary and sufficient condition on the Lagrangian for the validity of the Allard type rectifiability result. We are also able to prove that in codimension one this condition is equivalent to the strict convexity of the integrand with respect to the tangent plane. In the paper [45], we apply the main theorem of [42] to the minimization of anisotropic energies in classes of rectifiable varifolds. We prove that the limit of a minimizing sequence of varifolds with density uniformly bounded from below is rectifiable. Moreover, with the further assumption that all the elements of the minimizing sequence are integral varifolds with uniformly locally bounded anisotropic first variation, we show that the limiting varifold is also integral. 2. Stability in branched transport: Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. Given a flow (traffic path) that transports a given measure $μ^−$ onto a target measure $μ^+$, along a 1-dimensional network, the transportation cost per unit length is supposed in these models to be proportional to a concave power α ∈ (0,1) of the intensity of the flow. The transportation cost is called α-mass. In the paper [27] we address an open problem in the book [15] and we improve the stability for optimal traffic paths in the Euclidean space $R^n$ with respect to variations of the given measures ($μ^−$, $μ^+$), which was known up to now only for α > 1− $\frac 1n$. We prove it for exponents α > 1− $\frac{1}{n−1}$ (in particular, for every α ∈ (0,1) when n = 2), for a fairly large class of measures (\μ^+\) and (\μ^−\). The α- mass is a particular case of more general energies induced by even, subadditive, and lower semicontinuous functions H : R → [0,∞) satisfying H (0) = 0. In the paper [28], we prove that the lower semicontinuous envelope of these energy functionals defined on polyhedral chains coincides on rectifiable currents with the H -mass

On unique sums in Abelian groups

Let $A$ be a subset of the cyclic group $\mathbf{Z}/p\mathbf{Z}$ with $p$ prime. It is a well-studied problem to determine how small $|A|$ can be if there is no unique sum in $A+A$, meaning that for every two elements $a_1,a_2\in A$, there exist $a_1',a_2'\in A$ such that $a_1+a_2=a_1'+a_2'$ and $\{a_1,a_2\}\neq \{a_1',a_2'\}$. Let $m(p)$ be the size of a smallest subset of $\mathbf{Z}/p\mathbf{Z}$ with no unique sum. The previous best known bounds are $\log p \ll m(p)\ll \sqrt{p}$. In this paper we improve both the upper and lower bounds to $\omega(p)\log p \leqslant m(p)\ll (\log p)^2$ for some function $\omega(p)$ which tends to infinity as $p\to \infty$. In particular, this shows that for any $B\subset \mathbf{Z}/p\mathbf{Z}$ of size $|B|<\omega(p)\log p$, its sumset $B+B$ contains a unique sum. We also obtain corresponding bounds on the size of the smallest subset of a general Abelian group having no unique sum