**Toplu Ders İçerikleri**

**Title of the course:** Bilim ve bilimsel gelişmeler

**Instructor:** Dr. Vedat Tanrıverdi

**Institution:** ODTÜ

**Dates:** 9-15 July 2018

**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:** Group Theory I

**Instructor:** Asst. Prof. Roghayeh Hafezieh

**Institution:** Gebze Teknik Üniversitesi

**Dates:** 9-15 July 2018

**Prerequisites:** Algebra I

**Level:** Undergraduate

**Abstract:** In group theory I, we will discuss: "The notion of a group, subgroup, cosets, Cyclic groups, generators, normal subgroups, Isomorphism laws, conjugation, permutations, symmetric group and alternating group."

**Language: **EN

**Title of the course:** Linear Algebra and Its Applications

**Instructor:** Asst. Prof. Seyfi Türkelli

**Institution:** Western Illinois University

**Dates:** 9-15 July 2018

**Prerequisites:** -

**Level:** Graduate, advanced undergraduate, beginning undergraduate

**Abstract:** This is a one-week course on Linear Algebra and Its Applications. In this class, we will see:

Vectors and Operations on Vectors

Matrices and Operations on Matrices

Linear Systems, Elementary Row Operations on Matrices, Gauss-Jordan Elimination Method

Vector Spaces and Subspaces

Examples of Subspaces

GPS Systems

**Language: **EN

**Title of the course:** Introduction to Ring Theory

**Instructor:** Assoc. Prof. Özlem Beyarslan

**Institution:** Boğaziçi Ü.

**Dates:** 9-15 July 2018

**Prerequisites:** -

**Level:** Undergraduate, advanced undergraduate, graduate.

**Abstract:**

1. What is a Ring?

2. Factorization in Polynomial Rings

3. Ideals, Homomorphisms and Factor Rings

4. Unique Factorization Domains

5. Further Topics in Ring Theory

**Language: **TR, EN

**Title of the course:** Sonlu Grupların Temsilleri

**Instructor:** Assoc. Kağan Kurşungöz

**Institution:** Sabancı Üniversitesi

**Dates:** 9-15 July 2018

**Prerequisites:** lineer cebir, lisans seviyesinde soyut cebir.

**Level:** Lisans 3-4 ve yüksek lisans öğrencileri

**Abstract:** Temel kavramlar, matris temsilleri, G-modülleri ve grup cebiri, indirgenebilme, G-homomorfizmaları, grup karakterleri, grup cebirinin ayrışımı, indirgenmiş temsiller.

**Kaynak kitap:** The Symmetric Group (Bruce Sagan)

**Language:** Türkçe

**Title of the course:** Elementary Algebraic Geometry

**Instructor:** Dr. Christian Urech

**Institution:** Imperial College

**Dates:** 9-15 July 2018

**Prerequisites:** Linear algebra, basic commutative algebra such as rings, ideals etc.

**Level:** Advanced undergraduate, graduate

**Abstract:** The aim is to give a very gentle and elementary introduction to algebraic geometry. I will try to cover the following subjects: Noetherian rings, Zariski topology, Hilbert Nullstellensatz, affine varieties and their coordinate rings, morphisms between varieties.

**Language: **EN

**Eğitmen:** Prof. Ali Nesin

**Kurum:** İstanbul Bilgi Ü.

**Tarih: **9-22 Temmuz 2018

**Dersin Adı:** Simetriler

**İçerik:** Konumuz matematiksel yapıları koruyan dönüşümler olacak. Daha çok klasik sayı kümelerine, çizgelere, geometrik şekillere odaklanacağız. Çok boyutlu uzaylara da geçeceğiz, örneğin n boyutlu küplerin simetrilerini bulacağız.

**Eğitmen:** MSc. Kübra Dölaslan

**Kurum:** ODTÜ

**Tarih: **9-22 Temmuz 2018

**Dersin Adı:** Problem Saati

**İçerik:** Bu derste, gün boyunca işlenen konularla ilgili problemler çözülecek, öğrencilerin soruları yanıtlanacaktır. Not: Problemler kolay olmayacaktır ve öğrenciden aktif katılım beklenecektir.

**Eğitmen:** Prof. Yusuf Ünlü

**Kurum:** Yeditepe Ü.

**Tarih: **9-22 Temmuz 2018

**Dersin Adı: **Polinomlar

**İçerik:** Bölme algoritması, Cebirsel türev, Bézout Teoremi, esas ideal bölgeleri, Tek türlü çarpnalara ayrılabilme bölgeleri, simetrik polinomlar.

**Eğitmen:** Doç. Dr. Kağan Kurşungöz

**Kurum:** Sabancı Ü.

**Tarih: **9-22 Temmuz 2018

**Dersin Adı:** Soyut Cebir

**İçerik:** Temel bilgiler (kümeler, bağıntı, fonksiyon, işlem, tam sayılar, modüler aritmetik), gruplar (grubun tanımı, devirli grup, alt grup, eşküme(koset), normal alt grup, permütasyon grupları, abelyen gruplar), homomorfizmalar, grup etkileri, (vakit kalırsa) halka ve cisimler.

**Kullanabilecek kaynaklar:** Elements of Abstract Algebra (Allan Clark), A first course in abstract algebra (John B. Fraleigh), Örneklerle Soyut Cebir (Prof. Dr. Fethi Çallıalp)

**Title of the course:** Examples in Group Theory

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 9-22 July 2018

**Prerequisites:** None

**Level:** Undergraduate, advanced undergraduate, graduate.

**Abstract:** We will give lots of examples of groups and prove and apply the theorems on these examples.

**Language: **TR, EN

**Title of the course:** Topolojiye Giriş

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 9-22 July 2018

**Prerequisites:** Temel kümeler kuramı.

**Level:** Her seviye.

**Abstract:** Açık ve kapalı kümeler ve komşuluk. Kapanış ve iç kavramları. Hausdorff uzayları. Sürekli fonksiyonlar. Topoloji üretmek. Kartezyen topoloji. Topolojinin kısıtlanması. Metrik uzaylar. Ultrametrik. Tıkız kümeler. Tychonoff teoremi.

**Language: **EN, TR

**Title of the course:** Temel türevli (differential) denklemlerin fizikte kullanımı

**Instructor:** Dr. Vedat Tanrıverdi

**Institution:** ODTÜ

**Dates:** 16-22 July 2018

**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:** Group Theory II

**Instructor:** Asst. Prof. Roghayeh Hafezieh

**Institution:** Gebze Teknik Üniversitesi

**Dates:** 16-22 July 2018

**Prerequisites:** Algebra I

**Level:** Graduate and advanced undergraduate

**Abstract:** In group theory II, we will discuss: "Elementray results on group theory, Direct product, Group action, Cayley theorem, Sylow theorems and their proofs, application of sylow theorems, simple groups, some classification."

**Language: **EN

**Title of the course:** Leibniz Cebirlerine Giriş

**Instructor:** Asst. Prof. Nil Mansuroğlu

**Institution:** Ahi Evran Üniversitesi

**Dates:** 16-22 July 2018

**Prerequisites:** -

**Level:** Graduate, advanced undergraduate

**Abstract:**

1. Day : Lie cebirlerine giriş

2. Day: Leibniz cebirlerinin temel kavramları

3. Day: Leibniz cebirleri ile ilgili örnekler

4. Day: -

5. Day: Yapı sabitleri

6. Day: 1, 2, 3 boyutlu Leibniz cebirleri

7. Day: Nilpotent Leibniz cebirleri

**Language: **TR

**Title of the course:** Introduction to Field Theory

**Instructor:** Assoc. Prof. Özlem Beyarslan

**Institution:** Boğaziçi Ü.

**Dates:** 16-22 July 2018

**Prerequisites:** Linear Algebra

**Level:** Advanced undergraduate, graduate.

**Abstract:** Field extensions, algebraic extensions, automorphisms of fields, finite fields.

**Language: **TR, EN

**Title of the course:** Young Tabloları

**Instructor:** Assoc. Kağan Kurşungöz

**Institution:** Sabancı Üniversitesi

**Dates:** 16-22 July 2018

**Prerequisites:** Lisans cebirine biraz göz aşinalığı

**Level:** Lisans 3-4 ve yüksek lisans öğrencileri

**Abstract:** Young tablolarında muhtelif işlemler, kelimeler üzerinde işlemler, Robinson-Schensted-Knuth eşlemesi, (vakit kalırsa) simetrik polinomlar.

**Kaynak Kitap:** Young Tableaux (William Fulton)

**Language:** Türkçe

**Title of the course:** Geometric group theory

**Instructor:** Dr. Christian Urech

**Institution:** Imperial College

**Dates:** 16-22 July 2018

**Prerequisites:** Basic notions in group theory (normal subgroups, quotients, isomorphism theorems etc.)

**Level:** Advanced undergraduate, graduate

**Abstract:** Generators and relations, free groups, graphs, Cayley graphs, group actions, trees, Nielsen Schreier theorem.

**Language: **EN

**Title of the course:** Group Actions and Sylow Theory

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 23-29 July 2018

**Prerequisites:** Basic group theory

**Level:** Advanced undergraduate, graduate.

**Abstract:** Our purpose will be to prove Sylow Theorems and to give some of their applications.

**Language: **TR, EN

**Title of the course:** Finite Fields

**Instructor:** Assoc. Prof. Özlem Beyarslan

**Institution:** Boğaziçi Ü.

**Dates:** 23-29 July 2018

**Prerequisites:** -

**Level:** Advanced undergraduate, graduate.

**Abstract:** Finite fields: existence, uniqueness, structure, explicit construction. Frobenius automorphisms. Galois’s Theory of finite fields.

**Language: **TR, EN

**Title of the course:** Elektromanyetik teoriye giriş

**Instructor:** Dr. Vedat Tanrıverdi

**Institution:** ODTÜ

**Dates:** 23-30 July 2018

**Prerequisites:** Temel differansiyel ve integral kavramları, temel elektrostatik bilgi

**Level:** Advanced undergraduate

**Abstract:** Coulumb yasası ile başlayıp Maxwell denklemlerinin differansiyel haline kadar işlenecektir.

**Language: **TR

**Title of the course:** Elements of Matrix Algebra

**Instructor:** Prof. Alexandre Borovik

**Institution:** Manchester U.

**Dates:** 23 -29 July 2018

**Prerequisites:** School level algebra

**Level:** Younger undergraduate students and fresh school leavers

**Abstract:** Entry level course which will focus on two topics usually not discussed in standard undergraduate courses:

What is reduced echelon form of matrix, really? What is its meaning and what it is doing?

What is the rank of a matrix?

**Language: **English

**Title of the course:** Introduction to Manifolds with Special Holonomy

**Instructor:** Dr. Özgür Kelekçi

**Institution:** Türk Hava Kurumu Ü.

**Dates:** 23-29 July 2018

**Prerequisites:** Basic Differential Geometry (not a must but preferable).

**Level:** Graduate, advanced undergraduate

**Abstract:** Manifolds with special holonomy attract significant interest in both mathematics and mathematical physics. They appear in many contexts in Riemannian geometry, particularly Ricci-flat and Einstein geometry, minimal submanifold theory and the theory of calibrations, and string theory. We will start with basics of Riemannian geometry. The emphasis will be on discussing the Ricci-flat geometries that occur, then the holonomy classifications will be studied.

**Language: **EN

**Title of the course:** Finans Matematiği

**Instructor:** Assoc. Prof. Deniz Ünal

**Institution:** Çukurova Ü.

**Dates:** 23-30 July 2018

**Prerequisites:** -

**Level:** Advanced undergraduate, beginning undergraduate

**Abstract:**

Temel kavramlar, basit/bileşik faiz-iskonto

Eşdeğer Senetler

Dış/iç İskonto

Ani Faiz

Paranın bugünkü/gelecekteki değeri

Annüiteler (düzenli ödemeli, değişken ödemeli, sonlu/sonsuz, sürekli ödemeli)**
Language: **TR

**Title of the course:** Lectures On G_2 Geometry

**Instructor:** Assoc. Prof. Mustafa Kalafat

**Institution:** -

**Dates:** 23-29 July 2018

**Prerequisites:** Linear Algebra, Riemannian Geometry (not a must but preferable)

**Level:** Graduate, advanced undergraduate

**Abstract:** In this lecture series we present a combinatorial approach to the exceptional Lie group G2. We give a survey of various results about the algebraic structure. For the sake of completeness we decided to present them in a self-contained way to be easily accessible for future usage. We also present some applications to geometry. Manifolds with G2 structures, Decomposition of the exterior algebra into irreducible G2 representations, Metric of a G2 structure, 16 classes of G2 structures, Deformations of G2 structures. Lie algebra of G2, roots and their spaces, order, Killing form.

**Textbook or/and course webpage:**

1. Karigiannis, Spiro - Deformations of G2 and Spin(7) structures. Canad. J. Math. 57 (2005), no. 5, 1012–1055. Author's Ph.D. Thesis also available on the Arxiv.org.

2. Anthony W. Knapp. Lie groups beyond an introduction, volume 140 of Progress in Mathematics. Birkhauser Boston, Inc., Boston, MA, second edition, 2002.

**Language: **TR, EN

**Title of the course:** Polytopes

**Instructor:** Assoc. Prof. Ali Özgür Kişisel

**Institution:** ODTÜ

**Dates:** 23-29 July 2018

**Prerequisites:** Linear Algebra

**Level:** Advanced Undergraduate

**Abstract:** We will discuss essential features of polytopes in all dimensions. Several families of examples will be presented. Some topics in focus will be regular polytopes, simple and simplicial polytopes, Schlegel diagrams, Gale transform, Euler's equation and Dehn-Sommerville equations.

**Language:** English, Turkish

**Title of the course:** Lie algebras

**Instructor:** Assoc. Prof. Şükrü Yalçınkaya

**Institution:** İstanbul Ü.

**Dates:** 23 July - 5 August 2018

**Prerequisites:** Good knowledge on linear algebra

**Level:** Advanced undergraduate, graduate

**Abstract:** Definitions and examples. Solvable and nilpotent Lie algebras. Semisimple Lie algebras. Root systems of semisimple Lie algebras.

**Language:** Turkish, English

**Eğitmen:** Prof. Ali Nesin

**Kurum:** İstanbul Bilgi Ü.

**Tarih: **23 Temmuz – 5 Ağustos 2018

**Dersin Adı:** Gerçel Sayılar Yapısı

**İçerik:** Gerçel sayılar aksiyomatik olarak tanımlayıp, "Arşimet Özelliği" ve "tamlık" gibi gerçel sayıların en temel özelliklerini kanıtlayacağız. Bunu yapmak için Cauchy dizilerini ve yakınsak dizileri tanımlamalıyız. Bu klasik konulardan sonra gerçel sayıların biricikliğini tartışacağız. Son günlerde sonsuz küçük elemanların olduğu ve gerçel sayılara çok benzeyen yapılar tanımlayacağız.

**Eğitmen:** Doç. Dr. Özlem Beyarslan

**Kurum:** Boğaziçi Ü.

**Tarih: **23 Temmuz – 5 Ağustos 2018

**Dersin Adı:** Diziler ve seriler

**İçerik:** 1. Gerçel sayıların özellikleri 2. yakınsak gerçel sayı dizileri 3. Limitin biricikliği 4.Yakınsak dizilerle işlemler 5. Monoton diziler 6. Cauchy dizileri 7. Gerçel sayıların tamlığı 8. Bolzano Weierstrass teoremi 9. Seriler 10. Serilerde yakınsaklık 11. Kıyaslama teoremleri 12. Yakınsaklık testleri

**Eğitmen:** Yard. Doç. Saadet Özer

**Kurum:** İstanbul Bilgi Ü.

**Tarih: **23 Temmuz – 5 Ağustos 2018

**Dersin Adı:** Kalkülüs

**İçerik:** Fonksiyonlar, Limit, Süreklilik, Türev, Türevin Uygulamaları, İntegral ve uygulamaları.

**Eğitmen:** MSc. Kübra Dölaslan

**Kurum:** ODTÜ

**Tarih: **23 Temmuz – 5 Ağustos 2018

**Dersin Adı:** Problem Saati

**İçerik:** Bu derste, gün boyunca işlenen konularla ilgili problemler çözülecek, öğrencilerin soruları yanıtlanacaktır. Not: Problemler kolay olmayacaktır ve öğrenciden aktif katılım beklenecektir.

**Title of the course:** An introduction to social choice theory

**Instructor:** Prof. Remzi Sanver

**Institution:** CNRS, Universite Paris Dauphine

**Dates:** 30 July – 5 August 2018

**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: **EN, TR

**Title of the course:** Fractal dimensions in Analysis

**Instructor:** Asst. Prof. Kemal Ilgar Eroğlu

**Institution:** İstanbul Bilgi Ü.

**Dates:** 30 July – 5 August 2018

**Prerequisites:** Undergraduate-level measure theory is recommended.

**Level:** Graduate, advanced undergraduate

**Abstract:** In this course we will go over various kinds of (fractal) dimensions that are used in analysis and study their elementary properties.

**Language: **TR, EN

**Title of the course:** Introduction to Ring Theory

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 30 July – 5 August 2018

**Prerequisites:** None

**Level:** Undergraduate, advanced undergraduate, graduate.

**Abstract:** Definition and examples of rings. Subrings. Ideals. Ideals generated by a subset. Quotient ring. Fundamental theorem of ring theory. Chinese remainder theorem.

**Language: **TR, EN

**Title of the course:** Galois Theory

**Instructor:** Assoc. Prof. Özlem Beyarslan

**Institution:** Boğaziçi Ü.

**Dates:** 30 July – 5 August 2018

**Prerequisites:** -

**Level:** Advanced undergraduate, graduate.

**Abstract:** Field extensions. Minimal polynomial. Construction of simple algebraic extension from an irreducible polynomial. The field of algebraic numbers. Splitting field, Normal and separable extensions, Fundamental theorem of Galois Theory.

**Language: **TR, EN

**Title of the course:** q-series and applications to number theory

**Instructor:** Prof. Mehmet Cenkci

**Institution:** Akdeniz Ü.

**Dates:** 30 July – 5 August 2018

**Prerequisites:** Mathematical analysis

**Level:** Undergraduate, graduate

**Abstract:** q-series and theta functions, Fundamental theorems about q-series and theta functions, Sums of squares and triangular numbers, Congruences for partition function.

**Language: **TR, EN

**Title of the course:** Numbers and Polynomials

**Instructor:** Prof. Alexandre Borovik

**Institution:** Manchester U.

**Dates:** 30 July - 5 August 2018

**Prerequisites:** Zero. Suitable for fresh entrants to university.

**Level:** Younger undergraduate students

**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:** English

**Title of the course:** Basic knot theory

**Instructor:** Asst. Prof. Georgios Dimitroglou Rizell

**Institution:** Uppsala University

**Dates:** 30 July – 5 August 2018

**Prerequisites:** Calculus. Elementary group theory and elementary topology are very helpful, but not required.

**Level:** Graduate, advanced undergraduate, beginning Undergraduate

**Abstract:**This is a course on the theory of knots in three dimensions for beginners. We will study classical topological notions such as the linking number, Seifert surfaces, and the knot group. In addition we will introduce certain knot polynomials (e.g. the Alexander, Jones, and A-polynomials), which are more modern invariants of knots.

**Language: **EN

**Title of the course:** Introduction to games and strategic behavior

**Instructor:** Dr. Ali Ozkes

**Institution:** Aix-Marseille School of Economics

**Dates:** 30 July - 5 August 2018

**Prerequisites:** High school mathematics

**Level:** Undergraduate

**Abstract:** The course introduces basic concepts of game theory through (economic) applications and presents models of bounded rationality. Basic concepts include finite two-person (non-) zero-sum games, mixed strategies, Nash equilibrium, games with imperfect information, repeated games, and voting games. Some models of structural deviation from equilibrium thinking are discussed.

**Language:** English

**Title of the course:** Ayrık Matematik

**Instructor:** Asst. Prof. Ayhan Dil

**Institution:** Akdeniz Ü.

**Dates:** 30 July - 12 August 2018

**Prerequisites:** Calculus

**Level:** Undergraduate (1-2)

**Abstract:**

Recurrent Problems

Sums

Integer Functions

Number Theory

Binomial Coefficients

Special Numbers

Generating Functions

**Language: **TR

**Eğitmen:** Prof. Haluk Oral

**Kurum:** -

**Tarih: **6-12 Ağustos 2018

**Dersin Adı:** Matematik Sohbetleri

**İçerik:** Sayma problemleri, reel sayıların bazı özellikleri, Binom teoremi, Öklid algoritması, şifreleme.

**Eğitmen:** Prof. Melih Boral

**Kurum:** -

**Tarih: **6-12 Ağustos 2018

**Dersin Adı:** Sayılar Kuramı

**İçerik:** Tamsayıların özellikleri, kongruens, Fermat ve Euler teoremleri, lineer kongruensler, kuadratik residüler.

**Eğitmen:** Prof. Melih Boral

**Kurum:** -

**Tarih: **6-12 Ağustos 2018

**Dersin Adı:** Sayılar Kuramı

**İçerik:** Tamsayıların özellikleri, kongruens, Fermat ve Euler teoremleri, lineer kongruensler, kuadratik residüler.

**Title of the course:** Elliptic curves

**Instructor:** Dr. Dino Festi

**Institution:** JGU Mainz

**Dates:** 6-12 August 2018

**Prerequisites:** Basic commutative algebra and analysis

**Level:** Graduate, advanced undergraduate

**Abstract:** This course is meant to be an introduction to the study of elliptic curves. We will approach the subject from both an algebraic point of view and a complex analytic one. The final goal of the course is to show that the categories of complex tori, lattice of rank 2, and elliptic curves over the complex field are equivalent. More details can be find in the syllabus of the course given last year (see attachment).

**Textbook or/and course webpage:**

Silverman, `the arithmetic of elliptic curves’.

Course’s notes (see attachment).

**Language: **EN

**Title of the course:** Algebraic Curves

**Instructor:** Dr. Davide Cesare Veniani

**Institution:** Johannes Gutenberg-Universität Mainz

**Dates:** 6-12 August 2018

**Prerequisites:** Basic ring theory (ideals, polynomials, fields, algebraic closure)

**Level:** Graduate, advanced undergraduate, beginning undergraduate

**Abstract:** This course is intended as an invitation to algebraic geometry. The theory of algebraic curves already shows many of the complexities of this field, but it is an accessible topic that will help the students build a strong intuition. The main focus will be on affine and projective plane curves, but we will also develop fundamental algebraic tools such as Hilbert’s Nullstellensatz and discrete valuation rings. Our main result will be Bézout’s Theorem. This course will be a complement to Dino Festi’s course on elliptic curves.

**Textbook or/and course webpage:** “Algebraic Curves” by W. Fulton, chapters 1-5

(http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf)

**Language: **EN

**Title of the course:** Topics in algebraic number theory

**Instructor:** Dr. Pınar Kılıçer

**Institution:** Oldenburg University

**Dates:** 6-12 August 2018

**Prerequisites:** Basic algebra, linear algera, field theory, module theory.

**Level:** Graduate, advanced undergraduate

**Abstract:**

- Number rings, ideals in number rings

- Explicit ideal factorization in number rings

- Geometry of numbers and applications

- Computing units and class groups

**Textbook or/and course webpage:** Daniel A. Marcus – Number Fields

**Language: **EN

**Title of the course:** Bernoulli Numbers and Zeta Functions

**Instructor:** Prof. Mehmet Cenkci

**Institution:** Akdeniz Ü.

**Dates:** 6-12 August 2018

**Prerequisites:** Basic complex analysis including residue theorem and analytic continuation.

**Level:** Undergraduate, graduate

**Abstract:** Bernoulli numbers; Stirling numbers; Clausen and von Staudt Theorem and Kummer’s Congruence; Generalized Bernoulli numbers; The Euler-Maclaurin Summation Formula and the Riemann zeta function; Special values and complex integral representation of L-functions.

**Language: **TR, EN

**Title of the course:** Stochastic Optimization

**Instructor:** Dr. Tatiana González Grandón

**Institution:** Humboldt Universität Berlin

**Dates:** 6-12 August 2018

**Prerequisites:** Linear Optimization

**Level:** Graduate, advanced undergraduate

**Abstract:** Stochastic optimization refers to a collection of methods for minimizing/maximizing an objective function when randomness is present. We will study the types of problems when uncertainty is present in the constraint functions. Over the last few decades these methods have become essential tools for economics, engineering, business and computer science. We will look at applications and essential theory to solve this type of problems.

1. Optimization with random variables Introduction to Optimization Problems with uncertainties and to Probability Tools.

2. Probability Constraints Examples and Solutions.

3. Continuity & Differentiability of Probability Constraints

4. When do we have convexity? Prékopa's Theorem

5. Numerical Solutions and Discussion of Optimization Examples.

**Textbook or/and course webpage:
**1. Stochastic Programming. Andras Prékopa.

**2. Lectures on Stochastic Programming (Modeling and Theory). A. Shapiro, D. Dentcheva.**

**Language:**EN

**Title of the course:** Modular Forms and L functions in Analytic Number Theory

**Instructor:** Dr. Eren Mehmet Kıral

**Institution:** -

**Dates:** 6-12 August 2018

**Prerequisites:** Complex Analysis, Fourier analysis.

**Level:** Graduate, advanced undergraduate

**Abstract:**

**Day 1:** The Gauss Circle Problem, Voronoi Summation formula

**Day 2:** The jacobi theta function, every number can be written as a sum of four squares.

**Day 3:** The Riemann Zeta function, Riemann's memoir (a proof of Prime number theorem if Riemann Hypothesis were true)

**Day 4:** Kloosterman's original paper on representing integers by quadratic forms in four variables where a Kloosterman sum is first introduced

**Day 5:** Modular forms on the upper half plane, (for those also taking the Elliptic curve class) what did Andrew Wiles' modularity theorem which led to a proof of Fermat's Last Theorem actually say (no proof).

**Day 6:** Eisenstein series, Poincare series, Petersson Trace formula (more Kloosterman sums!). (Or some other topic you might ask during the week, but give me early notice)

**Language: **TR, EN

**Title of the course:** Quadratic Forms and the Hasse-Minkowski Theorem

**Instructor:** Dr. Martin Djukanovic

**Institution:** Ulm University

**Dates:** 6-12 August 2018

**Prerequisites:** Basic algebra, basic analysis helps but is not necessary.

**Level:** Graduate, advanced undergraduate

**Abstract:** We introduce the concept of a Diophantine equation, with some classical results and examples (Pythagorean triples, theorems of Fermat and Legendre). We prove the Quadratic Reciprocity Law, introduce p-adic numbers and Hensel's Lemma (without a formal proof), and we prove the basic properties of the Hilbert symbol. Finally, if time permits, we state and prove the Hasse-Minkowski theorem for ternary quadratic forms.

**Language: **EN

**Title of the course:** Introduction to Quantum Computing

**Instructor:** Asst. Prof. Ahmet Çevik

**Institution:** JSGA, ODTÜ

**Dates:** 6-12 August 2018

**Prerequisites:** Linear algebra

**Level:** Graduate and advanced undergraduate

**Abstract:** This is a concise introduction course in quantum computing for mathematicians and computer scientists. We aim to cover some fundamental topics in quantum computing such as the mathematical representation of quantum systems, quantum teleportation, superdense coding, some basic quantum algorithms including the Deutsch algorithm, Deutsch-Josza algorithm, Simon’s periodicity algorithm, and quantum models of computation. Only elementary linear algebra is required for this course. We will not assume for the auidiance to have any background on quantum mechanics, though some familiarity would certainly help.

**Language: **TR, EN

**Title of the course:** Elementary Number Theory

**Instructor:** Dr. Cihan Pehlivan

**Institution:** -

**Dates:** 6-12 August 2018

**Prerequisites:** -

**Level:** Beginning undergraduate

**Abstract:** Congruences, Chinese Remainder Theorem, Primitive Roots, Quadratic Residues and Reciprocity, Arithmetic Functions, Diophantine Equations.

Primitive roots, quadratic residues and reciprocity, Euler Phi Function.

**Language: **TR

**Title of the course:** Topics in Number Theory

**Instructor:** Asst. Prof. Haydar Göral

**Institution:** Dokuz Eylül Ü.

**Dates:** 6-19 August 2018

**Prerequisites:** Calculus

**Level:** Graduate, advanced undergraduate, beginning undergraduate

**Abstract:** The course will be two weeks but each week will be independent from each other.

First week: Prime numbers, sieve methods, zeta functions

Second week: Additive number theory, some special functions and numbers

**Language: **TR, EN

**Eğitmen:** Ali Törün

**Kurum:** -

**Tarih: **6-19 Ağustos 2018

**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:** MSc. Kübra Dölaslan

**Kurum:** ODTÜ

**Tarih: **6-19 Ağustos 2018

**Dersin Adı:** Problem Saati

**İçerik:** Bu derste, gün boyunca işlenen konularla ilgili problemler çözülecek, öğrencilerin soruları yanıtlanacaktır. Not: Problemler kolay olmayacaktır ve öğrenciden aktif katılım beklenecektir.

**Eğitmen:** Prof. Ali Nesin

**Kurum:** İstanbul Bilgi Ü.

**Tarih: **13-19 Ağustos 2018

**Dersin Adı:** Kombinasyon Hesapları ve Olasılık

**İçerik:** Asıl amacımız olasılık kuramı olacak. Olay kümemiz sonlu ya da sonsuz, ayrık ya da sonsuz olabilecek. Ama çoğu kez olasılık problemlerinde kombinasyon hesapları da gerekir. Kombinasyon hesaplarını ihtiyacımız olduğu kadar inceleyeceğiz. Her gün bir ya da iki probleme odaklanılacak. Öğrencilerin bu derste aktif olmaları beklenmektedir.

**Eğitmen:** Dr. Tülay Ayyıldız Akoğlu ve Dr. Kemal Akoğlu

**Kurum:** Karadeniz Teknik Ü.

**Tarih: **13-19 Ağustos 2018

**Dersin Adı:** Algoritmalarla Matematik

**İçerik:** Bu dersin amacı bir matematik probleminin algoritmalar yoluyla nasıl çözülebileceğini öğretmektir. Dersteki algoritmalar “pseudocode” şeklinde öğretilecek, daha sonra herhangi bir programlama dili ile bilgisayarda uygulama yapabilmesinin altyapısı oluşturulmuş olacaktır. İlk derste algoritma nedir ve matematik bilgisayara nasıl öğretilir, bilgisayar yardımıyla matematik problemleri nasıl çözülebilir soruları cevaplanacaktır.

Bir algoritmanin bileşenleri açıklanacak, kullanılabilecek veri tipleri ve yapıları (for, while, if/ then gibi temel enstrumanlar) tanıtılacaktır.

Daha sonra sıralama ve arama (searching/sorting) algoritmalarından birer örnek verilecektir, bu algoritma sayısal birer örnek ile öğrencilere uygulanacaktır.

Algoritma dizaynı ve uygulaması interaktif şekilde olmak üzere aşağıdaki konular üzerinde çalışılacaktır:

- Bir tam sayının faktoriyelini hesaplama,

- Tam sayıların ebob ve ekoklarını bulma,

- Belirli bir aralıktaki asal sayıları bulma,

- İki şehir arasında en kısa yolu bulma (iki farklı algoritma ile).

**Title of the course:** Orders of Finite Simple Groups

**Instructor:** Prof. Ayşe Berkman

**Institution:** MSGSÜ

**Dates:** 13-19 August 2018

**Prerequisites:** Basic knowledge of group theory including Sylow Theorems.

**Level:** Graduate, advanced undergraduate

**Abstract:** This will be more like a workshop centered on the question “Which positive integers are orders of finite simple groups?” I shall teach the necessary concepts, theorems, and techniques of group theory, and students will be assigned to make short presentations at the board.

**Language: **TR, EN

**Title of the course:** Numerical Semigroups

**Instructor:** Prof. Halil İbrahim Karakaş

**Institution:** Başkent Ü.

**Dates:** 13-19 August 2018

**Prerequisites:** Elementary number theory

**Level:** Graduate, advanced undergraduate

**Abstract:** Monoids and monoid homomrophisms, Multiplicity and embedding dimension, Frobenius number, genus, conductor and pseudo frobenius numbers, Symmetric and pseudo-symmetric numerical semigroups.

**Textbook:** J. C. Rosales, P. E. Garcia-Sanchez. Numerical Semigroups, Springer, 2009.

**Language: **TR, EN

**Title of the course:** Introduction to Module Theory

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 13-19 August 2018

**Prerequisites:** Ring theory

**Level:** Advanced undergraduate, graduate.

**Abstract:** Definition and examples. Homomorphisms. Fundamental theorem. Linear independence and sets of generateors. Free modules.

**Language: **TR, EN

**Title of the course:** Continued fractions

**Instructor:** Prof. David Pierce

**Institution:** MSGSÜ

**Dates:** 13-19 August 2018

**Prerequisites:** None. We shall study some topics that may be studied in a second-semester number-theory course; however, no specific results are required from a first-semester course.

**Level:** Graduate, advanced undergraduate

**Abstract:** We accept from childhood that multiplication of whole numbers is commutative; but Euclid gives a rigorous proof based on what we now call the Euclidean algorithm. This algorithm can be used to write any real number as a continued fraction. The continued fraction repeats when the real number is the solution of a quadratic equation. This case yields the solutions of a so-called Pell equation, x^2 - dy^2 = 1, an example of a Diophantine equation.

**Textbook or/and course webpage:** http://mat.msgsu.edu.tr/~dpierce/Courses/Sirince/ (in preparation)

**Language: **TR; EN

**Title of the course:** Philosophy of Mathematics

**Instructor:** Asst. Prof. Ahmet Çevik

**Institution:** JSGA, ODTÜ

**Dates:** 13-19 August 2018

**Prerequisites:** Curiosity

**Level:** Advanced undergraduate, beginning Undergraduate

**Abstract:** This is an introductory course in philosophy of mathematics. We will cover some fundamental subjects and various philosophical views concerning the ontology, epistemology and methodology of mathematics, including mathematical realism (Platonism), intuitionism, logicism, and formalism if time permits. No background is assumed, the course will be self-contained.

**Textbook or/and course webpage:** A. Çevik, Matematik Felsefesi ve Matematiksel Mantık, submitted.

**Language: **TR, EN

**Title of the course: **Dış cebire giriş ve Riemann geometrindeki bazı uygulamaları

**Instructor:** Prof. Dr. Muzaffer Adak

**Institution:** Pamukkale Üniversitesi

**Dates:** 13-19 August 2018

**Prerequisites:** Lineer Cebir ve Diferansiyel Denklem konularında temel bilgilere sahip olunması önerilir.

**Level:** Fizik ve Matematik bölümlerinde 3. ve 4. sınıf öğrencileri ile lisansüstü öğrencileri için uygundur.

**Abstract:** Manifoldlar, Vektörler ve Kovektörler, Tensörler, Metrik Tensörü, Dış Cebir ve Uygulamaları, Stokes Teoremi, İki Boyutlu Eğrisel Yüzeyler, Metrik ve Eğrilik, Paralel Taşıma ve Türev, Kovaryant Dış Türev, Riemann Geometrisi, Otoparalel Eğriler, Jeodezik, Jeodezik Sapma Denklemi

**Language:** Türkçe

**Title of the course:** Linear Representations of Finite Groups

**Instructor:** Dr. Şermin Çam Çelik

**Institution:** Özyeğin Ü.

**Dates:** 13-19 Ağustos 2018

**Prerequisites:** Linear algebra, basic group theory

**Level:** Graduate, advanced Undergraduate

**Abstract:** Definitions(Representations, Subrepresentations, Irreducible Representations, Equivalent Representations), Schur’ s Lemma, Character of a Representation, Orthogonality Relations for Characters, Number of Irreducible Characters, Some Basic Examples.

**Language: **EN, TR

**Title of the course:** Amenable Groups

**Instructor:** Prof. Ayşe Berkman, MSc. Barış Bektaş

**Institution:** MSGSÜ

**Dates:** 20-26 August 2018

**Prerequisites:** A course on group theory (or abstract algebra including group theory) is necessary. A course on topology will be very useful.

**Level:** Graduate, advanced undergraduate

**Abstract:** Measures and means on groups. Some properties of amenable groups. Examples and non-examples. Relation to the Banach-Tarski Paradox. Tarski numbers of groups.

We strongly recommend for the students who are interested in geometric group theory to take Growth of Groups, Self-Similar Groups and Groups Generated by Automata, and Amenable Groups together. Even though we will try to make the three courses independent, following all three will certainly enhance understanding of the material.

**Language: **TR, EN

**Title of the course:** Growth of Groups

**Instructor:** Dr. Dilber Koçak

**Institution:** ODTÜ

**Dates:** 20-26 August 2018

**Prerequisites:** A course on group theory (or abstract algebra including group theory) is necessary.

**Level:** Graduate, advanced undergraduate

**Abstract:**

• Growth functions and growth series of groups (Preliminaries, general notions and history)

• Growth of nilpotent groups

• Growth of solvable groups

• Groups of polynomial growth (outline of Gromov’s Theorem)

• Groups of intermediate growth (Grigorchuk groups)

• Summary, final remarks and open problems.

**Language: **TR, EN

**Title of the course:** Self-similar groups and groups generated by automata

**Instructor:** Asst. Prof. Mustafa Gökhan Benli

**Institution:** ODTÜ

**Dates:** 20-26 August 2018

**Prerequisites:** A course on group theory (or abstract algebra including group theory) is necessary.

**Level:** Graduate, advanced undergraduate

**Abstract:**

• Rooted trees and their automorphisms

• Self-similar groups

• Automata and groups generated by automata

• Examples and sources of self-similar groups

• Relations with the Burnside problem

• Contracting and branch groups

• Relations with amenability and growth

**Language: **TR, EN

**Title of the course:** Prime numbers

**Instructor:** Prof. David Pierce

**Institution:** MSGSÜ

**Dates:** 20-26 August 2018

**Prerequisites:** Some elementary number theory

**Level:** Graduate, advanced undergraduate

**Abstract:** We shall work through the Prime Number Theorem, that the probability that a given number N is prime is about 1/log(N).

**Language: **TR, EN

**Textbook or/and course webpage:** http://mat.msgsu.edu.tr/~dpierce/Courses/Sirince/ (in preparation)

**Title of the course:** Introduction to general topology

**Instructor:** Dr. Matteo Paganin

**Institution:** Sabancı Ü.

**Dates:** 20-26 August 2018

**Prerequisites:** Just the basics.

**Level:** Graduate, advanced undergraduate

**Abstract:** In this course we plan to give definitions, examples, and basic properties of topological spaces; closed and open subsets; neighbourhoods; metric spaces; basis for topologies; continuous, open, and closed functions, homeomorphism; closure, interior, accumulation points; limits; separation axioms; compactness and connectedness. The aim of the course will be to state and sketch the proof of the Weierstrass and Heine-Borel theorems.

**Language: **EN, TR

**Title of the course:** Automorphism groups of some structures

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 20-26 August 2018

**Prerequisites:** None

**Level:** Undergraduate, advanced undergraduate, graduate

**Abstract:** Symmetric groups. Automorphism of some graphs. Automorphism groups of number systems. Isometries. Linear maps. Automorphism groups of Z/nZ. Automorphism groups of groups.

**Language: **TR, EN

**Title of the course:** Incidence geometry and buildings

**Instructor:** Prof. Michel Lavrauw

**Institution:** Sabancı Ü.

**Dates:** 20-26 August 2018

**Prerequisites:** Basic algebra knowledge

**Level:** Advanced undergraduate, graduate

**Abstract:** Starting from basic point-line geometries satisfying a set of axioms, the students are introduced to projective planes, polar spaces, generalised polygons, and buildings. This course is about the relationship between groups and geometries, and is inspired by the work of Abel price winner Jacques Tits; in particular his work on the ecoding of the algebraic structure of linear groups in geometric terms. The course is intended as a gentle introduction to this theory, and accessible to any graduate student with basic algebra knowledge.

Most of the course will be based some of the chapters contained in the Handbook of Incidence Geometry [1] edited by Francis Buekenhout, a former PhD student of Jacques Tits. Below is a short description of the contents of [1].

This Handbook deals with the foundations of incidence geometry, in relationship with division rings, rings, algebras, lattices, groups, topology, graphs, logic and its autonomous development from various viewpoints. Projective and affine geometry are covered in various ways. Major classes of rank 2 geometries such as generalized polygons and partial geometries are surveyed extensively. More than half of the book is devoted to buildings at various levels of generality, including a detailed and original introduction to the subject, a broad study of characterizations in terms of points and lines, applications to algebraic groups, extensions to topological geometry, a survey of results on diagram geometries and nearby generalizations such as matroids.

[1] Handbook of Incidence Geometry, Buildings and Foundations (edited by F. Buekenhout) North Holland 1995.

**Language: **EN

**Title of the course:** Molecular dynamics, Monte Carlo dynamics

**Instructor:** Prof. François Dunlop

**Institution:** Université de Cergy Pontoise

**Dates:** 27 August – 2 September 2018

**Prerequisites:** Calculus

**Level:** Graduate, advanced undergraduate

**Abstract:** The course is about time evolution of extended systems, with emphasis on what to do and understand mathematically before going on the computer.

1. Time in dynamics and algorithms

2. Molecular dynamics: Runge Kutta and Verlet algorithms

3. Molecular dynamics: ergodicity

4. Monte Carlo dynamics as a Markov chain

5. Monte Carlo: algorithms for particle systems

6. Cellular automata

**Language: **EN

**Title of the course:** Limits, Sequences and Series

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 27 August – 2 September 2018

**Prerequisites:** TBA

**Level:** Advanced undergraduate, graduate

**Abstract:** TBA

**Language: **EN, TR

**Title of the course:** Field Theory

**Instructor:** Assoc. Prof. Özlem Beyarslan

**Institution:** Boğaziçi Ü.

**Dates:** 20 August – 9 September 2018

**Prerequisites:** Basic abstract algebra

**Level:** Advanced undergraduate, graduate.

**Abstract:** Fields and field extensions. Theory of finite fields. Algebraic closure. Automorphisms of fields. Galois theory. Galois completion. Algebraic closure of finite fields.

**Language: **TR, EN

**Eğitmen:** Asst. Prof. Salih Durhan

**Kurum:** -

**Tarih: **20 Ağustos – 2 Eylül 2018

**Dersin Adı:** Kombinasyon hesapları ve olasılık

**İçerik:** Asıl amacımız olasılık kuramı olacak. Olay kümemiz sonlu ya da sonsuz, ayrık ya da sonsuz olabilecek. Ama çoğu kez olasılık problemlerinde kombinasyon hesapları da gerekir. Kombinasyon hesaplarını ihtiyacımız olduğu kadar inceleyeceğiz. Her gün bir ya da iki probleme odaklanılacak. Öğrencilerin bu derste aktif olmaları beklenmektedir.

**Eğitmen:** Doç. Dr. Özlem Beyarslan

**Kurum:** Boğaziçi Ü.

**Tarih: **20 Ağustos – 2 Eylül 2018

**Dersin Adı:** Polinomlar

**İçerik:** Polinom halkası ve özellikleri, formal türev, polinomların diskriminantı.

**Eğitmen:** Prof. Ali Nesin

**Kurum:** İstanbul Bilgi Ü.

**Tarih: **20 Ağustos – 2 Eylül 2018

**Dersin Adı:** Oyunlar Kuramı

**İçerik:** Oyunların matematiğine odaklanacağız. Şans oyunları, şansın olmadığı oyunlar, sonlu oyunlar, potansiyel olarak sonlu oyunlar, sonsuz oyunlar... Varsa oyunların kazanan stratejilerini, yoksa kazanma olasılığımızı artıran stratejileri bulacağız. Kombinasyon hesapları ve olasılık konularına doğal olarak eğileceğiz.

**Eğitmen:** MSc. Kübra Dölaslan

**Kurum:** ODTÜ

**Tarih: **20 Ağustos – 2 Eylül 2018

**Dersin Adı:** Problem Saati

**İçerik:** Bu derste, gün boyunca işlenen konularla ilgili problemler çözülecek, öğrencilerin soruları yanıtlanacaktır. Not: Problemler kolay olmayacaktır ve öğrenciden aktif katılım beklenecektir.

**Title of the course:** Free Modules, Abel Groups and Vector Spaces

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 20 August – 2 September 2018

**Prerequisites:** Basic module theory

**Level:** Advanced undergraduate, graduate.

**Abstract:** Free modules. Homomorphisms from/onto free modules. Matrices and algebra of matrices. Free modules over PID's.

**Language: **TR, EN

**Title of the course:** Logic for Artificial Intelligence

**Instructor:** Mr. Arif Mardin

**Institution:** -

**Dates:** 27 August – 2 September 2018

**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. Introduction to probabilistic reasoning and decision making under uncertain knowledge.

**Language: **EN

**Textbook:** Hodges, W.: “Logic”, Penguin Books, 1977,

Nilsson, N.J.: “Principles of Artificial Intelligence”, Morgan Kaufmann, 1980,

Genesereth, M. and Nilsson, N.J.: “Logical Foundations of Artificial Intelligence”,

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:** Topics in Complex Function Theory, a detour around interpolation theorems

**Instructor:** Assoc. Prof. Uğur Gül

**Institution:** Hacettepe Ü.

**Dates:** 27 August – 2 September 2018

**Prerequisites: **A solid background in Graduate level Real and Complex Analysis and Functional Analysis

**Level:** Graduate

**Abstract:**

I. Infinite products, Blaschke condition, Blaschke products, Inner Functions.

II. Schwarz lemma, Schwarz-Pick lemma, Poincaré metric, Pick interpolation theorem.

III. Poisson-Jensen Formula, Hardy-Nevanlinna classes, Inner-outer factorization of Hardy Functions.

IV. An application of Blaschke products: Müntz-Szasz approximation theorem.

V. Carleson interpolation theorem.

**References:** "Bounded Analytic Functions" by J. B. Garnett, "Functional Analysis" by P. Lax

**Language: **TR, EN

**Title of the course:** Introduction to Real Analysis

**Instructor:** Assoc. Prof. Özgür Martin

**Institution:** MSGSÜ

**Dates:** 27 August – 2 September 2018

**Prerequisites:** Calculus

**Level:** Advanced undergraduate, graduate

**Abstract:** Riemann integration, pointwise and uniform convergence of sequences of functions, Lebesgue measure and integration.

**Language:** TR, EN

**Title of the course:** Category Theory

**Instructor:** Dr. Matteo Paganin

**Institution:** Sabancı Ü.

**Dates:** 27 August – 2 September 2018

**Prerequisites:** Some group theory, some topology, the more the better, to have examples.

**Level:** Graduate, advanced undergraduate

**Abstract:** 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. The aim of the course will be to show how most of the common contructions in Mathematics are adjoints.

**Language: **TR, EN

**Title of the course:** Advanced Calculus: Differential Form Approach

**Instructor:** Dr. Çağrı Dinler

**Institution:** Boğaziçi Ü.

**Dates:** 27 August – 2 September 2018

**Prerequisites:** Lineer Cebir ve Advanced Kalkülus

**Level:** Undergraduate-Graduate

**Abstract:** Bu ders ileride differensiyel geometri konularını öğrenmek isteyen öğrenciler için Calculus ve Lİneer Cebir dersleri ile Geometri arasındaki boşluğu doldurmayı hedeflemektedir. Calculus ve Lİneer Cebir konularını "differensiyel formlar" ile öğrenmek daha sonra öğrenilebilecek "Manifold ve Riemann Uzayları" konularının anlaşılmasını kolaylaştıracaktır. Derste temel olarak, çok-boyutlu integraller, Stoke's teoremleri, lineer cebir, kapalı fonksiyon teoremi (implicit function theorem), max-min problemleri, Lagrange multiplier gibi konular "differensiyel formlar" kullanılarak işlenecektir. Bu ders H. Edward'ın "Advanced Calculus: Differential Form Approach" adlı kitabının ilk beş konusuna değinmeyi hedeflemektedir

**Language:** TR, EN

**Title of the course:** Complex Analysis

**Instructor:** Prof. Sten Kaijser

**Institution:** Uppsala University

**Dates:** 27 August – 9 September 2018

**Prerequisites:**

First week: Calculus, linear algebra, complex numbers.

Second week: A first course in complex analysis.

**Level:** University student/graduate student (depending on audience)

**Abstract:**

First week: Complex numbers, analytic functions up to Cauchy formula

Second week: Some important theorems of complex analysis

**Language: **EN

**Eğitmen:** Doç. Dr. Özlem Beyarslan

**Kurum:** Boğaziçi Ü.

**Tarih: **3-9 Eylül 2018

**Dersin Adı:** Sayılar Kuramı

**İçerik:**

Bölünebilme, Öklid algoritması

Asallar

Aritmetiğin Esas Teoremi

En büyük ortak bölen

Bezoult özdeşliği

Fermat ve Wilson Teoremleri

Euler fonksiyonu ve Euler teoremi

En küçük ortak kat

Modüler aritmetik

Bölenlerin sayısı

Bölenlerin toplamı

Mükemmel sayılar

**Title of the course:** Hilbert’s Axioms

**Instructor:** Assoc. Prof. Emre Coşkun

**Institution:** ODTÜ

**Dates:** 3-9 September 2018

**Prerequisites:** Familiarity with Euclidean geometry.

**Level:** Graduate, advanced undergraduate, beginning undergraduate

**Abstract:** A quick overview of the first few books of Euclid’s Elements. Hilbert’s Axioms.

**Language: **EN, TR

**Textbook or/and course webpage**: Robin Hartshorne, Euclid and Beyond (ch. 2)

**Title of the course:** Statistical Mechanical Models on the Lattice

**Instructor:** Mr. Arif Mardin

**Institution:** -

**Dates:** 3-9 September 2018

**Prerequisites:** Some familiarity with random walk models on the lattice in d-dimensional space and discrete probabilistic models would be helpful. Basic notions of equilibrium statistical mechanics at the undergraduate level should also be useful.

**Level:** Advanced undergraduate, graduate

**Abstract:** The aim of this course is to offer a mathematically rigorous introduction to the Ising, percolation and self-avoiding walk models on the lattice in d-dimensional space. The course will focus on some of the important results obtained until now and the methods used to achieve them.

**Language: **EN

**Textbook:** Friedli, S. and Velenik, Y.: “Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction”, Cambridge University Press, 2017,

G. Grimmett: “Percolation”, 2nd edn., Springer Verlag, 1999,

Madras, N. and Slade, G.: “The Self-Avoiding Walk”, Birkhäuser, 1993.

Madras, N., and Slade, G.,: "The Self-Avoiding Walk", Birkhäuser, 1993.

**Title of the course:** Transform Methods for Differential Equations – A unified approach

**Instructor:** Dr. Konstantinos Kalimeris

**Institution:** University of Cambridge

**Dates:** 3-9 September 2018

**Prerequisites:** Basic knowledge of Complex Variables (An introduction to basic Complex Analysis themes will be given in the first lectures).

**Level:** Advanced undergraduate

**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:** Paradoxical decompositions: The Banach-Tarski paradox and others

**Instructor:** Asst. Prof. Burak Kaya

**Institution:** ODTÜ

**Dates:** 3-9 September 2018

**Prerequisites:** Basic group theory knowledge is sufficient. Some familiarity with group actions is suggested. The rest of the course will be self-contained.

**Level:** Advanced undergraduate, beginning Undergraduate

**Abstract:** We shall cover the Banach-Tarski paradox and some other paradoxical decompositions. If time permits, we may learn about amenable groups and prove some basic facts.

**Textbook or/and course webpage:** Stan Wagon, The Banach-Tarski Paradox, Cambridge University Press. The instructor will also provide some lecture notes.

**Language: **TR, EN

**Title of the course:** Introduction to Polyhedral Geometry

**Instructor:** Asst. Prof. Zafeirakis Zafeirakopoulos

**Institution:** GTÜ

**Dates:** 3-9 September 2018

**Prerequisites:** Knowledge of basic linear algebra and basic calculus.

**Level:** Graduate, advanced undergraduate

**Abstract:**

1. Definition of Polyhedra and friends

2. H-rep, V-rep , faces and complexity

3. Combinatorics of Polytopes

4. Generating Function of cones

5. Generating Function of polyhedra

6. Applications

**Language: **EN

**Title of the course:** Yakınsama Türleri ve İlişkileri

**Instructor:** Prof. Şafak Alpay

**Institution:** ODTÜ

**Dates:** 3-9 September 2018

**Prerequisites:** -

**Level:** Lisans 3 ve üstü

**Abstract:** Fonksiyonların noktasal, düzgün ve hemen her yerde yakınsaması, L_p(l <= p < \infty) uzayında yakınsama, ölçümde yakınsama, Egoroff Teoremi, Vitali Yakınsama Teoremi, Ascoli-Arzela Teoremi, L_p uzayının Banach uzayı özellikleri.

**Language: **TR, EN

**Eğitmen:** Prof. Ali Nesin

**Kurum:** İstanbul Bilgi Ü.

**Tarih: **3-16 Eylül 2018

**Dersin Adı:** Oyun ve Olasılık

**İçerik:** Oyunların matematiğine odaklanacağız. Şans oyunları, şansın olmadığı oyunlar, sonlu oyunlar, potansiyel olarak sonlu oyunlar, sonsuz oyunlar... Varsa oyunların kazanan stratejilerini, yoksa kazanma olasılığımızı artıran stratejileri bulacağız. Kombinasyon hesapları ve olasılık konularına doğal olarak eğileceğiz.

**Eğitmen:** Prof. Melih Boral

**Kurum:** -

**Tarih: **3-16 Eylül 2018

**Dersin Adı:** Polinomlar ve Cebirsel Sayılar

**İçerik:** Polinomlar ve polinom fonksiyonları, bölme algoritması, polinomların rasyonel kökleri, cebirsel sayılar ve polinomları, indirgenemez polinomlar.

**Eğitmen:** MSc. Kübra Dölaslan

**Kurum:** ODTÜ

**Tarih: **3-16 Eylül 2018

**Dersin Adı:** Problem Saati

**İçerik:** Bu derste, gün boyunca işlenen konularla ilgili problemler çözülecek, öğrencilerin soruları yanıtlanacaktır. Not: Problemler kolay olmayacaktır ve öğrenciden aktif katılım beklenecektir.

**Eğitmen:** Prof. Haluk Oral

**Kurum:** -

**Tarih: **10-16 Eylül 2018

**Dersin Adı:** Matematik Sohbetleri

**İçerik:** Sayma problemleri, reel sayıların bazı özellikleri, Binom teoremi, Öklid algoritması, şifreleme.

**Title of the course:** Classical Construction Problems in Geometry

**Instructor:** Assoc. Prof. Emre Coşkun

**Institution:** ODTÜ

**Dates:** 10-16 September 2018

**Prerequisites:** Familiarity with Euclidean geometry is a must, experience with fields and field extensions is a plus.

**Level:** Graduate, advanced undergraduate, beginning undergraduate

**Abstract:** The three classical problems in geometry: quadrature of the circle, trisection of the angle, duplication of the cube. Fields and field extensions. Constructible numbers. Impossibility of the solutions of the three classical problems with ruler and compass.

**Language: **EN, TR

**Textbook or/and course webpage: **Robin Hartshorne, Euclid and Beyond (ch. 6)

**Title of the course:** Construction of the real numbers

**Instructor:** Dr. Denis Chéniot

**Institution:** Université de Marseille

**Dates:** 10-16 September 2018

**Prerequisites:** First year of university

**Level:** Advanced undergraduate

**Abstract:** Construction of the real numbers by cuts. Theorem of the least upper bound, convergence of bounded monotone sequences, convergence of Cauchy sequences. Construction of the real numbers by Cauchy sequences, equivalence of the two constructions. Characterization of the order type of the set of real numbers as having a countable dense subset, no maximum or minimum and no gaps.

**Language: **EN, TR

**Title of the course:** Axiom of determinacy and some of its consequences

**Instructor:** Asst. Prof. Burak Kaya

**Institution:** ODTÜ

**Dates:** 10-16 Eylül 2018

**Prerequisites:** Exposure to topology of metric spaces is required. Little exposure to Lebesgue measure is suggested but not required, as we shall go over the construction. It may help to be acquainted with the axiom of choice and its variants, though, knowledge on axiomatic set theory is not required.

**Level:** Graduate, advanced undergraduate

**Abstract:** We shall first introduce infinite games on natural numbers and the axiom of determinacy (AD) which states that all such games have winning strategies for one of the two players. We will then cover some consequences of AD, without assuming the axiom of choice. Finally, we will cover some determinacy results such as the Gale-Stewart theorem, assuming the axiom of choice.

**Textbook or/and course webpage:** Relevant parts of the book “Set Theory, The Third Millenium Edition” by Thomas Jech. The instructor will also provide some lecture notes.

**Language: **TR, EN

**Title of the course:** Introcduction to Convex Optimization

**Instructor:** MSc. Oğuzhan Yürük

**Institution:** Technische Universität Berlin

**Dates:** 10-16 September 2018

**Prerequisites:** Basic Linear Algebra

**Level:** Advanced undergraduate, beginning Undergraduate

**Abstract:** This will be an introductory course on convex optimization. Participants are expected to have a basic knowledge of linear algebra. A background on optimization is not required however it can be useful. The main goal of this course is to provide participants with the required understanding of convex optimization in order to study conic programming. During the 6 days we will cover the following:

• Day 1: The notation that will be used throughout the course will be set. Preliminary results coming from the linear algebra will be given

• Day 2: The notions of affine hull, convex hull and conic hull will be introducted and we will see the Carathedory’s theorem on convex hulls.

• Day 3: We will cover the operations that preserve convexity. Then we will prove an important result of convexity, seperating hyperplane theorem.

• Day 4: Definition of a convex function and operations that preserve convex functions will be given. The first and second order convexity conditions for functions will be shown and used to verify some important examples of convex functions.

• Day 5: The notion of convex optimization problem will be introduced with the notation that comes with it. First order optimality conditions will be proven.

• Day 6: We will focus on generalized cone inequalities and then understand what is a conic programming problem why it is a convex problem.

**Language: **EN

**Title of the course:** Banach Uzayları ve Üzerlerindeki Sınırlı Doğrusal Dönüşümler

**Instructor:** Prof. Şafak Alpay

**Institution:** ODTÜ

**Dates:** 10-16 September 2018

**Prerequisites:** -

**Level:** Lisans 4 ve üstü

**Abstract:** Sonlu boyutlu normlu uzaylar, Banach-Steinhaus Teoremi, Açık Gönderim Teoremi, Hahn-Banach Teoremi, L_p uzaylarının Banach uzayı özellikleri.

**Language: **EN, TR

**Title of the course:** Percolation on Lattices

**Instructor:** MSc. Pal Galicza

**Institution:** Renyi Institute, Budapest

**Dates:** 10-16 September 2018

**Prerequisites:** A course in probability theory and basic knowledge of calculus and graph theory is necessary.

**Level:** Advanced undergraduate, graduate

**Abstract:** Percolation is perhaps the simplest model of statistical physics exhibiting phase transition. The main focus of the course will be on plane lattices, but if there is interest we shall investigate different graphs and groups. We will prove that the critical probability on the square lattice is 1/2 and that at criticality there is no percolation. Finally, we shall sketch the proof of Smirnov's celebrated theorem on the conformal invariance of the limit of percolation probabilities.

**Language:** EN

**Textbook:** Bollobas -Riordan: Percolation, Grimmett: Percolation

**Title of the course:** Ölçüm Teorisi ve İntegral

**Instructor:** Prof. Zafer Ercan

**Institution:** AİBÜ

**Dates:** 10-16 September 2018

**Prerequisites:** -

**Level:** Lisans 3 ve üstü

**Abstract:** Sigma cebirlerinde ölçüm kavramı, ölçülebilir kümeler ve fonksiyonlar, Lebesgue ölçüm, çeşitli yakınsaklık kavramları ve İntegrallenebilir fonksiyonlar.

**Language: **EN, TR

**Title of the course:** "Learning-Teaching-Researching Mathematics"

**Instructor:** Prof. Alexandre Borovik

**Institution:** Manchester U.

**Dates:** 3 - 16 September 2018

**Prerequisites:** Interest to mathematics

**Level:** All levels, from schoolchildren to teachers

**Abstract:** The course is based on my books and papers about mathematical thinking and mathematical practice.

**Language:** English

**Title of the course:** Singular curves (Tekil eğriler)

**Instructor:** Prof. Ferit Öztürk

**Institution:** Boğaziçi Ü.

**Dates:** 10-16 September 2018

**Prerequisites: **Analysis, algebra

**Level:** higher undergrad, grad

**Abstract:** Analytic curves in C^2 are 2-manifolds. In this course, we investigate the topology of those curves with an isolated singularity at the origin. We will talk about the Newton polygon, the Puiseux series and a bit of knots. The course will be self-contained. (C^2'de karmaşık analitik eğriler, iki boyutlu manifoldlardır. Orijinde tek bir tekilliği olan eğrilerin topolojisi hakkında konuşacağız. Newton poligonu, Puiseuex serileri ve bir miktar düğümler içeren bu ders, ihtiyacı olan her topolojik/geometrik kavramı kendi kuracak.)

**Language:** TR or ENG, depending on the audience

**Title of the course:** Introduction to probability theory

**Instructor:** Prof. Eduard Emelyanov

**Institution:** ODTÜ

**Dates:** 10-16 September 2018

**Prerequisites:** Basic calculus and linear algebra

**Level:** Undergraduate

**Abstract:** Random vectors and sequences.

Law of large numbers. Markov chains with finite number of states. Problems solving.

**Language: **EN

**Title of the course:** Introduction to Dynamic System Modelling

**Instructor:** Yağmur Denizhan

**Institution:** Boğaziçi University, Electrical and Elctronics Engg. Dept.

**Dates:** 10-16 September 2018

**Prerequisites:** Basic knowledge on differential equations

**Level:** Advanced undergraduate, graduate

**Abstract:** This course is meant to be an introduction to the mathematical modelling of dynamic systems and some associated philosophical problems. Due to time limitation the analyses will mainly focus on continuous-time systems. Additional to some common examples with simple models, it will be discussed how more complex systems can be approached. During the 6 days the following issues will be addressed:

• Day 1: What is a system? A dynamic system? State variables and state-space representation. Poincaré’s geometric approach. Equilibrium behaviours.

• Day 2: Dissipativeness. Attractors. Stability. Lyapunov theorems.

• Day 3: Modelling periodic behaviour. Coupled systems and synchronisation.

• Day 4: Deterministic chaos. Limited predictability. Chaotic attractors.

• Day 5: Chaos control. Synchronisation of coupled chaotic systems.

• Day 6: Discussion of how to approach complex dynamic phenomena.

**Language:** X

**Title of the course:** Transfinite induction and well-ordered sets.

**Instructor:** Dr. Denis Chéniot

**Institution:** Université de Marseille

**Dates:** 17-23 September 2018

**Prerequisites:** First year of university

**Level:** Advanced undergraduate

**Abstract:** Examples of the use of transfinite induction in mathematics. Definition and elementary properties of the well-ordered sets. Proof and definition by transfinite induction, with an emphasis on the justification of the latter. Comparability of well-ordered sets. Proof of the well-orderability of every set (Zermelo's theorem).

**Language: **EN

**Title of the course:** Solvable and Nilpotent Groups

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 17-23 September 2018

**Prerequisites:** A good course in Group Theory.

**Level:** Advanced undergraduate, graduate.

**Abstract:** Commutators, commutator calculus. Derived subgroups. Central series. Solvable and nilpotent groups. Examples of linear groups: Borel subgroup and unipotent subgroups.

**Language: **TR, EN

**Title of the course:** Quadratic Forms

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 17-23 September 2018

**Prerequisites:** Linear Algebra.

**Level:** Advanced undergraduate, graduate.

**Abstract:** We will classify quadratic forms over real numbers, complex numbers and finite fields.

**Language: **TR, EN

**Title of the course:** Invariants of Permutation Groups

**Instructor:** MSc. Hülya Duyan

**Institution:** Central European University

**Dates:** 17-23 September 2018

**Prerequisities:** Basic group theory and graph theory

**Level:** Undergraduate, graduate

**Abstract:** The base size of a group and the metric dimension of a graph were introduced around 40 years ago. They are used in different areas such as computational group theory and the graph isomorphism problem. Since the introduction of distinguishing number of permutation groups, many connections between base size, metric dimension and distinguishing number have been discovered. In this course, we shall study these concepts, their relations and cover some applications.

**Language:** TR, EN

**Title of the course:** Categorificaiton and diagrammatic algebra

**Instructor:** Dr. Can Ozan Oğuz

**Institution:** University of Southern California

**Dates:** 17-23 September 2018

**Prerequisites:** No prerequisites for diagrammatic algebra, however for categorification, some experience with algebraic structures such as vector spaces is necessary. No knowledge of category theory is required. We will introduce the necessary notions along the way.

‘Görsel cebir’ için hiç bir önkoşul yoktur, ancak ‘categorification’ için vektör uzayları gibi cebirsel yapılara aşinalık gerekir. Kategori teorisi bilgisine gerek yoktur. Derste gereken yerlerde ilgili kavramları tanımlayacağız.

**Level:** Graduate, advanced undergraduate

**Abstract:** People understand the world as objects, and relations between these objects. Category theory adopts the same approach and works with mathematical objects, and relations between those objects. However some mathematical theories are blind to relations, they just focus on objects. Categorification is the term used for introducing an extra layer of relations between these objects.

A suitable notation and language for this purpose is diagrammatic algebra, where algebraic variables are denoted as embedded curves, and have a topological flavor. The course will have two parts, diagrammatic algebra and categorification, which can be studied independently.

İnsanların dünya anlayışı nesneler, ve nesneler arasındaki ilişkilerden oluşur. Kategori teorisi bu fikir üzerine kurulmuştur. Bir kategoride matematiksel nesneler ve aralarındaki ilişkiler vardır. Ancak bazı teorilerde sadece nesneleri görürüz. ‘Categorification’ böyle bir teoriyi alıp, nesneler arasındaki ilişkileri tanımlama anlamına gelen bir kavramdır.

Bu işlemi gerçekleştirmek için ‘görsel cebir’ hem uygun bir dil, hem de iyi bir notasyondur. Bu notasyonda cebirsel değişkenleri uzaya gömülü eğriler olarak çizilir ve topolojik özellikleri vardır. Eğrileri izotopi altında değiştirseniz de aynı cebirsel elemanı temsil ederler. Ders ‘categorification’ ve ‘görsel cebir’ diye, birbirinden bağımsız anlaşılabilen iki kısımdan oluşacaktır.

**Language: **TR, EN

**Title of the course:** Lie Gruplari ve Lie Cebirleri

**Instructor:** Dr. Ezgi Kantarcı

**Institution:** University of Southern California

**Dates:** 17-23 September 2018

**Prerequisites:** Linear Algebra

**Level:** Graduate, advanced undergraduate

**Abstract:** This course will be an introduction to Lie Groups and Lie Algebras, through matrices. We will work on classical Lie groups and their corresponding algebras, loosely following the first few chapters of Lie Groups: An Introduction through Linear Groups by Wulf Rossmann.

**Language: **EN

**Title of the course:** Symmetric Polynomials and Young Tableaux

**Instructor:** Dr. Ezgi Kantarcı

**Institution:** University of Southern California

**Dates:** 17-23 September 2018

**Prerequisites:** Undergraduate Algebra

**Level:** Graduate, advanced undergraduate, undergraduate

**Abstract:** Symmetric polynomials and Young Tableux are at the core of a lot of research in Algebraic Combinatorics. The purpose of this course is to teach the basics of this area, and then introduce three popular branches: quasisymmetric functions, shifted tableaux and crystal graphs.

**Language: **EN