**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:** A quick course in knot theory

**Instructor:** Dr. Neslihan Güğümcü

**Institution:** National Technical University of Athens

**Dates:** 9-15 July 2018

**Prerequisites:** -

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

**Abstract:** Knots have been objects of mathematics since the 18th century. Today the theory of knots is a developed theory lying in the branch of algebraic topology and it interrelates with many other areas such as mathematical physics, physics, biology and chemistry. In this course, we will learn the basics of the theory of knots. The outline of the course will be mostly as follows.

Day 1: Basic concepts of algebraic & geometric topology

Day 2: Knots and links in mathematics: an overview discussion and formal definitions and basic theorems

Day 3: Simple invariants of knots and links

Day 4: Some polynomial invariants of knots and links

Day 5: The Jones polynomial and the Kauffman bracket

Day 6: Other knotted objects; An introduction to the theory of braids and tangles, Applications of the knot theory to biology, physics and chemistry

**Language: **TR, 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:** 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:** 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

**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:** 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:** Topics in Number Theory

**Instructor:** Dr. Haydar Göral

**Institution:** Koç 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 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 “pseudo code” 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:** Topics in Ring Theory

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

**Institution:** Boğaziçi Ü.

**Dates:** 13-19 August 2018

**Prerequisites:** Undergraduate algebra

**Level:** Advanced undergraduate, graduate.

**Abstract:** Representations of finite groups

**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:** 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

**Cv: **xxx

**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:** 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:** Prof. Ali Nesin

**Kurum:** İstanbul Bilgi Ü.

**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:** Semisimple rings

**Instructor:** Assoc. Prof. Müge Taşkın

**Institution:** Boğaziçi Ü.

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

**Prerequisites:** Ring theory and module theory

**Level:** Graduate

**Abstract:** After defining and studying basic properties of simple and semisimple rings, we will focus on Artinian semisiple rings. We'll prove the Artin-Wedderburn theorem which provides the classification of Artinian semisiple rings and the modules over them.

**Language:** İngilizce

**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:** Real Analytic Functions

**Instructor:** Dr. Derya Çıray

**Institution:** Universitat Konstanz

**Dates:** 3-9 September 2018

**Prerequisites:** Basic analysis

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

**Abstract:**

The aimed lecture will basically consists of the following parts:

1. Elementary properties of Real Analytic functions (Power series,analytic continuation and more)

2. Implicit function theorem

3.The Weierstrass Preparation Theorem

**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:** Henstock-Kurzweil integral

**Instructor:** Asst. Prof. Serge Randriambololona

**Institution:** Galatasaray Ü.

**Dates:** 3-9 September 2018

**Prerequisites:** A good understanding of a first course of analysis in one variable

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

**Abstract:** The Henstock-Kurzweil integral is an integral which is more general than Riemann integral (in the sense that some non Riemann integrable functions may happen to be Henstock-Kurzweil integrable), but does not require measure theory for its definition (as opposed to Lebesgue integral).

We will quickly review the theory of Riemann integral and point-out some of its limitations. We will then discuss how the Henstock-Kurzweil integral allows one to deal with some of these limitations. If time permits, we will discuss some relations to the Lebesgue integral.

**Textbook or/and course webpage: **

https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/kurzweil.pdf (part of it)

**Language: **EN

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

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

**Institution:** GTÜ

**Dates:** 3-9 September 2018

**Prerequisites:** -

**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.

**Title of the course:** Tensor, Exterior Products and Determinents

**Instructor:** Prof. Ali Nesin

**Institution:** İstanbul Bilgi Ü.

**Dates:** 3-16 September 2018

**Prerequisites:** A solid course on abstract algebra.

**Level:** Advanced undergraduate, graduate.

**Abstract:** We will define and prove fundamental results about tensor products and tensor algebras.

**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:** 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