12 Haz TMD Lisans ve Lisansüstü Yazokulu Tüm Derslerin İçerikleri
Toplu Ders İçerikleri
Title of the course: Bilim ve bilimsel gelişmeler
Instructor: Dr. Vedat Tanrıverdi
Institution: –
Dates: 15-21 July 2019
Prerequisites: –
Level: Beginning undergraduate
Abstract: 14’üncü yüzyıla kadar bilimsel gelişmelerin kısa bir özeti ile başlanacak Avrupa’da bilimsel gelişmelerin başlama süreci ile devam edilecektir. Bilimin ve bilimsel yöntemin tanımı ve bilimsel etiğin nasıl olması gerektiğine dair konulara değinilecektir.
Language: TR
Title of the course: Lie ve Leibniz cebirlerine giriş
Instructor: Dr. Nil Mansuroğlu
Institution: Ahi Evran Üniversitesi
Dates: 15-22 July 2018
Prerequisites: –
Level: Graduate, advanced undergraduate
Abstract:
15.07.19 Lie cebir, Lie alt cebir tanımı ve örnekleri
16.07.19 Lie cebirlerinde ideal tanımı ve özellikleri
17.07.19 Nilpotent Lie cebirleri, Çözülebilir Lie cebirleri
18.07.19 –
19.07.19 Leibniz cebir, Leibniz alt cebir tanımı ve örnekleri
20.07.19 Leibniz cebirlerinde ideal tanımı ve özellikleri, Lie cebirleri ile Leibniz cebirleri arasındaki farklar ve benzerlikler
21.07.19 Nilpotent Leibniz cebirleri, Çözülebilir Leibniz cebrleri
Language: TR
Title of the course: Ağaçlar
Instructor: Prof. Ali Nesin
Institution: İstanbul Bilgi Ü.
Dates: 15-21 July 2019
Prerequisites: –
Level: Beginning undergraduate, undergraduate
Abstract: Bazı kavramlar matematiğin hemen her dalında karşımıza çıkarlar. “Ağaç” kavramı bunlardan biridir. Örneğin, her aşamada iki karardan birini verdiğiniz bir oyun düşünün, bu aslında bir ağaçtır. Ya da bir yol düşünün, her 100 metrede bir üçe ayrılıyor, ya da bazen ikiye bazen üçe ayrılıyor, ama hiç geriye dönmüyor. Bu da bir ağaçtır. Bu derste, ağaçlarla matematiğin çeşitli konuları arasındaki bağı göreceğiz. Reel sayılar, tam sayılar, oyunlar, p-sel sayılar, olasılık, polinomlar içinde ağaç göreceğimiz konulardan bazıları olacak.
Language: TR, EN
Title of the course: Scissors Congruence
Instructor: Prof. Murad Özaydın
Institution: University of Oklahoma
Dates: 15-21 July 2019
Prerequisites: Calculus, abstract math, linear algebra
Level: Graduate, advanced undergraduate, beginning Undergraduate
Abstract: Hilbert’s 3rd problem (1900) asked if two polyhedra with the same volume can be subdivided into a finite number of smaller polyhedra so that each piece of the first polyhedron is congruent to one of the second. Two such polyhedra are said to be Scissors Congruent. The corresponding statement for polygons in the plane was probably known in antiquity, but the first known proof is Wallace (1807). Dehn (1901) proved that a cube and a regular tetrahedron of the same volume are not Scissors Congruent and the invariant he defined for this purpose along with volume was shown by Sydler (1965) to be a complete set of invariants of Scissors Congruence in 3-space. A similar result in 4 dimensions was proven by Jessen (1972). In higher dimensions analogous questions are still open.
I plan to cover the theorems of Wallace (Scissors Congruence = Area in the plane) and Dehn (solution to Hilbert’s 3rd problem) and hope to mention some more recent developments.
Language: EN, TR
Title of the course: Ramsey Theory
Instructor: Dr. Jeffrey Bergfalk
Institution: UNAM Morelia
Dates: 15-21 July 2019
Prerequisites: Some level of experience and comfort with proofs.
Level: Graduate, advanced undergraduate, beginning undergraduate
Abstract: We intend firstly a survey of classical Ramsey theory —- of Ramsey’s original theorem(s), of the Van der Waerden, Hales-Jewett, and Hindman theorems, for example. We hope to suggest such theorems’ importance in some broader mathematical questions as well. Time permitting, we’ll also touch on colorings of uncountable cardinalities.
Textbook or/and course webpage: The standard reference Ramsey Theory, by Graham, Rothschild, and Spencer, gives a good picture of what classical material we have in mind.
Language: EN
Eğitmen: Doç. Dr. Özlem Beyarslan
Kurum: Boğaziçi Ü.
Tarih: 15-21 Temmuz 2019
Dersin Adı: Sayılar Kuramı
İçerik: Doğal Sayılar, tamsayılar, tümevarım, bölünebilme, en büyük ortak bölen, öklid algoritması, asallar, aritmetiğin temel teoremi, modüler aritmetik, aritmetik fonksiyonlar ve özellikleri.
Title of the course: Introduction to Group Theory, Examples, Problem Hours
Instructor: MSc. Kübra Dölaslan
Institution: ODTÜ
Dates: 15-28 July 2019
Prerequisites: The student should attend the course with the same name given by Ali Nesin.
Level: Beginning undergraduate, undergraduate
Abstract: We will discuss some examples and problems about the concepts taught in the main lecture by Ali Nesin.
Language: TR, EN
Title of the course: Introduction to Axiomatic Set Theory
Instructor: Prof. Alexei Muravitsky
Institution: Northwestern State University
Dates: 15-28 July 2019
Prerequisites: There are no special technical prerequisites; however, familiarity with the grammar and terminology of set-theoretic language and that of formal logic would be a great help.
Level: Graduate, advanced undergraduate
Abstract: The course adopts an intuitive stance to the subject (set theory) before launching into formal axiomatic development. This policy is supposed to engender an easy feel for set-theoretic concepts. Prerequisites: There are no special technical prerequisites; however, familiarity with the grammar and terminology of set-theoretic language and that of formal logic would be a great help. E-copies of the books that are listed below will be provided.
Language: EN
Textbooks: (Main text) K. Hrbacek and T. Jech, Introduction to Set Theory, third edition, revisited, and expanded, Marcel Dekker, 1999.
(supplementary texts)
• R. L. Vaught, Set Theory, An Introduction, second edition, Birkhäuser, 1995.
• J. Barwise (editor), Handbook of Mathematical Logic, Elsevier, 1993.
• I. Lavrov and L. Maksimova, Problems in Set Theory, Mathematical Logic and the Theory of Algorithms, Kluwer Academic/Plenum Publishers, 2003.
• A. Levy, Basic Set Theory, Springer-Verlag, 1979.
• J. Malitz, Introduction to Mathematical Logic: Set Theory, Computable Functions, Model Theory, Springer-Verlag New York, Inc., 1979.
• P. Halmos, Naïve Set Theory, Springer-Verlag, 1974.
• M. Hallett, Cantorian Set Theory and Limitation of Size, Clarendon Press, 1984.
Eğitmen: Prof. Ali Nesin
Kurum: İstanbul Bilgi Ü.
Tarih: 15-28 Temmuz 2019
Dersin Adı: Grup örnekleri ve temel grup teori
İçerik: Grup teori oldukça zor bir konudur, kolay kolay özümsenmez genellikle. Bu derste onlarca grup örneği vererek grup teorisini öğrenci için daha kolay anlaşılır bir hâle getirmeye çalışacağım. Başlangıçta hemen hiç teori yapmayacağız. Zaman ilerledikçe teorinin dozunu artıracağım. Yoğunlaşma gerektiren bir ders olacak.
Eğitmen: Prof. Yusuf Ünlü
Kurum: Yeditepe Ü.
Tarih: 15-28 Temmuz 2019
Dersin Adı: Halka Örnekleri
İçerik: Tam sayılar halkası. Z’de Bölme algoritması. Z[i] Gauss halkası. Z[i] de bölme algoritması. Polinomlar. Q rasyonel sayılar halkası. Q[X] polinom halkası.
Q[X] de bölme algoritması. Öklidyen bölgeler. Tek türlü çarpanlara ayrılabilme. Gauss önsavı. Z[X] de tek türlü çarpanlara ayrılabilme.
Eğitmen: MSc. Kübra Dölaslan
Kurum: ODTÜ
Tarih: 15-28 Temmuz 2019
Dersin Adı: Problem Saati
İçerik: Lise programındaki derslerde öğretilen kavramlar üzerine problemler sorup birlikte tartışacağız.
Title of the course: Temel türevli (differential) denklemlerin fizikte kullanımı
Instructor: Dr. Vedat Tanrıverdi
Institution: –
Dates: 22-28 July 2019
Prerequisites: –
Level: Beginning undergraduate
Abstract: Türev ve integralin temel tanımı, Newton’un ikinci yasası, kütleçekim yasası, harmonik hareket, dalga hareketi.
Language: TR
Title of the course: Elektrik
Instructor: Dr. Vedat Tanrıverdi
Institution: –
Dates: 22-28 July 2019
Prerequisites: –
Level: Beginning undergraduate
Abstract: Coulumb yasası ile başlayıp Maxwell denklemlerinden elektirkle ilgili olanların differansiyel haline kadar işlenecektir
Language: TR
Title of the course: Numbers and Polynomials
Instructor: Prof. Alexandre Borovik
Institution: University of Manchester
Dates: 22-28 July 2019
Prerequisites: Reasonable mastery of secondary school algebra. Suitable for fresh entrants to university or even high school students (if they understand English).
Level: Beginning Undergraduate, highschool
Abstract: The course deals with basic number theory (that is, integers) and polynomials, as concrete, sensible, sound, familiar to students objects, but treats them with full proofs in an unifying approach.
1. Taking possibility of division (of integers and polynomials over a field) with remainder for granted, a sequence of results about greatest common divisors, uniqueness of factorisation, etc. — up to the Chinese Reminder Theorem and Lagrange’s Interpolation Formula, will be *proved*. More precisely, every theorem will be formulated in two versions: for integers and for polynomials, with only one of them being proved in a lecture, the other one left as an exercise for students.
2. This part of the course will be rounded up by an explanation that the Chinese Reminder Theorem and Lagrange’s Interpolation Formula are *one and the same thing*.
3. Then complex numbers and roots of unity will be introduced, and the Fundamental Theorem of Algebra stated, unfortunately, without proof.
4. And, finally a version of the Fermat’s Theorem will be proved:
The equation X^n + Y^n = Z^n, n >1, has no solutions in polynomials X = X(t), Y = Y(t), Z = Z(t) of non-zero degree, with some historical remarks.
Language: EN
Title of the course: Complex Analysis
Instructor: MSc. Andrea Tomatis
Institution: Beuth Hochschule fuer Technik Berlin
Dates: 22-28 July 2019
Prerequisites: Analysis in R^n, Linear Algebra
Level: Advanced undergraduate
Abstract: This is a basic course in complex analysis. We will treat holomorphic functions, the Cauchy-Riemann equations, conformal mappings, the Cauchy theorem, the residue theorem, Laurent series, the monodromy theorem, Picard’s theorem.
Language: EN
Textbook:
Complex Analysis, Eberhard Freitag and Rolf Busam, Springer Verlag 2009,
Complex Analysis 3rd Edition, Lars Ahlfors, McGraw Hill 1979
Title of the course: Analizde Bazı Problemler
Instructor: Prof. Mehmet Cenkci
Institution: Akdeniz Ü.
Dates: 22-28 July 2019
Prerequisites: Temel analiz dersleri.
Level: Lisans, lise (ileri).
Abstract: Fonksiyonlar, diziler ve limitler, eşitsizlikler, türev ve integral konularından ilginç problemler çözülecektir.
Language: TR
Textbook or/and course webpage:
1. H.İ. Karakaş, İ. Aliyev, Analiz ve Cebirde İlginç Olimpiyat Problemleri ve Çözümleri, Palme, 2012.
2. W.J. Kaczor, M.T. Nowak, Problems in Mathematical Analysis, Volumes I, II, and III, AMS, 2000, 2001, 2003.
3. B. Gelbaum, Problems in Analysis, Springer, 1982.
4. P.N. de Souza, J.-N. Silva, Berkeley Problems in Mathematics, Springer, 2004.
Title of the course: Analitik sayılar teorisinden seçme konular
Instructor: Dr. Cihan Pehlivan
Institution: –
Dates: 22-28 July 2019
Prerequisites: –
Level: Undergraduate, graduate
Abstract: Basic Notations, Some elementary Sieves, The normal order method, The Turan Sieve, The sieve of eratosthenes.
Language: EN, TR
Title of the course: Zeta Functions and the Heisenberg Group
Instructor: Dr. E. Mehmet Kıral
Institution: Sophia University
Dates: 22-28 July 2019
Prerequisites: Enough mathematical maturity to not be dismayed by seeing a Fourier transform, or hearing the phrase representation theory. We can define any desired term, but you should be comfortable with not necessarily knowing the whole theory that is behind them with full rigour.
Although no particular knowledge is absolutely necessary, it is a reasonable assumption that if ALL of the following phrases are completely new to you, then you would have a difficult time trying to follow the lectures (some new words is ok): Square integrable functions, Hilbert Space, Unitary Operator, Unitary Representation of a group, Riemann Zeta function, Functional Equation, Gamma function, holomorphic function (analytic continuation), Matrix Group, Diagonalization.
Level: Graduate, advanced undergraduate
Abstract: I would like to emphasize the role of harmonic analysis in the functional equations of the Riemann zeta function, the Hurwitz zeta functions and other L-functions. The Fourier transform and its relation to the functional equation is encoded in the representation theory of the Heisenberg group.
Further topics at the intersection of number theory and harmonic analysis can be discussed.
Language: TR, EN
Title of the course: Quaternions
Instructor: Prof. Adrien Deloro
Institution: Sorbonne Université
Dates: 22 July – 4 August 2019
Prerequisites: Real and complex numbers. Linear algebra. Some group theory
Level: Graduate, advanced undergraduate
Abstract: Quaternions were discovered in the XIXth century by Hamilton. They generalise complex numbers in a very interesting manner. The course will describe them, and see them in relation with linear algebra (matrix theory) and geometry (isometries of R^3).
Language: EN
Title of the course: Lie algebras
Instructor: Assoc. Prof. Şükrü Yalçınkaya
Institution: İstanbul Ü.
Dates: 22 July – 4 August 2019
Prerequisites: Good knowledge of linear algebra, basics of group theory.
Level: Graduate, advanced undergraduate
Abstract: In the first week, we study some of the fundamental facts about Lie algebras over the complex field. We start by various important examples of Lie algebras and continue with discussing exponential map, solvable and nilpotent Lie algebras and Cartan decomposition. In the second week, we study the root systems and the structure of their Weyl groups.
Language: TR, EN
Title of the course: Somut Matematik
Instructor: Dr. Öğr. Üyesi Ayhan Dil
Institution: Akdeniz Ü.
Dates: 22 July – 4 August 2019
Prerequisites: –
Level: Beginning undergraduate, highschool
Abstract:
Yineleme Problemleri
Toplamlar
Sonlu ve Sonsuz Kalkülüs
Biraz Sayılar Teorisi
Binom Katsayıları
Özel Sayılar
Üreteç Fonksiyonları
Hipergeometrik Fonksiyonlar
Language: TR
Title of the course: Black box fields
Instructor: Prof. Alexandre Borovik
Institution: University of Manchester
Dates: 29 July – 4 August 2019
Prerequisites: Linear algebra, basic abstract algebra
Level: Graduate and advanced undergraduate
Abstract:
* Algebraic cryptography: finite fields, the Diffie-Hellman key exchange, some other basic protocols.
* One-way functions, the Discrete Logarithm Problem.
* Homomorphic encryption.
* Basic concepts of black box algebra. Black box groups, rings, fields.
* Structural analysis of black box fields of characteristic 2.
Language: EN
Title of the course: Complex Analysis
Instructor: MSc. Andrea Tomatis
Institution: Beuth Hochschule fuer Technik Berlin
Dates: 22-28 July 2019
Prerequisites: Analysis in R^n, Linear Algebra
Level: Advanced undergraduate
Abstract: This is a basic course in complex analysis. We will treat holomorphic functions, the Cauchy-Riemann equations, conformal mappings, the Cauchy theorem, the residue theorem, Laurent series, the monodromy theorem, Picard’s theorem.
Language: EN
Textbook:
Complex Analysis, Eberhard Freitag and Rolf Busam, Springer Verlag 2009,
Complex Analysis 3rd Edition, Lars Ahlfors, McGraw Hill 1979
Title of the course: Analytic Tools in Number Theory
Instructor: Prof. Mehmet Cenkci
Institution: Akdeniz Ü.
Dates: 29 July – 3 August 2019
Prerequisites: Necessary background will be summarized during the course.
Level: Graduate, advanced undergraduate
Abstract: We will discuss topics in Chapter 9 of Cohen’s book, Number Theory Volume II: Analytic and Modern Tools. More specifically, Bernoulli numbers and polynomials, real and complex gamma functions, integral transforms and Bessel functions will be the topics to be covered.
Language: TR, EN
Textbook or/and course webpage:
1. H. Cohen, Number Theory, Volume II: Analytic and Modern Tools, Springer, 2007.
Title of the course: Group Actions and Sylow Theorems
Instructor: Prof. Ali Nesin
Institution: İstanbul Bilgi Ü.
Dates: 29 July – 4 August 2019
Prerequisites: Temel grup teorisi
Level: Graduate, advanced undergraduate
Abstract: As in the title.
Language: TR, EN
Title of the course: Group Actions and Sylow Theorems, Problem Hours
Instructor: MSc. Kübra Dölaslan
Institution: ODTÜ
Dates:29 July – 4 August 2019
Prerequisites: The student should attend the course with the same name given by Ali Nesin.
Level: Graduate, advanced undergraduate
Abstract: We will discuss some examples and problems about the concepts taught in the main lecture by Ali Nesin.
Language: TR, EN
Title of the course: Elementer Sayılar Teorisi
Instructor: Dr. Cihan Pehlivan
Institution: –
Dates: 29 July – 4 August 2019
Prerequisites: –
Level: Lisans 1, 2
Abstract: Özel aritmetik dizilerde asalların sonsuzluğu, Fermat Sayıları, Mersenne sayıları, Fermat’ın küçük teoremi ve genelleştirilmesi, çarpanlara ayırma algoritmaları… (Dersimiz öğrencilerle aktif bir şekilde işlenecek olup, isteyen öğrenciler bazı derslerde kısa sunumlar yapacaktır).
Language: TR, EN
Title of the course: Introduction to Randomised Algorithms
Instructor: Dr. Tuğkan Batu
Institution: London School of Economics
Dates: 29 July – 4 August 2019
Prerequisites: Familiarity with basic (discrete) probability theory is helpful, but necessary background will be covered in the course.
Level: Graduate, advanced undergraduate
Abstract: This course will be a brief introduction to randomised algorithms. We will start with reviewing some tools from discrete probability theory that are commonly used in the design and the analysis of randomised algorithms. We will then illustrate the use of randomisation in computation through examples.
Language: EN
Eğitmen: Prof. Ali Nesin
Kurum: İstanbul Bilgi Ü.
Tarih: 29 Temmuz – 11 Ağustos 2019
Dersin Adı: Kombinasyon Hesapları, Oyunlar, Olasılık
İçerik: Kombinasyon hesapları oldukça geniş bir konudur. Bu derste her liselinin bilmesi gerekenleri kanıtlarıyla beraber gösterdikten ve birçok örnek verdikten sonra, üreteç fonksiyonlarından hareketle çok daha derin sonuçlara varacağız. Ayrıca kombinasyopn hesaplarını olasılık ile harmanlayarak çeşitli oyun analizleri yapacağız.
Eğitmen: MSc. Kübra Dölaslan
Kurum: ODTÜ
Tarih: 29 Temmuz – 11 Ağustos 2019
Dersin Adı: Sayılar Kuramı
İçerik: TBA
Eğitmen: Doç. Dr. Özlem Beyarslan
Kurum: Boğaziçi Ü.
Tarih: 29 Temmuz – 11 Ağustos 2019
Dersin Adı: Çizge Kuramı
İçerik: Çizgeler teorisinin temel tanımları, güvercin yuvası prensibi, sayma kuralları, çizgelerde tümevarım, düzlemsel çizgeler, çizgeleri boyama, Ramsey sayıları, sonsuz Ramsey teoremi, Hall evlilik teoremi, ağaçlar.
Title of the course: Black Box rings
Instructor: Prof. Alexandre Borovik
Institution: University of Manchester
Dates: 5-11 August 2019
Prerequisites: Linear algebra, basic abstract algebra
Level: Graduate and advanced undergraduate
Abstract:
* Homomorphic encryption
* Basic concepts of black box algebra. Black box groups, rings, fields.
* Structural analysis of black box matrix rings over finite fields.
Language: EN
Title of the course: Probability with problems
Instructor: Dr. Kerem Altun
Institution: Işık Ü.
Dates: 5-11 August 2019
Prerequisites: High school level probability theory
Level: Beginning undergraduate
Abstract: We will discuss probability theory through problems available in the Android app titled “Probability Puzzles”. The app covers fundamental concepts in probability theory, starting from basic level dice-rolling problems to advanced level problems involving Markov chains and martingales. We will try to cover all problems in the app, pushing the students in class to their limits.
Language: TR, EN
Title of the course: Introduction to Ring Theory
Instructor: Prof. Ali Nesin
Institution: İstanbul Bilgi Ü.
Dates: 5-11 August 2019
Prerequisites: –
Level: Beginning undergraduate, undergraduate
Abstract: As in the title.
Language: TR, EN
Title of the course: Introduction to Ring Theory, Problem Hours
Instructor: MSc. Kübra Dölaslan
Institution: ODTÜ
Dates: 5-11 August 2019
Prerequisites: The student should attend the course with the same name given by Ali Nesin.
Level: Graduate, advanced undergraduate
Abstract: We will discuss some examples and problems about the concepts taught in the main lecture by Ali Nesin.
Language: TR, EN
Title of the course: Field Theory
Instructor: Assoc. Prof. Özlem Beyarslan
Institution: Boğaziçi Ü.
Dates: 5-11 August 2019
Prerequisites: Linear Algebra
Level: Graduate, advanced undergraduate
Abstract: Field extensions, splitting fields and normal extensions, algebraic closure, inseperable extensions, finite fields.
Language: TR, EN
Title of the course: Nonlinear Dynamics and Chaos
Instructor: Dr. Deniz Eroğlu
Institution: Kadir Has Ü.
Dates: 5-11 August 2019
Prerequisites: Single-variable calculus, curve sketching, Taylor series, separable differential equations, linear algebra.
Level: Graduate, advanced undergraduate
Abstract: Dynamical system theory is interested in the evolution of systems. It tries to understand the processes in motion and limitation (stability) of systems. Chaos in dynamics is one of the scientific revolutions of the twentieth century, that deepened our understanding of the nature of unpredictability. In this course, we will discuss the fundamentals of these theories.
Language: TR; EN
Title of the course: Introduction to Module Theory
Instructor: Prof. Ali Nesin
Institution: İstanbul Bilgi Ü.
Dates: 5-11 August 2019
Prerequisites: Ring theory
Level: Graduate, advanced undergraduate
Abstract: As in the title.
Language: TR, EN
Title of the course: Introduction to Module Theory, Problem Hours
Instructor: MSc. Kübra Dölaslan
Institution: ODTÜ
Dates: 5-11 August 2019
Prerequisites: The student should attend the course with the same name given by Ali Nesin.
Level: Graduate, advanced undergraduate
Abstract: We will discuss some examples and problems about the concepts taught in the main lecture by Ali Nesin.
Language: TR, EN
Title of the course: PIDs and UFDs
Instructor: Prof. Ali Nesin
Institution: İstanbul Bilgi Ü.
Dates: 12-18 August 2019
Prerequisites: Basic algebra
Level: Graduate, advanced undergraduate
Abstract: We will prove basic results about PID’s and UFD’s. We will try to give some nontrivial examples.
Language: TR, EN
Title of the course: PIDs and UFDs, Problem Hours
Instructor: MSc. Kübra Dölaslan
Institution: ODTÜ
Dates: 12-18 August 2019
Prerequisites: The student should attend the course with the same name given by Ali Nesin.
Level: Graduate, advanced undergraduate
Abstract: We will discuss some examples and problems about the concepts taught in the main lecture by Ali Nesin.
Language: TR, EN
Title of the course: Galois Theory
Instructor: Assoc. Prof. Özlem Beyarslan
Institution: Boğaziçi Ü.
Dates: 12-18 August 2019
Prerequisites: Linear algebra and familiarity with algebraic structures, like groups rings fields.
Abstract: Galois extensions, examples and applications,, cyclic extensions solvable and radical extensions.
Level: Graduate, advanced undergraduate
Language: TR, EN
Title of the course: Coxeter Groups
Instructor: Prof. Piotr Kowalski
Institution: Uniwersytet Wrocławski
Dates: 12-18 August 2019
Prerequisites: Undegraduate group theory and linear algebra.
Level: Graduate, advanced undergraduate
Abstract: We will define Coxeter groups and discuss examples including finite reflection groups and affine Weyl groups. We will proceed to the Coxeter-Dynkin diagrams and discuss the connections with platonic solids, linear matrix groups and finite simple groups.
Language: EN
Title of the course: Projective Geometry
Instructor: Prof. Michel Lavrauw
Institution: Sabancı Ü.
Dates: 12-25 August 2019
Prerequisites: The students are assumed to be familiar the basic notions from algebra such as matrices, vector spaces, fields, rings, and groups.
Level: Graduate, advanced undergraduate
Abstract: We will start from first principles and treat the basics of projective geometry over fields. On the way we will touch upon the axiomatic approach to the subject, its connections to the theory of projective planes, incidence geometry and Tits buildings. Some first examples leading towards concepts from algebraic geometry will also be given.
Language: EN
Eğitmen: Mr. Ali Törün
Kurum: –
Tarih: 12-25 Ağustos 2019
Dersin Adı: Meşhur Problemler ve Hikâyeleri
İçerik: Matematik tarihinde yer almış bazı meşhur popüler matematik problemlerinin kısa hikâyesi ve ünlü matematikçilerin bu problemlere getirdikleri çözümlerin incelenmesi.
Eğitmen: Prof. Ali Nesin
Kurum: İstanbul Bilgi Ü.
Tarih: 12-25 Ağustos 2019
Dersin Adı: Oyun ve Matematik
İçerik: Şans oyunları dahil, oyunların içindeki matematikten sözedeceğiz. Tabii ki olasılık işin işine girecek, ama şans oyunları dışındaki oyunları da irdeleyip kazanan strateji konusunu irdeleyeceğiz. Nash dengesi de işleyeceğimiz konulara dahil olacak, ama daha çok örneklerden yola çıkacağımız için izlemesi kolay bir ders olacağını düşünüyorum.
Eğitmen: Doç. Dr. Özlem Beyarslan
Kurum: Boğaziçi Ü.
Tarih: 12-25 Ağustos 2019
Dersin Adı: Sayılar Kuramı
İçerik: Doğal Sayılar, tamsayılar, tümevarım, bölünebilme, en büyük ortak bölen, öklid algoritması, asallar, aritmetiğin temel teoremi, modüler aritmetik, aritmetik fonksiyonlar ve özellikleri.
Eğitmen: MSc. Kübra Dölaslan
Kurum: ODTÜ
Tarih: 12-25 Ağustos 2019
Dersin Adı: Problem Saati
İçerik: Lise programındaki derslerde öğretilen kavramlar üzerine problemler sorup birlikte tartışacağız.
Title of the course: An introduction to social choice theory
Instructor: Prof. Remzi Sanver
Institution: CNRS, Universite Paris Dauphine
Dates: 19-25 August 2019
Prerequisites: Temel mantık kavramlarına aşina olmak.
Level: Graduate ile advanced undergraduate arası
Abstract: The course introduces basic concepts and results of the theory of decision making, including May’s characterization of majoritarianism and Arrow’s impossibility theorem.
Language: TR, EN
Title of the course: Valued Fields
Instructor: Prof. Martin Hils
Institution: Universität Münster
Dates: 19-25 August 2019
Prerequisites: Field theory and galois theory
Level: Graduate, advanced undergraduate
Abstract: Valuations, places and valuation Rings, Discrete valuations. Extensions of valuations. Decomposition and inertia group, linear disjointness of fields.
Language: TR, EN
Title of the course: From primes to prime ideals, between algebra and geometry
Instructor: Dr. Matteo Paganin
Institution: Sabancı Ü.
Dates: 19-25 August 2019
Prerequisites: A basic knowledge of ring, ideals and polynomial rings is enough.
Level: Advance Undergradate / Graduate
Abstract: The aim of this course is to give a concise summary of the process of substituting prime numbers with prime ideals (in the explicit context of number rings and rings of polynomials) and finally to switch to the geometric point of view defining the spec of a ring and giving a brief correspondence between the geometrical and algebraic points of views.
Language: TR, EN
Title of the course: Analytic Number Theory
Instructor: Dr. Şermin Çam Çelik
Institution: Özyeğin Ü.
Dates: 19-25 August 2019
Prerequisites: Calculus
Level: Graduate, advanced undergraduate, beginning undergraduate
Abstract: Aritmetical Functions, Dirichlet Product, Multiplicative Functions, Euler’s Summation Formula, Chebyshev’s Functions, Some Equivalent Forms of the Prime Number Theorem.
Textbook: Introduction to Analytic Number Theory, Tom M. Apostol
Language: TR, EN
Title of the course: Operator Theory on Hilbert Spaces- an Introduction
Instructor: Dr. Elif Uyanık, Dr. Başak Koca, Assoc. Prof. Uğur Gül
Institution: Yozgat Bozok Ü., İstanbul Ü., Hacettepe Ü.
Dates: 19 August – 8 September 2019
Prerequisites: Yüksek lisans düzeyinde reel analiz bilgisi
Level: Yükseklisans
Abstract:
I. week (Elif UYANIK)
I. Functional Analysis Preliminaries: normed spaces, Banach spaces, bounded operators
II. Hahn-Banach theorem, dual maps
III. Baire Category theorem and consequences.
IV. Inner product spaces, Cauchy-Schwarz inequality, Hilbert spaces.
V. Hilbert Spaces, Orthonormal systems, basis
II. week (Başak KOCA)
I. Banach algebras, spectrum, spectral radius, unitization, Neumann series.
II. Gelfand-Mazur theorem, Spectral mapping theorem, spectral Radius formula.
III. Closed ideals, maximal ideals, maximal ideal space.
IV. Commutative Banach algebras, Gelfand transform.
V. C*-algebras, commutative C*-algebras, Gelfand-Naimark Theorem.
III. week (Uğur GÜL)
I. Operators on Hilbert spaces, Self-adjoint, unitary, normal operators, Isometries.
II. Spectral resolution of normal operators, L\infty functional calculus.
III. Topologies on B(H): weak operator topology, strong operatör topology, Von-Neumann bicommutant theorem, Von Neumann algebras
IV. projections and projection lattices in B(H)
V. Commutative Von-Neumann algebras.
References: 1. “Introduction to Functional Analysis” by R. Meise and D. Vogt, Oxford Science Publications, 1997
2. “C*-algebras and Operator Theory” by G. Murphy, Academic Press, 1990
3. “Fundamentals of the Theory of Operator Algebras” by R. Kadison and J. R. Ringrose, Academic Press, 1983.
Language: EN
Title of the course: Basic 3-Manifold Topology
Instructor: Dr. Agustin Moreno
Institution: University of Augsburg
Dates: 19 August – 1 September 2019
Prerequisites: Point set topology, basic algebraic topology
Level: Graduate, advanced undergraduate
Abstract: In this lecture, we will cover basic topics in 3-mfld topology, prime decomposition, torus decomposition, Seifert manifolds, JSJ Decomposition, Geometrization Conjecture.
Textbook or/and course webpage: Allen Hatcher-Notes on Basic 3-Manifold Topology
Language: EN
Title of the course: Classification of surfaces
Instructor: Dr. Merve Seçgin
Institution: TED Ü.
Dates: 19 August – 1 September 2019
Prerequisites: Point set topology, Basic algebraic topology
Level: Graduate, advanced undergraduate
Abstract: In this lecture, we will touch upon the classification of surfaces, triangulation of surfaces and relation with knot theory.
Textbook or/and course webpage: An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman
Language: TR, EN
Title of the course: Topics in Number Theory
Instructor: Asst. Prof. Haydar Göral
Institution: Dokuz Eylül Ü.
Dates: 19-25 August 2019
Prerequisites: Calculus
Level: Graduate, advanced undergraduate, beginning Undergraduate
Abstract: Infinitude of primes, sums over primes, arithmetic progressions, additive number theory.
Language: TR, EN
Title of the course: Arf Rings
Instructor: Prof. Ali Sinan Sertöz
Institution: Bilkent Ü.
Dates: 26 August – 1 September 2019
Prerequisites: Arf uses only elementary techniques but with an admirable degree of mathematical maturity. Therefore a must prerequisite is willingness to learn something new. For undergrads, if you know what a ring is then you are set to go. If not, I will define that too!
Level: Graduate, advanced undergraduate
Abstract: The aim of the course is to explain the original definition and the use of Arf rings as Arf himself did. We will start by defining the basics of curve singularities, their resolutions and the multiplicities involved. The motivating problem is to find these multiplicities that occur during the resolution process. I will explain Du Val’s geometric solution. At this point Du Val poses the challenge of finding these multiplicities once the local parametrization of the singularity is given. Arf solves this algebraic problem by defining some special rings which he called canonical rings and which are today known as Arf rings. I will go step by step through Arf’s original article and explain his solution and calculations. At the end, if time permits I may talk about Lipman’s generalization of Arf rings in the language of commutative algebra.
Language: EN, TR
Title of the course: Homological Algebra
Instructor: Asst. Prof. Ben Walter
Institution: ODTÜ Kıbrıs
Dates: 26 August – 1 September 2019
Prerequisites: Basic knowledge of groups, rings, and modules
Level: Basic knowledge of groups, rings, and modules
Abstract:
(We will follow Weibel’s homological algebra book.)
Week 1 will be chapters 1—3 (and possibly parts of 4.)
Chain Complexes: definitions, operations, long exact sequences, chain homotopies, mapping cones and cylinders.
Derived Functors: projective and injective resolutions, left and right derived functors, adjoints.
Tor and Ext: Basics, derived functors of inverse limit, universal coefficient theorems.
(Time permitting, we will include a bit of Koszul complexes from chapter 4.)
Language: EN
Textbook or/and course webpage: Weibel, C. An Introduction to Homological Algebra
Title of the course: Kısmi Türevli Denklemlerde Seçme Konular
Instructor: Assoc. Prof. Erhan Pişkin
Institution: Dicle Ü.
Dates: 2-8 September 2019
Prerequisites: –
Level: Lisans
Abstract: Birinci Mertebeden Kısmi Diferansiyel Denklemler, Yüksek Mertebeden Sabit Katsayılı Kısmi Diferansiyel Denklemler, Dalga Denklemi, D’Alembert Çözümü, Duhamel Prensibi, Değişkenlere Ayırma Yöntemi.
Language: TR
Title of the course: Ultrafilters and how to use them
Instructor: Asst. Prof. Burak Kaya
Institution: ODTÜ
Dates: 26 August – 1 September 2019
Prerequisites: No prior knowledge will be needed except some mathematical maturity. For the second half of the course only, some familiarity with algebraic structures and topological concepts are suggested, but not required.
Level: Graduate, advanced undergraduate, beginning Undergraduate
Abstract: In this course, we shall learn about ultrafilters and their applications in mathematics. The first half of the course will cover the basics (i.e. the construction, properties and different types of ultrafilters) and the second half of the course will cover some applications (e.g. the ultraproduct and ultralimit constructions, Stone-Cech compactification of a discrete space.)
Language: TR, EN
Title of the course: A Rigorous Introduction to Basic Probability Theory
Instructors: Arif Mardin
Institution: –
Dates: 26 August – 1 September
Prerequisites: Familiarity with sets, elementary operations on them, as well as some basic properties of combinatorics are what is needed to follow this course.
Level: Undergradute
Abstract: The aim of this course is to present the calculus of discrete probability through the basic axioms of probability theory as formulated by A.N.Kolmogorov. Fundamental notions such as probability space, sigma-algebra of events, random variables, independence, Borel-Cantelli lemmas, different forms of convergence of random variables, weak and strong laws of large numbers, the central limit theorem will be studied.
References:
(i) Uluğ Çapar: “Olasılık Teorisinin Gelişimi II-VI”, Matematik Dünyası, sayı 101-105; 2014-2015;
(ii) J. R. Rosenthal: “A First Look at Rigorous Probability Theory”, 2nd edn., World-Scientific, 2006;
(iii) A.N.Shiryaev: “Probability”, 2nd edn., Springer Verlag, 1996; Chapter I, pages: 1-130.
Title of the course: Introduction to Category Theory
Instructor: Dr. Matteo Paganin
Institution: Sabancı Ü.
Dates: 26 August – 1 September 2019
Prerequisites: Some group theory, some topology, the more the better, to have examples.
Level: Advance Undergradate / Graduate
Abstract: The aim of the course is to show how most of the common contructions in Mathematics are can be described with a common language, turning out to be so called adjoints.
In this course we plan to give definitions, (plenty of) examples, and basic properties of categories, morphisms, isomorphisms, monomorphisms and epimorphisms, initial, terminal, and zero objects, functors, morphisms of functors, representable functors, and adjoints.
Language: TR, EN
Title of the course: Introduction to dynamical systems
Instructor: Prof. François Dunlop
Institution: Université de Cergy-Pontoise
Dates: 26-31 August 2019
Prerequisites: Calculus
Level: Graduate, advanced undergraduate
Abstract: A dynamical system can be thought of as a function which is composed with itself over and over again. What is the behaviour of the n’th iterate as n goes to infinity? The course will try to give some answers and introduce the basic concepts through examples: circle rotations, expanding maps of the circle, shifts and subshifts, quadratic maps, etc…
Language: EN
Eğitmen: Prof. Haluk Oral
Kurum: –
Tarih: 2-8 Eylül 2019
Dersin Adı: Matematik Sohbetleri
İçerik: Sayma problemleri, reel sayıların bazı özellikleri, Binom teoremi, Öklid algoritması, şifreleme.
Title of the course: Differential Geometry
Instructors: Ali Ulaş Özgür Kişisel, Kadri İlker Berktav
Institution: METU
Dates: 26 August – 8 September 2019
Prerequisites: Linear algebra, multivariable calculus
Level: Advanced undergraduate, graduate
Abstract: Geometry of manifolds equipped with a Riemannian or Lorentzian metric will be explored. The course will focus on both the essential concepts and computations. The tools developed in this course will also be used in the course titled “Gravitation” which will be concurrently offered by Bayram Tekin and İlker Berktav in the village.
Contents: Differentiable manifolds, vector fields, differential forms, tensors, integration of forms, Stokes’ theorem. Covariant derivative, connections, parallel transport, geodesics, Riemann curvature tensor, Ricci, Einstein and Weyl tensors, Maurer-Cartan equations, isometries and Killing vectors.
Language: TR, EN
References: “Foundations of Differentiable Manifolds and Lie Groups”, Frank W. Warner “Riemannian Geometry”, Peter Petersen
“Lecture Notes on General Relativity”, Sean M. Carroll
Title of the course: General Relativity: The Theory of Gravitation
Instructors: Bayram Tekin, Kadri İlker Berktav
Institution: METU
Dates: 26 August – 8 September 2019
Prerequisites: Vector and Tensor Calculus on Manifolds
Level: Advanced undergraduate, graduate
Abstract: The theory of gravitation is General Relativity. We shall give a basic level introduction to the subject. A “Differential Geometry” course that will be given by Özgür Kişisel and İlker Berktav, concurrently, must be taken by students who lack the proper background on manifolds and related matters. Contents:
1. Broad overview of General Relativity
2. Special relativity
3. Vectors and tensors on flat spacetime
4. Mathematical description of matter and fields: the energy-momentum tensors
5. Curvature
6. Einstein Field equations
7. Gravitational Waves
8. Black Holes
9. Cosmology
Language: TR, EN
References: “A First Course in General Relativity” by B. Schutz,
“Lecture Notes on General Relativity” by Sean M. Carroll
Eğitmen: Prof. Dr. Melih Boral
Kurum: –
Tarih: 26 Ağustos – 8 Eylül 2019
Dersin Adı: Sayılar kuramı
İçerik: Bölünebilme, asal sayılar, aritmetiğin temel teoremi, karelerin toplamı olarak sayılar, polinomlar ve cebirsel sayılar, sayılar kuramında ilginç problemler.
Eğitmen: Prof. Ali Nesin
Kurum: İstanbul Bilgi Ü.
Tarih: 26 Ağustos – 8 Eylül 2019
Dersin Adı: Ağaçlar
İçerik: Bazı kavramlar matematiğin hemen her dalında karşımıza çıkarlar. “Ağaç” kavramı bunlardan biridir. Örneğin, her aşamada iki karardan birini verdiğiniz bir oyun düşünün, bu aslında bir ağaçtır. Ya da bir yol düşünün, her 100 metrede bir üçe ayrılıyor, ya da bazen ikiye bazen üçe ayrılıyor, ama hiç geriye dönmüyor. Bu da bir ağaçtır. Bu derste, ağaçlarla matematiğin çeşitli konuları arasındaki bağı göreceğiz. Reel sayılar, tam sayılar, oyunlar, p-sel sayılar, olasılık, polinomlar içinde ağaç göreceğimiz konulardan bazıları olacak.
Eğitmen: Doç. Dr. Özlem Beyarslan
Kurum: Boğaziçi Ü.
Tarih: 26 Ağustos – 8 Eylül 2019
Dersin Adı: Çizgeler Kuramı
İçerik: Çizgeler teorisinin temel tanımları, güvercin yuvası prensibi, sayma kuralları, çizgelerde tümevarım, düzlemsel çizgeler, çizgeleri boyama, Ramsey sayıları, sonsuz Ramsey teoremi, Hall evlilik teoremi, ağaçlar.
Eğitmen: MSc. Kübra Dölaslan
Kurum: ODTÜ
Tarih: 26 Ağustos – 1 Eylül 2019
Dersin Adı: Problem Saati
İçerik: Lise programındaki derslerde öğretilen kavramlar üzerine problemler sorup birlikte tartışacağız.
Title of the course: An Introduction to Module Theory
Instructor: Asst. Prof. Roghayeh Hafezieh
Institution: Gebze Teknik Ü.
Dates: 2-8 September 2019
Prerequisites: Algebra I
Level: Graduate and advanced undergraduate
Abstract: We will discuss: Definition of module and submodule, R-module homomorphisms and exact sequences, Direct Product and Direct sums, Hom, and tensor product.
Language: EN
Title of the course: Modular numbers and p-adics
Instructor: Assoc. Prof. Özlem Beyarslan
Institution: Boğaziçi Ü.
Dates: 2-8 September 2019
Prerequisites: familiarity with concepts of algebra
Level: Beginning undergraduate, advanced undergraduate, graduate
Abstract: definitions of p-adic numbers in different contexts, Hensel’s lemma, p-adic metric and completion.
Language: TR, EN
Title of the course: Introduction to Group and Lie Algebra Homology
Instructor: Asst. Prof. Ben Walter
Institution: ODTÜ Kıbrıs
Dates: 2-8 September 2019
Prerequisites: Basic knowledge of groups, rings, and modules
Level: Basic knowledge of groups, rings, and modules
Abstract: (Ee will follow Weibel’s homological algebra book.)
Week 2 will be chapters 6—7 (with parts of 5 included as needed.)
Group Homology and Cohomology: Definitions and properties, Shapiro’s Lemma, bar resolution, universal central extensions.
Lie Algebra Homology and Cohomology: Definitions, universal enveloping algebras, H^1 and H_1, H^2 and extensions, Chevalley-Eilenberg Complex, universal central extensions.
(Spectral Sequences: To be included as needed.)
Language: EN
Textbook or/and course webpage: Weibel, C. An Introduction to Homological Algebra
Title of the course: ZFC and Vopenka’s alternative set theory
Instructor: Dr. Alena Vencovska
Institution: University of Manchester
Dates: 2-8 September 2019
Prerequisites: –
Level: Advanced undergraduate
Abstract: Basics of ZF, axiom of choice and its equivalents, Vopenka’s alternative set theory, role of nonstandard analysis within ZFC and within Vopenka’s theory.
Language: EN
Title of the course: Logic for Artificial Intelligence
Instructors: Arif Mardin
Institution: –
Dates: 2-8 September 2019
Prerequisites: Apart from motivation to follow what is going on, and familiarity with the basics of logical reasoning, no particular familiarity with any subject is needed.
Level: Undergraduate
Abstract: Logic (more precisely propositional logic and predicate logic) as a method of representation of knowledge in artificial intelligence. Well-formed formulas, unification, resolution strategies for the resolution of problems.
References:
Ali Nesin: “Önermeler Mantığı”, Nesin Matematik Köyü Kitaplığı, 2014.
Hodges, W.: “Logic”, Penguin Books, 1977.
Nilsson, N.J.: “Principles of Artificial Intelligence”, Morgan Kaufmann, 1980.
Russell, S. and Norvig, P.: “Artificial Intelligence: A Modern Approach”, 3rd edn., Prentice Hall, 2013.
Genesereth, M., and Nilsson, N.J.: “Logical Foundations of Artificial Intelligence”, Morgan Kaufmann, 1987.
Title of the course: Fundamental Mathematics for Life Sciences: Probabilities
Instructor: Dr. Andrés Aravena
Institution: İstanbul Ü.
Dates: 2-8 September 2019
Prerequisites: High school algebra. Curiosity.
Level: Advanced undergraduate, beginning graduate
Abstract: This course teaches the mathematical concepts that young biologists will need to do high-impact science in the 21st century.
Data is cheap and abundant. Scientific value comes from extracting meaningful information from this big data. People have developed several computational tools to do this data mining, but computers alone do not solve problems. Mathematics solves problems, and then computers do it faster. The question is not how to find the correct program. Instead, we have to find the correct model.
In this new situation, young scientists need different training than previous generations. They need to understand, use, and sometimes create mathematical models that allow them to interpret their results and add value to their science. In this course we will teach the concepts that are the base of many important models. We will teach classical logic and how it can be extended to become probabilities. Then we will talk about the meaning of conditional probability and independence, join probability and Bayes Theorem. Next, we will talk about expected values, variance and entropy. Next content is the Law of large numbers. We will show some Classical distributions, such as Bernoulli, Binomial, and Hypergeometric. A short discussion on the Central Limit Theorem will lead us to show the Normal distribution. If time allows, we will talk about Parametric statistical inference, in particular, confidence intervals and Hypothesis testing.
As a final project, we analyze gene expression and determine confidence intervals for the differential expression of genes.
Language: EN
Textbook or/and course webpage:
• “Probability Theory: The Logic of Science” by E. T. Jaynes, 1999
• “How to Solve It” by G. Polya, 1945
• “Introduction to Mathematical Thinking” by Keith Devlin, 2012
Title of the course: Mathematical Tools for Life Sciences: Matrices and Linear Algebra
Instructor: Dr. Andrés Aravena
Institution: İstanbul Ü.
Dates: 2-8 September 2019
Prerequisites: High school algebra. Curiosity.
Level: Advanced undergraduate, beginning graduate
Abstract: This course teaches the mathematical tools and concepts that young biologists will need to do science in the 21st century.
Experimental Sciences have changed a lot in the last decades, and they keep changing quickly. In old times, measuring a few variables was expensive and time-consuming, thus just carrying on an experiment had enough intrinsic value. This is no longer true, in all sciences in general, and in particular for molecular biology. Today experiments are performed by machines (DNA sequencing, microarrays, real-time PCR) or require cheap repetitive manual labor. Producing huge amounts of data is inexpensive and easy. The issue today, and even more in the future, is to extract meaningful information and new knowledge from the available experimental data.
The new generation of scientists need to understand how, when, and why to use applied mathematical tools, such as analytic geometry, matrices, and graphs. In other words, scientists need to know the fundamentals of linear algebra.
In this course we will learn about dynamical systems (discrete time, linear). We use vectors to represent the system state, and matrices to represent transitions and transformations of this system. Matrix multiplication then represents the composition of transitions. We will talk about the analytic geometrical interpretation of vectors and operations, such as dot product, determinants, cross products. Then we will talk about Identity matrices, matrix inversion, linear independence, and dimension. The main applications that we will discuss are descriptive statistics and multivariate linear regression using least squares. If time allows us, we will talk also about the analysis of graphs, eigenvalues, and Markov chains.
Textbook:
• “No Bullshit Guide to Linear Algebra” by Ivan Savov, 2017
• “How to Solve It” by G. Polya, 1945
• “Introduction to Mathematical Thinking” by Keith Devlin, 2012
• “Doing Math with Python” by Amit Saha, 2015
Language: EN
Title of the course: Sequences of polynomials
Instructor: Prof. Sten Kaijser
Institution: Uppsala University
Dates: 2-8 September 2019
Prerequisites: Only some mathematical maturity.
Level: Graduate, advanced undergraduate
Abstract: There are two theories about sequences of polynomials, an older theory about orthogonal polynomials and a newer theory called umbral calcculus. There are interesting connections given by the fact that some families of orthogonal polynomials can also be investigated in the context of umbral calcculus. I will talk about both theories.
Textbook or/and course webpage: There are good web pages on Wikipedia about both theories. Those pages are good enough for the course.
Language: EN
Title of the course: Hahn-Banach Teoremleri
Instructor: Prof. Zafer Ercan
Institution: AİBÜ
Dates: 2-8 September 2019
Prerequisites: –
Level: Graduate, advanced undergraduate, undergraduate
Abstract: Hahn-Banach Teoremi yoksa fonksiyonel analiz yoktur! Bu teoremin farklı biçimleri verildikten sonra, denkleri olan Sandwich Teoremi, Mazur-Orlicz Teoremi verilecek. Ayrıca bu teoremlerin normlu uzaylarda bazı uygulamalarına yer verilecek.
Language: TR, EN
Title of the course: Quadratic Forms
Instructor: Prof. Ali Nesin
Institution: İstanbul Bilgi Ü.
Dates: 9-15 September 2019
Prerequisites: Basic algebra and linear algebra.
Level: Graduate, advanced undergraduate
Abstract: We will classify quadratic forms over reals, complexes and finite fields.
Language: TR, EN
Title of the course: Finite Fields and Their Algebraic Closure
Instructor: Assoc. Prof. Özlem Beyarslan
Institution: Boğaziçi Ü.
Dates: 9-15 September 2019
Prerequisites: TBA
Level: Graduate, advanced undergraduate
Abstract: TBA
Language: TR, EN
Title of the course: Metrik Geometri
Instructor: Dr. Mehmet Kılıç
Institution: –
Dates: 9-15 September 2019
Prerequisites: Temel Analiz bilgisi ve metrik uzaylara aşinalık.
Level: Graduate, advanced undergraduate
Abstract: Bu derste, uzunluk uzayları ve jeodezik uzaylar tanıtılacaktır. Ascoli Teorem’i ifade ve ispat edilecek ve bunun sonucu olarak bir ”proper” metrik uzayda, sonlu uzunluğa sahip bir yolla birleştirilebilen iki nokta arasında, minimal uzunluğa sahip bir yolun var olduğu gösterilecektir. Daha sonra metrik uzaylarda bazı konvekslik kavramları incelenecektir.
Language: TR
Textbook: Metric Spaces, Convexity and Nonpositive Curvature; Athanase Papadopoulos.
A Course in Metric Geometry; Dmitri Burago, Yuri Burago, Sergei Ivanov.
Title of the course: Model theory via homogeneous structures
Instructor: Asst. Prof. Nick Ramsey
Institution: University of California Los Angeles
Dates: 9-15 September 2019
Prerequisites: None for the first week, either the first week or some exposure to logic for the second.
Level: Graduate, advanced undergraduate
Abstract: In the first week, we intend to introduce the language of model theory through the special case of homogeneous structures: we will develop the basic vocabulary, describe the key examples, and aim for a proof of Fraïssé’s theorem that homogeneous structures correspond to Fraïssé classes of finite structures.
Language: EN
Title of the course: Tensor and other products of modules
Instructor: Prof. Ali Nesin
Institution: İstanbul Bilgi Ü.
Dates: 16-29 September 2019
Prerequisites: Basic module theory
Level: Graduate, advanced undergraduate
Abstract: Tensor products of modules, symmetric and alternating products. Determinents.
Language: TR, EN
Title of the course: Group Actions
Instructor: Prof. Ali Nesin
Institution: İstanbul Bilgi Ü.
Dates: 16-29 September 2019
Prerequisites: Basic group theory
Level: Graduate, advanced undergraduate
Abstract: Group actions, n-transitive and sharply n-transitive groups. Applications to group theory.
Language: TR, EN
Title of the course: Transform Methods for Differential Equations – A unified approach
Instructor: Dr. Konstantinos Kalimeris
Institution: University of Cambridge
Dates: 9-15 September 2019
Prerequisites: Basic knowledge of Complex Variables (An introduction to basic Complex Analysis themes will be given in the first lectures).
Level: Advanced undergraduate and graduate
Abstract: In this course we will study a class of differential equations which appear in a plethora of physical phenomena and include the heat, the wave, the Laplace, and the Helmholtz equations. Such equations for simple geometries and simple boundary conditions are traditionally solved via separation of variables or transform methods. However, these methods are limited, and even in the particular cases that they are applicable, they have several disadvantages.
After reviewing in a coherent way some of the classical transform methods, we will present a new approach, which has been acclaimed as the first major breakthrough in the solution of linear PDEs, since the discovery of the Fourier transform in the 18th century. In this course, this Unified Transform Method will be presented, making usage of basic mathematical tools and methods of complex analysis. The students will be given an overview of the several analytical and numerical advantages that this unified approach possesses. At the last lectures of the course the students will have the opportunity to apply this knowledge in a series of problems related to differential equations.
The course will also include an introduction of complex variables.
Language: EN
Title of the course: Finite Fields
Instructor: Dr. José-Ibrahim Villanueva-Gutiérrez
Institution: Universität Heidelberg
Dates: 16-22 September 2019
Prerequisites: Linear algebra, Algebra I
Level: Beginners
Abstract: Finite fields are present in many areas of mathematics. They frequently appear in number theory: the residue field of non-archimedian local fields are precisely finite fields. Finite fields provide a very mild environment to work on. For instance, the extensions of degree pn of a finite field of characteristic p, are pairwise isomorphic. Finite fields belong to the part of the nice puzzle of mathematics, where all pieces fall straightforward.
In this course we will study the main algebraic properties of finite fields. In particular, we will study their group of automorphisms. More generally we will study the group of automorphisms of the algebraic closure Fp of a finite field Fp. To this end we will dig in the topological Galois theory. We will show that the group of such automorphisms is isomorphic to the pro-finite completion bZ of Z.
Language: EN
Textbook: Short notes and exercices to be published in https://www.mathi.uni-heidelberg.de/~jgutierrez
Other references:
Serge Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002.
Robert B. Ash. Basic abstract algebra. Dover Publications, Inc., Mineola, NY, 2007. For graduate students and advanced undergraduates.
Title of the course: p-adic numbers
Instructor: Dr. Katharina Hübner
Institution: Universität Heidelberg
Dates: 16-22 September 2019
Prerequisites: Linear algebra, Algebra I
Level: Beginners
Abstract: The p-adic numbers Qp can be constructed by taking the completion of Q with respect to the p-adic norm. This is pretty much in analogy to taking the completion of Q with respect to the standard absolute value in order to obtain R.
At first sight Qp and R might seem more complicated than Q. But from an arithmetic viewpoint they are in fact much easier to handle than Q: For instance, if we want to find roots of a polynomial in R, we can do so by approximation, e.g. using Newton’s method.
The approximation process in Qp is even easier: If the polynomial is monic and has integral coefficients (i.e. in Zp), in most cases we just have to check whether there are solutions modulo p. This result is called Hensel’s lemma.
In this course we will define the p-adic numbers and investigate their basic properties. The final goal of the week will be to prove Hensel’s lemma.
Textbook: Short notes and exercices to be published in https://www.mathi.uni-heidelberg.de/~khuebner
Other references:
1. Svetlana Katok. p-adic analysis compared with real, volume 37 of Student Mathematical Library. American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2007
2. Fernando Q. Gouvêa. p-adic numbers. Universitext. Springer-Verlag, Berlin, second edition, 1997. An introduction.
3. Alain M. Robert. A course in p-adic analysis, volume 198 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000
Language: EN
Title of the course: Homogeneous structures and their automorphism groups
Instructor: Asst. Prof. Nick Ramsey
Institution: University of California Los Angeles
Dates: 16-22 September 2019
Prerequisites: None for the first week, either the first week or some exposure to logic for the second.
Level: Graduate, advanced undergraduate
Abstract: In the second week, we will build on this, showing how the combinatorial properties of a amalgamation class of finite structures can often be related to dynamical properties of their infinite limit, via results of Kechris-Rosendal and Kechris-Pestov-Todorcevic.
Language: EN
Title of the course: Axiom of Choice and Some of Its Consequences
Instructor: Prof. Ali Nesin
Institution: İstanbul Bilgi Ü.
Dates: 23-29 July 2019
Prerequisites: Mathematical maturity and abstract algebra
Level: Graduate, advanced undergraduate
Abstract: As in the title.
Language: TR, EN
Title of the course: Introduction to General Topology
Instructor: Dr. Ahmet Çevik
Institution: JSGA/ODTÜ
Dates: 23-29 September 2019
Prerequisites: Basic logic and sets. Preliminary real analysis is suggested but not required.
Level: Advanced undergraduate
Abstract: This is a standard course on general topology. Topics we shall cover will include metric spaces, open and closed sets, topological spaces, neighborhoods, closure, interior, basis, continous functions, homeomorphism, connectedness, compactness, separation axioms, Tychonoff’s Theorem, Urysohn’s Lemma.
Language: TR, EN
Title of the course: Introduction to Iwasawa theory
Instructor: Dr. José-Ibrahim Villanueva-Gutiérrez
Institution: Universität Heidelberg
Dates: 23-29 September 2019
Prerequisites: Algebra II, Topology, Analysis
Level: Advanced undergraduate
Abstract: The fundamental theorem of arithmetic states that each integer can be written in a unique way as a product of units and powers of prime numbers. This is equivalent to say that every ideal of Z is principal, equivalently that the ideal class group of Q is trivial.
If we take an arbitrary finite extension K of Q, i.e. a number field, the fundamental theorem of arithmetic might be no longer true in the ring of integers of K. Hence, the ideal class group might be non trivial. In the 50’s the Japanese mathematician K. Iwasawa figured out a way to describe the growth of the p-part of the ideal class group in some special towers of field extensions.
Nowadays Iwasawa theory has generalised in several ways. The purpose of this course is to give an insight on Iwasawa’s original approach which is a milestone on modern number theory. In particular we will study the following subjects
The Iwasawa algebra A
Structure Theorem of noetherian A-modules
Zp-extensions to prove the above mentioned classical theorem of Iwasawa.
Textbook: Short notes and exercices in https://www.mathi.uni-heidelberg.de/~jgutierrez
Suggested references:
L. C. Washington. Introduction to cyclotomic fields, volume 83 of Graduate Texts in Mathematics.
Springer-Verlag, New York, second edition, 1997
Serge Lang. Cyclotomic fields I and II, volume 121 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990. With an appendix by Karl Rubin
J. Neukirch, A. Schmidt, and K. Wingberg. Cohomology of number fields, volume 323 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, second edition, 2008
Language: EN
Title of the course: Quadratic forms
Instructor: Dr. Katharina Hübner
Institution: Universität Heidelberg
Dates: 23-29 September 2019
Prerequisites: Algebra, p-adic numbers
Level: Advanced undergraduate
Abstract: Suppose we want to solve a Diophantine equation, i.e. we are looking for rational solutions of a polynomial equation with integer coefficients. A standard approach is to first check whether there are solutions in Qp for varying p and in R, which is a much easier task. Obviously this is a necessary condition as every solution in Q is also a solution in Qp and in R.
The converse is not true in general: There are Diophantine equations that have solutions in Qp for all p and in R but not in Q. Although these are rather the exception, it is hard to prove when this is the case. If the polynomial has degree 2, however, it has a solution in Q if and only if it has some solution in each of the Qp’s and in R. This is the Hasse-Minkowski theorem.
In this course we will study quadratic forms (i.e. quadratic polynomials in several variables) and the corresponding bilinear forms. We first classify quadratic forms over the p-adic numbers and over the reals. Assembling all the information over the different Qp’s and over R enables us to study quadratic forms over Q and to prove the Hasse-Minkowski theorem.
Textbook: Short notes and exercices in https://www.mathi.uniheidelberg.de/~khuebner
Suggested references:
1. Jean-Pierre Serre. Cours d’arithmétique. Presses Universitaires de France, Paris, 1977. Deuxième édition revue et corrigée, Le Mathématicien, No. 2
2. J.-P. Serre. A course in arithmetic. Springer-Verlag, New York-Heidelberg, 1973. Translated from the French, Graduate Texts in Mathematics, No. 7
3. T. Y. Lam. The algebraic theory of quadratic forms. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1980. Revised second printing, Mathematics Lecture Note Series
Language: EN
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