2009 Lisans ve Lisanüstü TMD Matematik Yazokulu

  • Tarih13 Temmuz - 30 Ağustos 2009 
  • Amaç

    Lisans ve lisansüstü TMD yazokulu Şirince'de Matematik Köyü'nde 13 Temmuz - 30 Ağustos 2009 tarihleri arasında gerçekleşecektir. Değişikliğe tabi olabilecek ders programını aşağıda bulacaksınız.

    Bu yaz yapılacak lisans yazokulları için TÜBİTAK'tan ne yazık ki destek alınamamıştır.

    Bir matematik bölümünün birinci sınıfını başarıyla bitirmiş her öğrenci yazokuluna katılabilir; matematik bölümü dışından gelecekler ancak istisnai olarak ve yer varsa kabul edileceklerdir.

    2009 TMD Yazokulu başvuru formu icin tıklayın. Başvuru formunu aslicankorkmaz@nesinvakfi.org  adresine e-postayla yollamalısınız. Başvurunuzun ulaştığına dair bir onay mesajı gönderilecektir. Katılmak istediğiniz tarihleri başvuru formuna mutlaka yazmalısınız.

     
  • Ücret

    Yazokulunun ücreti, dört öğün yemek, konaklama, dersler ve her türlü temel ihtiyaçlar dahil kurumlara 70 TL, kişilere 50 TL`dir. Çadırlarda kalacaklara %50 indirim uygulanır. İhtiyacı olanlara burs vermekte imkânlarımız dahilinde cömert davranıyoruz.

    Destek: TMD ve TÜBİTAK`tan lisansüstü yazokulu için parasal destek aldık. Öte yandan bu yaz yapılacak lisans yazokullarının hiçbiri için TÜBİTAK`tan destek alamadık. Ege, Adnan Menderes ve İstanbul Bilgi üniversiteleriyle özel anlaşmamız vardır; bu kurumların öğrencileri bölüm başkanlarından gerekli bilgiyi alabilirler. Geçmiş tecrübelerimize dayanarak, kurumlardan parasal destek alalım ya da almayalım, ilkesel olarak her öğrenciden sembolik olarak günde 10 TL almaya karar verdik.
     
  • Genel Bilgi

    Süre: Katılım hafta sayısına göredir. Haftalar pazartesi sabahı başlar ve bir sonraki pazar akşamı biter. Köy‘e başlangıç tarihinden bir gün önce (bir pazar günü) gelinir ve bitiş tarihinde (gene bir pazar günü) Köy‘den ayrılınır. Haftalarımızda altı çalışma gün vardır, hafta ortasında, perşembe günü tatil yapılır ve topluca bir yere gidilir.

    Her ders 1,5 ile 2 saat arasında sürer. Genelde günde en az dört ders vardır. Aynı anda birkaç ders birden olabilir. Katılımcıların da seminer vermesi beklenir. Bazı dersler İngilizce olabilir; dolayısıyla katılımcıların matematik kitabı okuyacak ve ders dinleyebilecek kadar İngilizce bildikleri varsayılır.

    Köy‘de bol bol çadır kuracak yer vardır. Kısıtlı sayıda çadırımız vardır ancak çadırda kalacak katılımcıların çoğunun çadırlarını getireceklerini umuyoruz.

    Katılımcıların Köy‘de çamaşır, bulaşık, temizlik, yemek gibi gündelik işlerde çalışacakları varsayılır.

     
Kayıt ve başvuru işlemleriyle ilgili sorularınız için
aslicankorkmaz@nesinvakfi.org 
Nesin Matematik Köyü Etkinlikleri

Program (Değişebilir/Subject to modifications - Son günceleme tarihi 5 Ağustos 2009)

Ayrıntılı Ders İçerikleri

Title: Abelian Group Theory 
Instructor: Prof. Dr. Ali Nesin 
Institution: İstanbul Bilgi Üniversitesi 
Prerequisites: - 
Dates: 13-26 July 
Contents: Classification of finitely generated abelian groups. Torsion abelian groups. Free abelian groups and their subgroups. Classification of divisible abelian groups. Automorphisms of abelian groups.

Başlık: Analizden Kesitler 
Eğitmen: Prof. Dr. Yusuf Ünlü 
Kurum: Çukurova Üniversitesi 
Gereken: En az bir yıllık analiz dersi 
Tarih: 13-19 Temmuz 
İçerik:

Analiz‘in sayılar teorisine bazı uygulamaları

  • İrrasyonel ve transandant sayılar, Liouville Teoremi.
  • e sayısının transandantlığı.
  • π sayısının irrasyoleliği.

Sonsuz seriler.

  • Bernoulli polinomları ve sayıları.
  • Euler toplama formülü. Basel Problemi ve Euler‘in çözümü
  • Serilerin toplamının hesaplanmasında residü yöntemi
  • Serilerin toplamının hesaplanmasında kullanılan diğer bir takım önemli yöntemler.

Faktöriyel ve Gamma fonksiyonu

  • Genel bilgiler
  • Wallis çarpımı
  • Stirling Formülü

Title: Mathematical Foundations of Quantum Mechanics 
Instructor: Msc. E. Mehmet Kıral 
Institution: Boğaziçi Üniversitesi 
Prerequisites: Lineer Algebra 
Dates: 13-19 July 
Contents: In this short one week course we will not do quantum mechanics, but rather cover the mathematical background necessary for having a formal theory of it. So we will cover the theory of Hilbert Spaces with a view towards its use in Physics. We will accomplish this by following John Von Neumann‘s book; "Mathematical Foundations of Quantum Mechanics".

Title: Classical Mechanics 
Instructor: Dr. Ashna Sen 
Institution: Brockwood Park School 
Prerequisites: Lisans 
Dates: 13-26 July 
Contents:

Day 1: Introduction to classical mechanics – short history. Basic concepts of force, motion, mass and units of physical quantities used in laws of motion. Quick survey of laws of motion.

Day 2. Set up of all three laws of motion and introduction to Newton`s Laws. Implications of Newton`s laws, idea of inertial frame of reference and motion as momentum. Examination of F = p¢ (derivative of momentum) and third law: F(ij) = -F(ji). Full analysis of all three laws of motion and some problems related to them including motion down an inclined plane and some Tension/String/Pulley type problems

Day 3: Introduction of conservation of energy. F = ma continued, Concept of work as an integral quantity. F.dr = dW and integrating around integrals with one example using vector calculus and study of line integrals/paths of integration.

Day 4 &5: Continue with definition of work as line integral and conservative fields – meaning of conservative and non-conservative force fields. How the definition of work arises out of integrating T = mv2. Also, around a closed curve the line integral gives ‘zero` signifying a conservative field. Examples and several line integral problems worked out showing path independence.

Day 6&7: Curl of a conservative force field is zero. Why? Proof. Idea of potential, and potential energy of a force field and their difference. Motion in a general one dimensional potential including calculations of stability and equilibrium.

Day 8 More on momentum. Conservation of momentum. Angular momentum revisited. Some examples using vector calculus.

Day 9 Calculus of variations – introduction to famous problems like the `Brachistochrone`. Fermat`s principle and use of it for proofs of Snell`s law and problems arising out of them. Motivation: idea behind `extremalising`

Day 10 Lagrange`s formulation introduction: Principle of least action. Changing coordinate systems –  Setting up for Hamiltonian dynamics. Suggestions and more equations. Generalised momenta, Conditional variation including the lagrange multiplier method– catenary. What is the meaning of constraint.

Day 11&12 More indepth analysis of Hamiltonian and Lagrangian ideas and wrap-up of all concepts. 

Title: Random Graphs 
Instructor: Dr. Özlem Beyarslan 
Institution: Boğaziçi Üniversitesi 
Prerequisites: - 
Dates: 13-26 July 
Contents:

1. What is a random graph G(n, p). 
2. First order theories. 
3. Zero-one laws. 
4. Ehrenfeucht games 
5. What happens in G(n, p) when p is irrational.

Title: Real Closed Fields

Instructor: MSc. Demirhan Tunç 
Institution: University of Notre Dame 
Prerequisites: Some basic notions about fields and polynomial rings 
Dates: 20-26 July 
Content: Ordered Fields, Formally Real Fields, Real Closed Fields and Hilbert‘s 17th Problem.

Başlık: Hilbert`s 16th Problem 
Eğitmen: Dr. Nermin Salepçi Ferret 
Kurum: Koç Üniversitesi 
Gereken: Giriş seviyesi projektif geometri 
Tarih: 20-26 Temmuz 
İçerik: Reel projektif yüzey üzerindeki  reel cebirsel eğrilerin  topolojik özelliklerinin incelenmesi ve derecesi 6 ve daha küçük olan reel cebirsel eğrilerin sınıflandırılması.

Title: Polynomials 
Instructor: Mr. Doğa Güçtenkorkmaz 
Institution: İstanbul Bilgi Üniversitesi 
Prerequisites: At least one year of solid mathematics education. 
Dates: 20-26 July 
Contents: Algebraic properties of polynomials, irreducibility of some polynomials, field extensions and Galois groups.

Başlık: Morse Kuramı 
Eğitmen: Dr. Ferit Öztürk 
Kurum: Boğaziçi Üniversitesi 
Gereken: İleri düzeyde analiz 
Tarih: 20 Temmuz - 2 Ağustos 
İçerik: Her manifoldun üzerinde bir "morse fonksiyonu" vardır. Bu fonksiyonun kritik noktaları ve bunların çevresi, manifoldun topolojisini anlamak için yeterlidir.

Başlık: Analitik Sayılar Teorisinde Üreteç Fonksiyonları 
Eğitmen: Msc. Ayhan Dil 
Kurum: Akdeniz Üniversitesi 
Gereken: Temel Calculus kavramlarını bilmek. 
Tarih: 27 Temmuz - 2 Ağustos 
İçerik: Toplamlar ve rekürans bağıntıları. Sayılar teorisi ve kombinatorikteki bazı özel polinomlar ve sayılar hakkında temel bilgiler, bunların özelliklerinin üreteç fonksiyonları yardımıyla incelenmesi.

Title: Ordinals, Cardinals and Zorn`s Lemma 
Instructor: Prof. Dr. Ali Nesin 
Institution: İstanbul Bilgi Üniversitesi 
Prerequisites: None (or almost none) 
Dates: 27 July - 9 August 
Contents: Ordinaller, kardinaller, Yerleştirme Aksiyomu, Seçim Aksiyomu, Zorn Önsavi ve uygulamaları.

Title: Introduction to Probability Theory 
Instructor: Msc. Elif Yamangil 
Institution: Harvard University 
Prerequisites: Calculus with one variable and some familiarity with calculus with several (two) variables. 
Dates: 27 July - 2 August 
Content:

  • Counting (multiplication rule, sampling)
  • Probability space, conditional probability (Bayes‘ rule, law of total probability, independence)
  • Random variables (Bernoulli, Binomial, Geometric, Poisson)
  • Expected values (linearity, variance, standard deviation, Uniform, Exponential random variables)
  • Normal distribution (standardization)
  • Transformations of random variables (joint and marginal distributions, random vectors)
  • Sums of independent random variables (Gamma and Beta random variables)
  • Covariance and correlation (variance of the sum, Cauchy-Schwarz)
  • Conditional expectation (conditional variance, iterated expectation)
  • Moment generating functions, inequalities, limit theorems (law of large numbers, central limit theorem)

Title: Hyperbolic Manifolds 
Instructor: MSc. Özgür Evren 
Institution: CUNY 
Prerequisites: Advanced. 
Dates: 3-9 August 
Content: Hiperbolik Uzayin Tanimi (Ust Yari Uzay ve Birim Kure modelleri), Jeodezikler, Hiperbolik Uzaklik Fonksiyonu ve Hiperbolik Alan, Izometrilerin Siniflandirilmasi ve Isometri Gruplari, Duzgun Sureksiz Grup Etkileri, Ayrik Altgruplar, Yorunge Uzaylari, Hiperbolik Yuzey Ornekleri ve Poincare Teoremi.

Title: A Mathematical Introduction to Modal Logic 
Instructor: MSc. Can Başkent 
Institution: CUNY 
Prerequisites: Good level of abstract mathematics + mathematical maturity. 
Dates: 10-16 August 
Content: Introduction: Motivations and History. Syntax, Semantics and Proof Theory of Modal Logic. Topological and Game Theoretical Semantics. Truth Preserving Operations and Bisimulations. Completeness proofs. Different Modal Logics and corresponding defining formulae. Applications of Modal logics to philosophy and computer science.

Title: Around Sylow Theory 
Instructor: Prof. Dr. İsmail Güloğlu 
Institution: Doğuş Üniversitesi 
Prerequisites: At least one year of good undergraduate math education and the following concepts and results from group theory will be assumed as known (better as mastered): Group, subgroup ,coset, Lagrange`s Theorem, normal subgroup,  homomorphism, kernel of a homomorphism, isomorphism, automorphism, quotient group, isomorphism theorems ,direct product,  structure theorems for finite abelian groups, basic knowledge about permutations (sign, cycle decomposition.etc.) and symmetric group, Cayley`s theorem of abstract algebra. 
Dates: 3-9 August 
Content:

  1. Action of a group on a set, permutation representations of a group, Sylow theorems,
  2. Finite nilpotent groups,
  3. Elementary remarks about finite solvable groups,
  4. Some important characteristic subgroups,
  5. Schur-Zassenhaus and Hall` s theorems,
  6. Finite groups admitting fixed-point free automorphisms of prime order.

Title: Elementary Mathematics from a Higher Point of View 
Instructor: Prof. Dr. Alexandre Borovik 
Institution: Manchester University 
Prerequisites: None (or almost none) 
Dates: 3-23 August 
Content: Why is teaching mathematics so difficult? My course will be devoted to hidden structures and concepts of elementary mathematics which frequently remain unnoticed but seriously influence students‘ perception of mathematics.

I will try to develop some (time permitting) of the following themes.

1. Arithmetic of "named" numbers, like the problem of dividing 10 apples between 5 people; Laurent polynomial ring; dimensional analysis in physics, from Froude‘s Law of Steamship Comparison to Kolmogorov‘s "5/3 Law" for the energy distribution in turbulent flow.

2. Why is addition commutative? I will analise a real life story about a  girl aged 6 who could easily solve "put a number in the box" problems of the type 7 + [ ] = 12, by counting how many 1`s she had to add to 7 in order to get 12 but struggled with [ ] + 6 = 11, because she did not know where to start.

3. What is common and what is difference between induction and recursion?

4. Carry (remember what is it? According to Wikipedia, "carry is a digit that is transferred from one column of digits to another column of more significant digits" during addition of decimals) and cohomology. 10-adic and 2-adic numbers. Euler‘s sum

1 + 2 + 4 + 8 + 16 + ... = -1.

5. "Russian peasants‘ multiplication" and modules over commutative rings; exponentiation in modular arithmetic; its applications to cryptography: Diffie-Hellman key exchange and RSA; timing and power trace attacks on embedded cryptographis devices (like microchips in credit cards). Mathematics of binary trees.

6. Why are the Chinese Remainder Theorem in Number Theory and the Lagrange Interpolation Formula in Numerical analysis one and the same thing?

At least the beginning of the course will be relatively elementary. But the students in the course should be psychologically prepared for sudden jumps onto very abstract levels of mathematics.

Title: Groups and Geometry 
Instructor: Assoc. Prof. Ayşe Berkman 
Institution: ODTÜ / METU 
Prerequisites: At least one year of mathematical education 
Dates: 10-16 August 
Content: There are multiple connections between geometry and group theory, since geometry means symmetry and the properties of symmetries are studied by group theory. In this short course we will exploit this idea, from tilings and freezes to Coxeter groups (some of which can be considered to be the building structures of our universe!)

Title: Nonstandard Analysis 
Instructor: Assoc. Prof. David Pierce 
Institution: ODTÜ / METU 
Prerequisites: Some knowledge of analysis and abstract algebra. 
Dates: 10-16 Ağustos 
Content: Invented in the 17th century if not earlier, calculus can be understood in terms of ``infinitesimals``: Non-zero numbers whose absolute values are less than every fraction 1/n.  In this understanding, the region bounded by a curve is the sum of rectangles of infinitesimal width; the slope of a curve at a point is a ratio of infinitesimals. But there are no infinitesimals on the so-called "real number line".  In the usual "rigorous" treatment of calculus, invented in the 19th century, infinitesimals do not appear: they are replaced with the notion of a "limit".However, 20th-century logic shows that infinitesimals can be made just as "real" as the real numbers, so that the original intuitive approach to calculus is entirely justified.

Title: Elliptic Curves 
Instructor: Asst. Prof. Ayhan Günaydın 
Institution: Oxford University 
Prerequisites: At least two years of good undergraduate math education 
Dates: 10-16 August 
Content: Elliptic curves lie in the meeting point of three main areas of mathematics: arithmetic, geometry and complex analysis. The primary aim of this short course is to understand how these areas are connected via elliptic curves. We shall follow a very historical path to do this; we start by considering things as done by Gauss, Abel and Jacobi, and end up at working with elliptic curves in the framework of modern language of geometry and arithmetic. A more precise list of topics would be as follows:

1. Elliptic integrals and elliptic functions: basic definitions, inversion of elliptic functions by Gauss and Abel, the Weierstrass function, Abel‘s theorem.

2. Modular groups and modular functions.

3. Imaginary quadratic fields.

4. Arithmetic of elliptic curves.

We shall mostly focus on 1 and 3. If time allows some more advanced topics might be considered as well.

The audience will be assumed to be familiar with basic notions of algebraic geometry and manifold theory; and of course a good two semesters of complex analysis. A good source for this course would be "Elliptic Curves, H. McKean - V. Moll, Cambridge University Press, 1997".

Title: Discrete Valuation Rings 
Instructor: Prof. Dr. Ali Nesin 
Institution: İstanbul Bilgi Üniversitesi 
Prerequisites: Abstract algebra, some basic knowledge of ring and field theory 
Dates: 10-16 August 
Content: Generalities. Discrete valuation rings and their maximal ideals. Valuations on ℚ. Archimedean and non-Archimedean valuations. Dedekind domains. Localization. Power series, meromorphic functions, p-adic numbers and their extensions. Integral closure. Completion. Derivations. More if time allows.

Title: Short Course in Complex Analysis 
Instructor: Prof. Eduard Emelyanov 
Institution: ODTÜ / METU 
Prerequisites: Analysis 
Dates: 24-30 August 
Content: This course is devoted to a short presentation of basic ideas of geometric theory of functions of one complex variable including the Riemann theorem.

Title: Special Functions: Orthogonal Polynomials 
Instructor: Dr. Veronica Pillwein 
Institution: RISC, Linz 
Prerequisites: Basic knowledge in calculus and algebra. 
Dates: 17-23 August 
Content: Certainly one of the most prominent members in the class of Special Functions are orthogonal polynomials like Jacobi, Hermite or Laguerre polynomials. These polynomials gained their fame because of their various applications in, e.g., physics, numerical mathematics and chaos theory. One can take different viewpoints in the investigation of orthogonal polynomials: (complex) analytic, combinatorial, symbolic, ... 

In this course we will give a self-contained introduction to the theory of orthogonal polynomials covering several of the above aspects. It will contain the basic facts about orthogonal polynomials and provide an outlook to further interesting topics in this context which go beyond the scope of this course. Those interested in symbolic summation and corresponding RISC software will find that the methods treated in the lecture of Flavia Stan are applicable to various problems discussed in this course. 

As prerequisites it will be sufficient to have basic knowledge in calculus and algebra.

Title: Topics in Computer Algebra 
Instructor: Msc. Burçin Eröcal 
Institution: RISC, Linz 
Prerequisites: Linear algebra and some abstract algebra 
Dates: 17-23 August 
Content: After introducing some tools from computer algebra such as computing in homomorphic images and p-adic lifting, applications of these methods to obtain asymptotically fast algorithms for exact linear algebra over polynomial rings will be presented. We will use the open source computer algebra system Sage (http://sagemath.org) for demonstrations of the algorithms discussed.

Title: Symbolic Summation 
Instructor: MSc. Flavia Stan 
Institution: RISC, Linz 
Prerequisites: Basic knowledge from analysis and linear algebra. 
Dates: 17-23 August 
Content: Many of the topics discussed in the lecture can be found in the book ``Concrete Mathematics - A Foundation for Computer Science`` by R.L.Graham, D.E.Knuth und O.Patashnik (Addison-Wesley, 1994). A citation from its preface: 

``... But what exactly is Concrete Mathematics? It is a blend of CONtinuous and disCRETE mathematics. More concretely, it is the controlled manipulation of mathematical formulae, using a collection of techniques for solving problems. Once you dots have learned the material in this book, all you will need is a cool head, a large sheet of paper, and a fairly decent handwriting in order to evaluate horrendous looking sums, to solve complex recurrence relations, ...``
The main topics of the lecture are: recurrence relations, generating functions and summation algorithms. These methods have a wide applicability e.g. in dealing with special functions like classical orthogonal polynomials (more details in Veronika Pillwein‘s course). 

Moreover, out of enviromental reasons and to eliminate the ``decent handwriting`` factor, we will present implementations of these techniques in different software packages of the Algorithmic Combinatorics group at RISC such as fastZeil, MultiSum, GeneratingFunctions, ... and illustrate how these programs are useful in a mathematician‘s day-to-day life.

Title: Finite Rings 
Instructor: Prof. Oleg Belegradek 
Institution: İstanbul Bilgi Üniversitesi 
Prerequisites: Some abstract algebra 
Dates: 24-30 August 
Content: Finite simple rings will be classified.

Başlık: Analizden Kesitler 
Eğitmen: Prof. Dr. Ali Nesin 
Kurum: İstanbul Bilgi Üniversitesi 
Gereken: En az bir yıl matematik eğitimi almış olmak. 
Tarih: 24-30 Ağustos 
İçerik: Gama fonksiyonları, Liouvill sayıları ve analizden seçme konular.

Title: Philosophy of Mathematics 
Instructor: BSc. Brian Edwards 
Institution: Brockwood Park School 
Prerequisites: Interest in philosophy and mathematics 
Dates: 13-26 July 
Content:

Part 1—Ancient Greek Foundations of Philosophy and Mathematics

1.  Introduction + Journey to Pythagorean Revolution

2.  Pythagorean specifics: Form, Cosmos, Harmonics

3.     Platonic developments: Geometry as Virtue: Academy

4.     Plato`s problem: Being and Time

5.     Aristotle: Being ta mathematikos and University foundations.

6.  The Euclidean Point: Summary

Part 2—Modern Transformations of Ancient Problems

1.     Cosmos to Nature, Euclid to Galileo

2.     Renaissance synthesis: Bruno, Cusanus, and Infinity

3.     The rigorous Form: Bacon, Newton and Locke

4.     The Cartesian explosion of Doubt

5.     Ratio and Mathematics: Kant, Leibniz, Spinoza

6.     Back Home: Mathematics, Philosophy and the 20th century.

This is obviously ambitious for two weeks.  The course is thematic, so it may morph depending on the group and where the reflection naturally leads us.

Title: Evrende Neler Var? 
Instructor: Prof. Dr. Ali Alpar 
Institution: Sabancı Üniversitesi 
Prerequisites: None! 
Dates: 17-23 Ağustos 
Content: Gözlemsel olarak evrenin yapısı hakkında bildiklerimiz ve bunun arkasındaki fizik ve bilim tarihi.

Title: Engin Mermut

Instructor: Yard. Doç. Dr. Engin Mermut 
Institution: Dokuz Eylül Üniversitesi 
Prerequisies: An introductory course on algebra is enough. Familiarity wih modules and rings, some special rings, exact sequences will be useful. 
Dates: 17-23 Agust 
İçerik: This one week short course is an introductory course on homological algebra and aims to do as much of the main parts for the following fundamental topics.

1. Motivation: algebraic topology and presentations, projective (free) resolutions.

2. The adjoint pair of functors Hom and tensor product in the categories of modules.

3. Projective, injective and flat modules. Purity.

4. Homology of complexes of modules.

5. Projective, injective and flat resolutions.

6. Derived functors.

7. Ext and Tor, the derived functors of Hom and tensor product.

8. Projective dimension, injective dimension, flat dimension of modules.

9. The left global dimension, the right global dimension, the weak dimension of a ring.

10. Ext and extensions: the Baer sum of short exact sequences.

11. Some special rings characterized homologically: Semisimple rings, von Neumann regular rings, hereditary rings and Dedekind domains, semihereditary rings and Prüfer domains, ...

12. Further topics if time permits.

Title: Commutative Algebra 
Instructor: Doç. Dr. Feza Arslan 
Institution: ODTÜ 
Prerequisites: Herhangi bir temel cebir dersi almış olmak. Ring, ideal gibi kavramlara biraz aşina olmak. 
Dates: 24-30 Ağustos 
Content: Modules, Rings and modules of fractions, Primary decomposition, Integral dependence and integral closure. (Some examples from algebraic geometry and algebraic number theory will be the central objects of interest throughout the course.)

Title: Injective Modules 
Instructor: MSc. Sinem Odabaşı 
Institution: Dokuz Eylül Ü. 
Prerequisites: Modüller kategorisine aşina olmak. 
Dates: 10-16 Ağustos 
Content:

Injective modules

Divisibility

Essential extensions

Injective envelope

Indecomposable injective modules

Injective modules and chain conditions

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