You are cordially invited to the Number Theory Seminar organized by the Department of Mathematics.<\/p>\n
Speaker: Do\u011fa Can Sertba\u015f (Istinye \u00dcniversitesi) and Tomos Parry (Bilkent)<\/p>\n
\u201d A week of additive number theory\u201d<\/p>\n
Abstract: Take a natural number N. Can you write<\/p>\n
\t\t\tN=p_1+p_2\r\n<\/code><\/pre>\nwith p_1,p_2 primes?<\/p>\n
Additive number theory is concerned with questions about the natural numbers which are in some way \u201cadditive\u201d questions. We will give a series of seminars with the aim of giving a feel for some of the questions that motivate number theory research and to present some of the tools used to tackle them. No real prerequisites are required since we aim to outline more than get too involved in details.<\/p>\n
Ternary Goldbach: What about writing<\/p>\n
\t\t\tN=p_1+p_2+p_3\r\n<\/code><\/pre>\nthen? This is the ternary Goldbach conjecture. It was solved completely only recently, although most of it could be said to have been solved already a good while ago \u2013 Vinogradov (1930\u2019s) proved that all can be written in this form. Helfgott (2013) then made a number of innovations to bring the size down to a number a computer could verify. Vinogradov\u2019s proof uses the circle method, a major tool in additive number theory. We\u2019ll present this proof.<\/p>\n
Roth\u2019s theorem: Another additive problem is whether an arbitrary set \u2286 {1, 2, \u2026 , }contains a 3-term arithmetic progression. This was proven by Roth (1953) also using the circle method, and we present the proof.<\/p>\n
Although there are still some moderately interesting questions that are attacked to day with the circle method in its classical form presented in the Vinogradov talk, more modern research depends on considerable generalisations of these methods, a highlight being the Green-Tao theorem that the primes contain a 100-term arithmetic progression for example. And the main engine in Roth\u2019s proof has revealed itself to really be density
\nincrement. If time permits, we\u2019ll say a few words about these more current developments.<\/p>\n
Ramsey Theory & Sum-free sets: In 1947, Erd\u0151s gave a lower bound for the diagonal Ramsey numbers which are defined after the celebrated theorem of Ramsey. Erd\u0151s\u2019 proof contains purely probabilistic arguments and his approach can be considered as an example of a proof technique which is so called the probabilistic method. According to this method, one obtains the existence of a particular mathematical object in a non-constructive way. In these series of talks, we first introduce the Ramsey numbers for some given natural numbers \u2265 1, and then explain the proof of Erd\u0151s. After that, we discuss Schur\u2019s and van der Waerden\u2019s theorems, and using the probabilistic method we also give lower bounds for Schur and van der Waerden numbers. Moreover, we demonstrate how various probabilistic techniques can be applied to problems involving sum-free sets and sets with distinct sums. Finally, we will explore how the probabilistic method proves the existence of certain types of additive bases, if time permits.<\/p>\n
Date: August 5 \u2013 8, 2025
\nTime: 14:00
\nPlace: SB-Z11<\/p>\n","protected":false},"excerpt":{"rendered":"
You are cordially invited to the Number Theory Seminar organized by the Department of Mathematics. Speaker: Do\u011fa Can Sertba\u015f (Istinye \u00dcniversitesi) and Tomos Parry (Bilkent) \u201d A week of additive number theory\u201d Abstract: Take a natural number N. Can you write N=p_1+p_2 with p_1,p_2 primes? Additive number theory is concerned [\u2026] <\/p>\n