**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

**İstanbul Bilgi Ü.**

Institution:

Institution:

**5-11 August 2019**

Dates:

Dates:

**Ring theory**

Prerequisites:

Prerequisites:

**Graduate, advanced undergraduate**

Level:

Level:

**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

**İstanbul Bilgi Ü.**

Institution:

Institution:

**12-18 August 2019**

Dates:

Dates:

**Basic algebra**

Prerequisites:

Prerequisites:

**Graduate, advanced undergraduate**

Level:

Level:

**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

**İstanbul Bilgi Ü.**

Institution:

Institution:

**16-29 September 2019**

Dates:

Dates:

**Basic module theory**

Prerequisites:

Prerequisites:

**Graduate, advanced undergraduate**

Level:

Level:

**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