Kazım İlhan İkeda (Boğaziçi): Gündüz İkeda'nın Matematiğe Katkıları üzerine
Bu konuşma özellikle cebirsel sayılar kuramı üzerine yapmış olduğu çalışmaların ve elde etmiş olduğu sonuçların kısa bir özeti üzerine olacaktır.
Zafeirakis Zafeirakopoulos (GTÜ):Using Polyhedral Geometry
Polyhedral geometry is a wide research area that has many connections to different branches of mathematics, from algebra and algebraic geometry to
optimization and combinatorics or number theory. In this talk we will focus on algorithmic (generating function) methods in polyhedral geometry.
In particular, we will disucss lattice point counting and how to apply it for problems in combinatorics and number theory.
Hatice Boylan (İstanbul Üni.):What are Jacobi forms and what are they good for?
The theory of Jacobi forms was created in 80's of the last century by Eichler and Zagier. This theory turned out to have many useful applications
ranging from combinatorics over physics and algebraic geometry to number theory. It complements the classical theory of elliptic modular forms.
In this talk we try to explain the notion of Jacobi forms and its main applications in number theory and arithmetic geometry. In the second part
of the talk we indicate recent affords to extend the classical theory to a theory of Jacobi forms over number fields.
Susumu Tanabe (Galatasaray):Japon Matematiğinin Şafağı: Teiji Takagi (1875-1960) ve Class Field Teorisinin Doğuşu.
Dünya çapında önem taşıyan,
sayılar kuramının bir alanı olan Class Field teorisinin temel kısmı, Japon matematikçi T. Takagi’ye aittir. Japonya’da, Takagi’nin makalelerinden
önce yayınlanan makaleler (1915-1920’den önce) araştırma seviyesine ulaşamamış öğrencilerin alıştırma seviyesinde kalıyordu. Bu anlamda M.İkeda’nın hocası
K.Shoda’nın hocası olan T. Takagi Japonya’da matematiksel konular üzerinde gerçek anlamda araştırmayapmaya başlamış bir bilim adamı olarak kabul edilmekte.
Bu konuşmada, Galois teorisinin bir genişletmesi olarak kabul edilen Class Field teorisi üzerine sunulmuş, L.Kronecker, R.Dedekind, D.Hilbert gibi alman matematikçilerin
problematiklerini, Takagi’nin bunları nasıl dikkate aldığını ve onun ‘Kroneckers Jugendtraum- Kornecker’in Gençlik Rüyası ’ problemini çözmesi üzerine, E.Artin, H. Hasse gibi Takagi’den
sonra gelen matematikçiler tarafından verilmiş yeni formülleştirmeler üzerinde duracağız.
Ekin Özman (Boğaziçi): Sayı Teorisine Mütevazı bir Giriş
Mustafa Topkara (MSGSÜ):Düşük boyutlu topoloji: Kulp ayrışması ve diyagramlar
4 ve daha küçük boutlu manifoldların incelendiği alana "düşük boyutlu topoloji" denir. Bu konuşmada kulp ayrışması (Handle decomposition) yöntemini
kullanarak iki boyutlu manifoldların (yani yüzeylerin) betimleyici diyagramlarını oluşturacağız. Sonrasında, bu yöntemi 3 ve 4 boyutlu manifoldları
resmetmek üzere genişleteceğiz. Bu sayede, belli başlı düşük boyutlu manifoldları elde edecek ve kimi temel özellikleri üzerine konuşacağız.
Çağatay Altuntaş (Koç):Analytic Class Number Formula
An introduction will be made to the analytic class number formula.
Tarık Arabacı (Bilgi): MacWilliams identity for complete m-spotty Rosenbloom-Tsfasman weight enumerators of linear codes
In order to create a weight enumerator of a code, weights of all codewords in that code need to be calculated and classied. This is a laborious activity for codes containing a large number of codewords. However, the number of codewords in a code is inversely proportional to the number of codewords in its dual code. So the very large codes have very small dual codes. It is much easier and quicker to calculate the weight enumerator of the small dual code than to calculate the weight enumerator of a large code.
An identity between the weight enumerators of the codes and their duals was first established by Jessie Mac Williams using the classical vectorial inner product of Euclidean space and Hamming metric. Similar identities have been established for dierent metrics and inner products in subsequent years. These identities are called as Mac Williams identities in memory of Jessie MacWilliams.
There are several weight enumerators used in coding theory for dierent uses. While the Hamming metric and derived weight enumerators focus on the number of the errors/nonzero symbols in the vectors, the Rosenbloom-Tsfasman metric and derived weight enumerators deal with the position of that errors/nonzero symbols. Whereas both of them take codewords as a single unit and define the distance between them, m-spotty types of weight enumerators divide the codewords into pieces. Weight enumerators that maintain the information about pieces are called "complete" and can only be expressed by multivariable polynomials. In this study, Mac Williams identity will be established for complete m-spotty Rosenbloom-Tsfasman weight enumerators.
Gökhan Ayyıldız (MSGSÜ): Friz Grupları
Friz tanım , grupların sınıflandırılması ve örnekler
İlayda Barış (MSGSÜ):Longest Common Subsequence Problem
The longest common subsequence problem has came up in 1970s. We already know that the problem is np-hard in the most general case. But it can be solved in polynomial time using dynamic programming under some conditions. The topic has a wide application area, further it is related to some problems in combinatorics, probability and computer science. In this talk, I will introduce the LCSP, give some solutions and talk about its applications.
Bartu Bingöl (Boğaziçi): Kummer Yaklaşımıyla Fermat’nın Son Teoremi
Bu konuşmada; Cebirsel Sayılar Kuramının doğuşunda, daha genel anlamda ise Matematiksel düşünce ve düşünmenin olgunlaşmasında büyük katkısı olan Fermat’nın Son Teoremi ve ona yaklaşımlar içerisindeki mihenk taşlarından biri olan Kummer’in önerdiği kanıt tartışılacaktır.
Sabri Çetin(Bilkent): Fundamental Group of Knot Complements
Basic examples of knots will be given. We will talk about the fundamental groups of knot complements and a general method for computation of the fundamental grouo of a knot complement. The talk will be in English and a basic knowledge of algebraic topology, including the concept of homotopy and fundamental group will be assumed to be known in this talk.
Baran Çetin (Bilgi): Valuation Fields
Valued fields and examples, padic valuation, piadic metric and norm
Gülsemin Çonoğlu (MSGSÜ): Sonlu projektif düzlemler
Konuya dair Aksiyomlar-Önemli Gözlemler-Sonuç olarak 7 nokta ve 7 doğrudan oluşan Fano düzlemi-Sorular
Tugay Değirmenci (MSGSÜ):Goursat's Teoremi
Rastgele iki altgrup seçip ,bu iki altgrubun direkt çapımlarının altgruplarını saymak.
Serkan Doğan (Bilkent): Minkowski convex body theorem and its application
Theorem will be stated and hopefully proved in the presentation. Also if time permits I will prove some number theoretical results such as every prime number of the form 4k+1 can be written as a sum of two square numbers.
Kübra Dölaslan (ODTÜ): Hemen hemen her çizge asimetriktir / Almost all graphs are asymmetric
Otomorfi grubunda yalnızca birim otomorfi olan çizgelere asimetrik çizge denir. Bu konuşmada, Ali Nesin'in bir makalesinden hareketle, rastgele bir çizgenin asimetrik olma olasılığını inceleyeceğiz ve çizgenin köşe sayısı arttıkça bu olasılığın bire yaklaştığını kanıtlayacağız.
Melike Efe (Bilgi): Classification of Lattes Maps
The purpose of this presentation is to investigate Latt\`{e}s maps on $ \widehat{\mathbb{C}} $ which are holomorphically conjugate to an affine map on $ \mathbb{C}/\Lambda $. In this work, we introduce some notions and facts from dynamical systems, algebraic topology and complex analysis in order to examine these maps deeply. A part of our work concerns the results of John Milnor related to Latt\`{e}s maps. Specifically, he showed that the degree of a conjugating holomorphism is either $ 2,3,4 $ or $ 6 $. Following this, we introduce an explicit form of a conjugating holomorphism of a given degree by using the aforementioned results and the properties of an elliptic function which can be written as a rational function of Weierstrass’ elliptic function and its derivatives. Finally, we calculate the ramification behaviour of the Latt\`{e}s maps.
Buket Eren (GS): On the uniqueness conjecture for Markov numbers
The aim of this talk is to present an interesting open problem about Markov numbers, namely uniqueness conjecture. After a brief introduction about approximation theory of badly approximable numbers, i will introduce Markov’s theorem which combines approximation of irrationals and a Diophantine equation in a totally unexpected way. Finally, i will present an equivalent version of the conjecture in the context of Markov's result.
Alaittin Kırtışoğlu (hatay): The Game of Hex and Brouwer Fixed Point Theorem
Oyunun kuralları açıklanacak. Hex Theorem ve 1. oyuncu için kazanan stratejinin olduğu gösterilecek. Hex Theorem kullanılarak Brouwer Fix Point ispatlanacak.
Makale; The Game of Hex and Brouwer Fixed Point Theorem, David Gale, 1979
Zeynep Kısakürek (Bilgi): Infinite Fields of Finite Morley Rank
This talk will focus on Angus Macintyre's result which states an infinite field of finite Morley rank is algebraically closed. Despite the fact that this result has a model theoretic proof, we will deal with a proof using algebraic tools to highlight the interaction between model theory and algebra. To explain this result, I will present a brief introduction to the basic notions of model theory with relevant facts and we will bring some results from Galois theory.
Şeyda Köse (ODTÜ): Construction with Straigthedge and Compass
In this talk, we will show the impossibility of some geometric constructions as a simple application of results obtained by algebraic extensions. We will focus on three basic question;
1-Is it possible using only straightedge and compass to construct a cube with precisely twice the volume of a given cube?
2-Is it possible using only straightedge and compass to trisect any given angle?
3-Is it possible using only straightedge and compass to construct a square with precisely the area of a given circle?
It will be assumed that audience have basic knowledge of field extensions during the talk
Hamide Kuru (GS): The Fundamental Theorem of Galois Theory
Bu çalışmada Galois Teorisinin Temel Teoremi hakkında konuşacağız. Öncelikle polinom halkaları, cisim genişlemeleri, cebirsel eleman, minimal polinom gibi kavramların tanımlarından bahsedeceğiz. Daha sonra Galois Grup tanımını verip teoremi anlamaya hazır hale geleceğiz ve bir örnek ile sunumu sonlandıracağız.
Ebru Nayir (GS): Finite Groups of order less than 23
Sonlu grupları sınıflandırmaya da yarayan Sylow teoremlerini ve bir takım araçları kullanarak mertebesi 12 olan tüm grupları belirlemek.
Furkan Öztürk (Bilkent): Ortogonal Fonksiyonlar, Sturm-Liouville Teorisi ve Kuantum Mekaniğindeki Uygulamaları
Konuşma uygulamalı matematik, fonksiyonel analiz ve quantum mekaniği üzerine olacaktır. Linear vektör uzayları ve genelleştirilmiş Fourier serileri ile bir matematiksel başlangıç yapacağım. Ardından Hermitik işlemcilerin teorisini anlatıp bunlar ile ilgili fiziksel uygulamaları anlatacağım. Bunların hepsini içine alan daha genel bir teori olan Sturm-Liouville teorisine bir giriş yaptıktan sonra, Schrödinger denklemini, bu teorinin özel bir uygulaması olarak inceleyeceğim. Bu özdeğer denkleminin çözümleri olan özhal fonksiyonlarını matematiksel olarak inceledikten sonra, bunların fiziksel karşılığı olan dalga fonksiyonunun analizini yapacağım.
Merve Seçgin (Uludağ): Topological surgeries of 3-manifolds
It is known that every closed 3-manifold can be obtained by a topological (p/q)-surgery along a link. In this talk we will give definition of topological surgery and show how some 3-manifolds can be seen as a topological surgery along a knot.
Ergün Süer (Bilgi): Representations of SU(2)
The essence of harmonic analysis is to decompose complicated expressions into pieces that reflect the structure of a group action when there is one. The goal is to make some difficult analysis manageable. In the seventeenth and eighteenth centuries, the groups that arose in this connection were the circle group, the real line, and finite abelian groups. In the case of the circle, the decomposition is just the expansion of a periodic function on reals into its Fourier series. In this talk, we try to explain the Peter-Weyl theorem and classify the representations of the special unitary group SU(2).
Ersin Süer : A Classical Approach to the Class Number Problems
We will give a brief exposition of integral binary quadratic forms and a classical approach to the class number problems. Using the relation between form class group and ideal class group, if time permit, we will compute class number of some quadratic fields.
Yasin Emre Üsküplü (koç): Shortest Distance Between Skew Lines
We will show the existence of the shortest distance between skew lines and then we will derive an explicit formula for this distance. In order to understand this talk basic linear algebra and multivariable calculus(in 2-dimension) is enough.
Baran Zadeoğlu (Bilkent): An elementary proof of the nullstellensatz
Hilbert's nullstellensatz is one of the central theorems in algebraic geometry. Its proof usually involves using a fair amount of algebra. In 2006, an elementary proof of it got published by Enrique Arrondo. The proof uses basic algebra notions. I will present this proof.
I can also be found at: http://www.mat.ucm.es/~arrondo/monthly169-171-arrondo.pdf
