Senior Thesis at CMC is often described institutionally as the capstone of one’s undergraduate education. As such, I wanted my own to accurately capture and reflect how I’ve grown as a student and mathematician these past four years. What follows is my attempt to distill lessons I learned in mathematics outside the curriculum, written for incoming undergraduates and anyone with just a little bit of mathematical curiosity. In it, I attempt to dispel some common preconceptions about mathematics, namely that it’s uninteresting, formulaic, acultural, or completely objective, in favor of a dynamic historical and cultural perspective, with particular attention paid to the early twentieth century search to secure the foundations of mathematics and a detailed look at contemporary Hungarian mathematics. After doing so, I conclude that the scope of mathematics is not what one might expect but that it’s still absolutely worth doing and appreciating.

In 1946, Erdös posed the distinct distances problem, which asks for the minimum number of distinct distances that any set of n points in the real plane must realize. Erdös showed that any point set must realize at least &Omega(n1/2) distances, but could only provide a construction which offered &Omega(n/&radic(log(n)))$ distances. He conjectured that the actual minimum number of distances was &Omega(n1-&epsilon) for any &epsilon > 0, but that sublinear constructions were possible. This lower bound has been improved over the years, but Erdös' conjecture seemed to hold until in 2010 Larry Guth and Nets Hawk Katz used an incidence theory approach to show any point set must realize at least &Omega(n/log(n)) distances. In this thesis we will explore how incidence theory played a roll in this process and expand upon recent work by Adam Sheffer and Cosmin Pohoata, using geometric incidences to achieve bounds on the bipartite variant of this problem. A consequence of our extensions on their work is that the theoretical upper bound on the original distinct distances problem of &Omega(n/&radic(log(n))) holds for any point set which is structured such that half of the n points lies on an algebraic curve of arbitrary degree.

Many real-world network data can be formulated as graphs, where a binary relation exists between nodes. One of the fundamental problems in network data analysis is community detection, clustering the nodes into different groups. Statistically, this problem can be formulated as hypothesis testing: under the null hypothesis, there is no community structure, while under the alternative hypothesis, community structure exists. One is of the method is to use the largest eigenvalues of the scaled adjacency matrix proposed by Bickel and Sarkar (2016), which works for dense graph. Another one is the subgraph counting method proposed by Gao and Lafferty (2017a), valid for sparse network. In this paper, firstly, we empirically study the BS or GL methods to see whether either of them works for moderately sparse network; secondly, we propose a subsampling method to reduce the computation of the BS method and run simulations to evaluate the performance.

We study the Erdős- Sòs conjecture that states that ever graph of average degree greater than k-1 contains every tree of order k+1. While the conjecture was studied for some graphs, it still remains open and of interest after more than 40 years. We study the conjecture for graphs with no K2,s where, s ≥ 2 and k > 12(s-1). We use the fact that as G contains no K2,s, any two distinct vertices in G have at most s-1 neighbors in common in proving the results. We have answered in the affirmative that the Erdős- Sòs conjecture is true for graphs defined above, thus adding to the list of graphs for which the conjecture is true. We also study the Cayley Isomorphism Problem that states that for which finite groups H is it true that any two Cayley graphs of H are isomorphic if and only if they are isomorphic by a group automorphism of H ? (H is a CI-group with respect to graphs.) Determining whether or not a group is a CI-group with respect to graphs has received considerable attention over the last 40 or so years. In particular, we study the problem for (pq,r)-metacirculant color digraphs where p < q < r and pq/| α|. We use the fact that Γ is a CI-color digraph of H if and only if given a permutation γ ∈ SH such that γ-1HLγ ≤ Aut(Γ), HL and γ-1HLγ are conjugate in Aut(Γ). We consider the Cayley isomorphism problem for a nonabelian group of order pqr, where p, q, r are distinct primes such that pq/(r- 1). We show that the results are true, not only for Cayley graphs but for some related classes of non Cayley vertex transitive graphs, thus solving the problem for that case.

Let P = (p1, p2, . . . , pN ) be a sequence of points in the plane, where pi = (xi, yi) and x1 < x2 < · · · < xN . A famous 1935 Erdős-Szekeres theorem asserts that every such P contains a monotone subsequence S of √ N points. Another, equally famous theorem from the same paper implies that every such P contains a convex or concave subsequence of Ω(log N) points. First we define a (k + 1)-tuple K ⊆ P to be positive if it lies on the graph of a function whose kth derivative is everywhere nonnegative, and similarly for a negative (k + 1)-tuple. Then we say that S ⊆ P is kth-order monotone if its (k + 1)- tuples are all positive or all negative. In this thesis we investigate quantitative bound for the corresponding Ramsey-type result. We obtain an Ω(log(k−1) N) lower bound ((k − 1)-times iterated logarithm). We also improve bounds for related problems: Order types and One-sided sets of hyperplanes. 1

Ramsey-type results for ordered hypergraphs Martin Balko Abstract We introduce ordered Ramsey numbers, which are an analogue of Ramsey numbers for graphs with a linear ordering on their vertices. We study the growth rate of ordered Ramsey numbers of ordered graphs with respect to the number of vertices. We find ordered match- ings whose ordered Ramsey numbers grow superpolynomially. We show that ordered Ramsey numbers of ordered graphs with bounded degeneracy and interval chromatic number are at most polynomial. We prove that ordered Ramsey numbers are at most polynomial for ordered graphs with bounded bandwidth. We find 3-regular graphs that have superlinear ordered Ramsey numbers, regardless of the ordering. The last two results solve problems of Conlon, Fox, Lee, and Sudakov. We derive the exact formula for ordered Ramsey numbers of mono- tone cycles and use it to obtain the exact formula for geometric Ramsey numbers of cycles that were introduced by Károlyi et al. We refute a conjecture of Peters and Szekeres about a strengthening of the fa- mous Erd˝os-Szekeres conjecture to ordered hypergraphs. We obtain the exact formula for the minimum number of crossings in simple x-monotone drawings of complete graphs and provide a combinatorial characterization of these drawings in terms of colorings of ordered...

Le fil conducteur de cette thèse, composée de trois parties, est la notion de branchement.Le premier chapitre est consacré à l'arbre de Yule et à l'arbre binaire de recherche. Nous obtenons des résultats d'oscillations asymptotiques de l'espérance, de la variance et de la distribution de la hauteur de ces arbres, confirmant ainsi une conjecture de Drmota. Par ailleurs, l'arbre de Yule pouvant être vu comme une marche aléatoire branchante évoluant sur un réseau, nos résultats permettent de mieux comprendre ce genre de processus.Dans le second chapitre, nous étudions le nombre de particules tuées en 0 d'un mouvement brownien branchant avec dérive surcritique conditionné à s'éteindre. Nous ferons enfin apparaître une nouvelle phase de transition pour la queue de distribution de ces variables.L'objet du dernier chapitre est le graphe d'Erdős–Rényi dans le cas critique : $G(n,1/n)$. En introduisant un couplage et un changement d'échelle, nous montrerons que, lorsque $n$ augmente les composantes de ce graphe évoluent asymptotiquement selon un processus de coalescence-fragmentation qui agit sur des graphes réels. La partie coalescence sera de type multiplicatif et les fragmentations se produiront selon un processus ponctuel de Poisson sur ces objets. / This thesis is composed by three chapters and its main theme is branching processes.The first chapter is devoted to the study of the Yule tree and the binary search tree. We obtain oscillation results on the expectation, the variance and the distribution of the height of these trees and confirm a Drmota's conjecture. Moreover, the Yule tree can be seen as a particular instance of lattice branching random walk, our results thus allow a better understanding of these processes.In the second chapter, we study the number of particles killed at 0 for a Brownian motion with supercritical drift conditioned to extinction. We finally highlight a new phase transition in terms of the drift for the tail of the distributions of these variables.The main object of the last chapter is the Erdős–Rényi graph in the critical case: $G(n,1/n)$. By using coupling and scaling, we show that, when $n$ grows, the scaling process is asymptotically a coalescence-fragmentation process which acts on real graphs. The coalescent part is of multiplicative type and the fragmentations happen according a certain Poisson point process.

Le sujet principal de cette thèse est l’étude des valeurs moyennes et corrélations de fonctions multiplicatives. Les résultats portant sur ces derniers sont subséquemment appliqués à la résolution de plusieurs problèmes. Dans le premier chapitre, on rappelle certains résultats classiques concernant les valeurs moyennes des fonctions multiplicatives. On y énonce également les théorèmes principaux de la thèse. Le deuxième chapitre consiste de l’article “Mean values of multiplicative functions over the function fields". En se basant sur des résultats classiques de Wirsing, de Hall et de Tenenbaum concernant les fonctions multiplicatives arithmétiques, on énonce et on démontre des théorèmes qui y correspondent pour les fonctions multiplicatives sur les corps des fonctions Fq[x]. Ainsi, on résoud un problème posé dans un travail récent de Granville, Harper et Soundararajan. On décrit dans notre thése certaines caractéristiques du comportement des fonctions multiplicatives sur les corps de fonctions qui ne sont pas présentes dans le contexte des corps de nombres. Entre autres, on introduit pour la première fois une notion de “simulation” pour les fonctions multiplicatives sur les corps de fonctions Fq[x]. Les chapitres 3 et 4 comprennent plusieurs résultats de l’article “Correlations of multiplicative functions and applications". Dans cet article, on détermine une formule asymptotique pour les corrélations X n6x f1(P1(n)) · · · fm(Pm(n)), où f1, . . . ,fm sont des fonctions multiplicatives de module au plus ou égal à 1 ”simulatrices” qui satisfont certaines hypothèses naturelles, et P1, . . . ,Pm sont des polynomes ayant des coefficients positifs. On déduit de cette formule plusieurs conséquences intéressantes. D’abord, on donne une classification des fonctions multiplicatives f : N ! {−1,+1} ayant des sommes partielles uniformément bornées. Ainsi, on résoud un problème d’Erdos datant de 1957 (dans la forme conjecturée par Tao). Ensuite, on démontre que si la valeur moyenne des écarts |f(n + 1) − f(n)| est zéro, alors soit |f| a une valeur moyenne de zéro, soit f(n) = ns avec iii Re(s) < 1. Ce résultat affirme une ancienne conjecture de Kátai. Enfin, notre théorème principal est utilisé pour compter le nombre de représentations d’un entier n en tant que somme a+b, où a et b proviennent de sous-ensembles multiplicatifs fixés de N. Notre démonstration de ce résultat, dû à l’origine à Brüdern, évite l’usage de la “méthode du cercle". Les chapitres 5 et 6 sont basés sur les résultats obtenus dans l’article “Effective asymptotic formulae for multilinear averages and sign patterns of multiplicative functions," un travail conjoint avec Alexander Mangerel. D’après une méthode analytique dans l’esprit du théorème des valeurs moyennes de Halász, on détermine une formule asymptotique pour les moyennes multidimensionelles x−l X n2[x]l Y 16j6k fj(Lj(n)), lorsque x ! 1, où [x] := [1,x] et L1, . . . ,Lk sont des applications linéaires affines qui satisfont certaines hypothèses naturelles. Notre méthode rend ainsi une démonstration neuve d’un résultat de Frantzikinakis et Host avec, également, un terme principal explicite et un terme d’erreur quantitatif. On applique nos formules à la démonstration d’un phénomène local-global pour les normes de Gowers des fonctions multiplicatives. De plus, on découvre et explique certaines irrégularités dans la distribution des suites de signes de fonctions multiplicatives f : N ! {−1,+1}. Visant de tels résultats, on détermine les densités asymptotiques des ensembles d’entiers n tels que la fonction f rend une suite fixée de 3 ou 4 signes dans presque toutes les progressions arithmétiques de 3 ou 4 termes, respectivement, ayant n comme premier terme. Ceci mène à une généralisation et amélioration du travail de Buttkewitz et Elsholtz, et donne un complément à un travail récent de Matomäki, Radziwiłł et Tao sur les suites de signes de la fonction de Liouville. / The main theme of this thesis is to study mean values and correlations of multiplicative functions and apply the corresponding results to tackle some open problems. The first chapter contains discussion of several classical facts about mean values of multiplicative functions and statement of the main results of the thesis. The second chapter consists of the article “Mean values of multiplicative functions over the function fields". The main purpose of this chapter is to formulate and prove analog of several classical results due to Wirsing, Hall and Tenenbaum over the function field Fq[x], thus answering questions raised in the recent work of Granville, Harper and Soundararajan. We explain some features of the behaviour of multiplicative functions that are not present in the number field settings. This is accomplished by, among other things, introducing the notion of “pretentiousness" over the function fields. Chapter 3 and Chapter 4 include results of the article “Correlations of multiplicative functions and applications". Here, we give an asymptotic formula for correlations X n_x f1(P1(n))f2(P2(n)) · · · · · fm(Pm(n)) where f . . . ,fm are bounded “pretentious" multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions f : N ! {−1,+1} with bounded partial sums. This answers a question of Erdos from 1957 in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either f(n) = ns for Re(s) < 1 or |f(n)| is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of n = a + b where a,b belong to some multiplicative subsets of N. This gives a new "circle method-free" proof of the result of Brüdern. Chapters 5 and Chapter 6 are based on the results obtained in the article “Effective asymptotic formulae for multilinear averages and sign patterns of multiplicative functions," joint with Alexander Mangerel. Using an analytic approach in the spirit of Halász’ mean v value theorem, we compute multidimensional averages x−l X n2[x]l Y 16j6k fj(Lj(n)) as x ! 1, where [x] := [1,x] and L1, . . . ,Lk are affine linear forms that satisfy some natural conditions. Our approach gives a new proof of a result of Frantzikinakis and Host that is distinct from theirs, with explicit main and error terms. As an application of our formulae, we establish a local-to-global principle for Gowers norms of multiplicative functions. We reveal and explain irregularities in the distribution of the sign patterns of multiplicative functions by computing the asymptotic densities of the sets of integers n such that a given multiplicative function f : N ! {−1, 1} yields a fixed sign pattern of length 3 or 4 on almost all 3- and 4-term arithmetic progressions, respectively, with first term n. The latter generalizes and refines the work of Buttkewitz and Elsholtz and complements the recent work of Matomaki, Radziwiłł and Tao. We conclude this thesis by discussing some work in progress.