Connections Between Additive Combinatorics Graph Theory And Incidence Geometry
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| Author | : Mozhgan Mirzaei |
| Publisher | : |
| Total Pages | : 132 |
| Release | : 2020 |
| Genre | : |
| ISBN | : |
One of the Erd\H{o}s-like cornerstones in incidence geometry from which many other results follow is the celebrated Szemer\'edi-Trotter Theorem which states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this thesis, we study the effect of forbidding grids and short even cycles on the incidence graphs of point-line arrangements in the plane. Let \(A\) and \(B\) be two disjoint finite sets of points in the plane such that their union contains no three points on a line. We say that \(A\) \emph{avoids} \(B\) if no straight line determined by a pair of points in \(A\) intersects the convex hull of $B.$ $A$ and \(B\) are called mutually avoiding if \(A\) avoids \(B\) and \(B\) avoids \(A .\) Aronov et al. showed that any set of \(n\) points in general position in the plane contains a pair of mutually avoiding sets, each of size at least \(\Omega(\sqrt{n})\). Moreover, they proved that any set of \(n\) points in general position in \(\mathbb{R}^{d}\) contains a pair of mutually avoiding sets, each of size at least \(\Omega\left(n^{\frac{1}{d^{2}-d+1}}\right)\). In this thesis, we give a generalized version of mutually avoiding set theorem in the plane. Given an algebraic structure \(R\) and a subset \(A \subset R,\) define the sum set and the product set of \(A\) to be \(A+A=\{a+b: a, b \in A\}\) and \(A \cdot A=\{a \cdot b: a, b \in A\}\) respectively. Showing under what conditions at least one of \(|A+A|\) or \(|A \cdot A|\) is large has a long history of study that continues to the present day. By employing recent developments on the energy of polynomials over finite fields, we give the best-known lower bounds on $\max\{|A+A|, |f(A,A)|\}$, when $A$ is a small subset of $ \mathbb{F}_p,$ and $f$ is a quadratic nondegenerate polynomial in $\mathbb{F}_p[x,y].$
| Author | : Michael Tait |
| Publisher | : |
| Total Pages | : 113 |
| Release | : 2016 |
| Genre | : |
| ISBN | : |
This thesis studies problems in extremal graph theory, combinatorial number theory, and finite incidence geometry, and the interplay between these three areas. The first topic is the study of the Tur\'an number for $C_4$. F\"uredi showed that $C_4$-free graphs with $\mathrm{ex}(n, C_4)$ edges are intimately related to polarity graphs of projective planes. We prove a general theorem about dense subgraphs in a wide class of polarity graphs, and as a result give the best-known lower bounds for $\mathrm{ex}(n, C_4)$ for many values of $n$. We also study the chromatic and independence numbers of polarity graphs, with special emphasis on the graph $ER_q$. Next we study Sidon sets on graphs by considering what sets of integers may look like when certain pairs of them are restricted from having the same product. Other generalizations of Sidon sets are considered as well. We then use $C_4$-free graphs to prove theorems related to solvability of equations. Given an algebraic structure $R$ and a subset $A\subset R$, define the {\em sum set} and {\em product set} of $A$ to be $A+A = \{a+b:a,b\in A\}$ and $A\cdot A = \{a\cdot b: a,b\in A\}$ respectively. Showing under what conditions at least one of $|A+A|$ or $|A\cdot A|$ is large has a long history of study that continues to the present day. Using spectral properties of the bipartite incidence graph of a projective plane, we deduce that nontrivial sum-product estimates hold in the setting where $R$ is a finite quasifield. Several related results are obtained. Finally, we consider a classical question in finite incidence geometry: what is the subplane structure of a projective plane? A conjecture widely attributed to Neumann is that all non-Desarguesian projective planes contain a Fano subplane. By studying the structural properties of polarity graphs of a projective plane, we show that any plane of even order $n$ which admits a polarity such that the corresponding polarity graph has exactly $n+1$ loops must contain a Fano subplane. The number of planes of order up to $n$ which our theorem applies to is not bounded above by any polynomial in $n$.
| Author | : Yufei Zhao |
| Publisher | : Cambridge University Press |
| Total Pages | : 336 |
| Release | : 2023-07-31 |
| Genre | : Mathematics |
| ISBN | : 1009310933 |
Using the dichotomy of structure and pseudorandomness as a central theme, this accessible text provides a modern introduction to extremal graph theory and additive combinatorics. Readers will explore central results in additive combinatorics-notably the cornerstone theorems of Roth, Szemerédi, Freiman, and Green-Tao-and will gain additional insights into these ideas through graph theoretic perspectives. Topics discussed include the Turán problem, Szemerédi's graph regularity method, pseudorandom graphs, graph limits, graph homomorphism inequalities, Fourier analysis in additive combinatorics, the structure of set addition, and the sum-product problem. Important combinatorial, graph theoretic, analytic, Fourier, algebraic, and geometric methods are highlighted. Students will appreciate the chapter summaries, many figures and exercises, and freely available lecture videos on MIT OpenCourseWare. Meant as an introduction for students and researchers studying combinatorics, theoretical computer science, analysis, probability, and number theory, the text assumes only basic familiarity with abstract algebra, analysis, and linear algebra.
| Author | : Terence Tao |
| Publisher | : Cambridge University Press |
| Total Pages | : 18 |
| Release | : 2006-09-14 |
| Genre | : Mathematics |
| ISBN | : 1139458345 |
Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.
| Author | : Adam Sheffer |
| Publisher | : Cambridge University Press |
| Total Pages | : 263 |
| Release | : 2022-03-24 |
| Genre | : Mathematics |
| ISBN | : 1108832490 |
A thorough yet accessible introduction to the mathematical breakthroughs achieved by using new polynomial methods in the past decade.
| Author | : Jonathan M. Borwein |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 395 |
| Release | : 2013-05-16 |
| Genre | : Mathematics |
| ISBN | : 1461466423 |
“Number Theory and Related Fields” collects contributions based on the proceedings of the "International Number Theory Conference in Memory of Alf van der Poorten," hosted by CARMA and held March 12-16th 2012 at the University of Newcastle, Australia. The purpose of the conference was to promote number theory research in Australia while commemorating the legacy of Alf van der Poorten, who had written over 170 papers on the topic of number theory and collaborated with dozens of researchers. The research articles and surveys presented in this book were written by some of the most distinguished mathematicians in the field of number theory, and articles will include related topics that focus on the various research interests of Dr. van der Poorten.
| Author | : Terence Tao |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 305 |
| Release | : 2009 |
| Genre | : Mathematics |
| ISBN | : 0821848852 |
Focuses on geometry, topology, and partial differential equations. This book discusses a variety of topics, including expository articles on topics such as gauge theory, the Kakeya needle problem, and the Black-Scholes equation. It is suitable for graduate students and research mathematicians interested in broad exposure to mathematical topics.
| Author | : Larry Guth |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 287 |
| Release | : 2016-06-10 |
| Genre | : Mathematics |
| ISBN | : 1470428903 |
This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.
| Author | : Simao Herdade |
| Publisher | : |
| Total Pages | : 73 |
| Release | : 2015 |
| Genre | : Combinatorial analysis |
| ISBN | : |
A general problem in Extremal Combinatorics asks about the maximum size of a collection of finite objects satisfying certain restrictions, and an ideal solution to it presents to you the objects which attain the maximum size. In several problems, it is the case that any large set satisfying the given property must be similar to one of the few extremal examples. Such stability results give us a complete understanding of the problem, and also make the result more flexible to be applied as a tool in other mathematical problems. Stability results in additive combinatorics and graph theory constitute the main topic of this thesis, in which we solve a question of Erdös and Sarközy on sums of integers, and reprove a conjecture of Posa and Seymour on powers of hamiltonian cycles. Along the way we prove stronger structural statements that have as a corollary the optimal solution to these problems. We also introduce a counting technique and two graph theory tools which we believe to be of great interest in their own right. Namely the Shifting Method, the Connecting Lemma, and a robust version of the classic Erdos-Stone Simonovits theorem.
| Author | : Helge Holden |
| Publisher | : Springer |
| Total Pages | : 774 |
| Release | : 2019-02-23 |
| Genre | : Mathematics |
| ISBN | : 3319990284 |
The book presents the winners of the Abel Prize in mathematics for the period 2013–17: Pierre Deligne (2013); Yakov G. Sinai (2014); John Nash Jr. and Louis Nirenberg (2015); Sir Andrew Wiles (2016); and Yves Meyer (2017). The profiles feature autobiographical information as well as a scholarly description of each mathematician’s work. In addition, each profile contains a Curriculum Vitae, a complete bibliography, and the full citation from the prize committee. The book also includes photos for the period 2003–2017 showing many of the additional activities connected with the Abel Prize. As an added feature, video interviews with the Laureates as well as videos from the prize ceremony are provided at an accompanying website (http://extras.springer.com/). This book follows on The Abel Prize: 2003-2007. The First Five Years (Springer, 2010) and The Abel Prize 2008-2012 (Springer 2014), which profile the work of the previous Abel Prize winners.
