Title of the course - Nesin Matematik Köyü

Transkript

Title of the course: Hilbert Functions, Free Resolutions and Betti Numbers
Instructor: MSc. Selvi Kara
Institution: ITU
Dates: 9 – 15 July 2012
Prerequisites: Level: Undergraduate, Graduate
Abstract: In this course, we will first give an exposition about Hilbert functions, Hilbert
series and their combinatorial properties. We will also give the theory of free resolutions
and their connections with Hilbert functions. After defining simpicial complices, reduced
simplicial homology and Stanley-Reisner rings, we will give several important results
connecting all these .Finally, we will present Hochster's formula to compute Betti
numbers of given Stanley-Reinser ideal. Furthermore, we will construct an example that
explains the formula by doing the necessary computations using techniques of linear
algebra.
Başlık: Sezgisel Kümeler Kuramı
Eğitmen: Prof. Dr. Ali Nesin
Kurum: İstanbul Bilgi Üniversitesi
Tarih: 9 – 22 Temmuz 2012
İçerik: Bu derste aksiyomatik olmasa da oldukça derin kümeler kuramı yapacağız.
Amacımız fonksiyon, kartezyen çarpım, denklik ilişkisi, sıralama gibi matematiğin en
temel kavramlarını olabildiğince matematiksel biçimde tanımlamak ve sonsuz kümeleri
daha iyi anlamak olacak.
Kaynakça: Ali Nesin, Sezgisel Kümeler Kuramı, Nesin Yayıncılık 2012.
Başlık: Analitik Geometri
Eğitmen: Yard. Doç. Özlem Beyarslan
Kurum: Boğaziçi Üniversitesi
Tarih: 9 - 22 Temmuz 2012
İçerik: Koordinat düzlemi, Doğru denklemleri. Çember, elips, parabol hiperbol. Kutupsal
koordinatlar.
Başlık: Polinomlar
Eğitmen: MSc. Doğa Güçtenkorkmaz
İçerik: Tek dereceli polinomlar. İndirgenemez polinomlar. Tamsayılar üzerine polinomları
çarpanlarına ayırma. Kök bulma metodları. EBOB ve eşitlikler.
Title of the course: Fuchsian Groups
Instructor: MSc. Doğa Güçtenkorkmaz
Institution: İstanbul Bilgi University
Dates: 9 – 22 July 2012
Prerequisites: First week: Necessary definitions will be recalled. Second week: There
may be a short review of the first week. So an advanced undergrad can follow the course
after first week.
Level: Advanced undergraduate, beginning Undergraduate
Abstract: First week: Fractional linear transformations with their actions on the Poincare
upper half plane. Properly discontinuous actions and discrete subgroups of PSL(2,R).
Definition and existance of their fundamental domains.
Second week: Algebraic and geometric properties. Their connection with Riemann
surfaces. Arithmetic groups. Modular group and its subgroups.
Title of the course: Permutation Groups
Instructor: Asst. Prof. Seyfi Türkelli
Institution: Georgia University
Dates: 9 – 22 July 2012
Prerequisites: Level: Undergraduate.
Abstract: In this course, I will give a brief introduction to permutation groups. I plan to
cover almost all the topics in the first two chapters of the book “Permutation Groups by
Dixon and Mortimer”. Here is the tentative schedule:
Day 1: Groups and examples
Day 2: Symmetry and symmetric groups
Day 3: Group actions, Orbits and Stabilizers
Day 4: Blocks, primitivity and permutation representations
Day 5: Orbits revisited, fixed points and examples of permutation groups
Day 6: Actions and automorphism groups
Day 7: Graphs, Relations
Day 8: Direct products, semi-direct products, wreath products
Day 9: Imprimitive groups, Primitive wreath products
Day 10: Affine and projective groups
Day 11: Transitive groups of degree at most 7
Day 12: Groups acting on trees
Title of the course: Universal Groups
Instructor: Prof. Mahmut Kuzucuoğlu
Institution: ODTÜ
Dates: 16 – 22 July 2012
Prerequisites: Level: Graduate
Abstract: Direct limit, Hall Universal group, Universal groups, Large groups.
Title of the course: Bilinear Forms
Instructor: Prof. Feride Kuzucuoğlu
Institution: Hacettepe University
Dates: 16 – 22 July 2012
Prerequisites: Level: Beginning Undergraduate
Abstract: Linear Algebra: Vector sapces, Linear Transformaations and Bilinear forms
Title of the course: Algebraic Coding Theory
Instructor: Assoc. Prof. Cem Güneri
Institution: Sabancı University
Dates: 9 – 15 July 2012
Prerequisites: Linear algebra and acquaintance with basic algebra (groups, rings).
Level: Undergrad (3rd and 4th year), or graduate.
Abstract: Coding theory studies problems related to information transmission. The
subject has rich connections to different branches of mathematics. In this course, we will
study codes from algebraic point of view. The following are tentative lecture titles:
Finite fields.
Coding problem and bounds.
Linear codes and examples.
Cyclic codes.
Relations to algebraic curves over finite fields.
Title of the course: Uniform cell decompositon of p-adic numbers
Instructor: Ms. Türkü Özlüm Çelik
Dates: 16 - 29 July 2012
Prerequisites: Analysis, Basic Topology, Algebra.
Level: Advanced undergraduated, graduated.
Abstract: The aim of this lecture is to give a cell decomposition for p-adic fields, uniform
in p. This generalizes a cell decomposition for fixed p, proved by Denef. We also give
some applications of the cell decomposition (for example uniform quantifier elimination
for p-adic fields).
Eğitmen: Yard. Doç. Dr. Özlem Beyarslan
koordinatlar.
Title of the course: Algorithms for Systems of Linear Differential Equations
Instructor: Asst. Prof.Flavia Stan
Institution: INRIA-Microsoft Research Centre
Dates: 23 – 29 July 2012
Prerequisites: Level: Undergraduate
Abstract: We present algorithms for the classification of singularities and for the
symbolic resolution of ordinary linear differential systems x^p dy/dx= A(x)y where y is a
vector with n>1 components and A(x) is a nxn matrix with formal power series
coefficients. The transformation of an n-th order ordinary differential equation to a first
order differential system in companion form is well known. It is then possible to go from
the solution space of one to the solution space of the other. The drawback of rewriting
back an arbitrary linear differential system is a possible swell in the coefficients of the
resulting equation. This motivates the search for algorithms that are directly applicable to
first order systems.
We use linear algebra techniques to present several algorithms for reducing the rank p of
systems of this type at a point, say x=0, to its minimal value, also called the true
Poincaré rank. These algorithms go back to J. Moser(1960), A. Hilali and A. Wazner
(1987), M. Barkatou and E. Pfluegel (2009). After classifying the singularity of the
system, we compute the fundamental matrix solution given by its n linearly independent
formal solutions. We include here algorithms to find polynomial and rational solutions due
to M. Barkatou (1997, 1999). Additional material can be found in the book of W. Wasow
entitled "Asymptotic expansions for ordinary differential equations".
Title of the course: Symbolic solutions of linear ordinary differential equations
Instructor: Asst. Prof. Burçin Eröcal
Institution: TU Kaiserslautern
Dates: 23 – 29 July 2012
Prerequisites: Basic (polynomial) Galois theory
Level: Advanced undergraduate.
Abstract: We present algorithms based on Galois theory for finding closed form solutions
of linear ordinary differential equations. After an elementary introduction to the Galois
theory of differential equations, we will present algorithms to find power series, rational
function and Liouvillian solutions of differential equations. The course will have a strong
computational aspect with implementations of the described algorithms and examples
presented in the open source computer mathematics system Sage.
Title of the course: Variational method for image processing and computer vision
Instructor: Asst. Prof. Jérôme Boulanger
Institution: CNRS-Institut Curie
Dates: 23 – 29 July 2012
Prerequisites: Basic analysis.
Level: Advanced undergraduate.
Abstract: Introduction to the problems and variational methods encountered in image
processing. Links with probabilitic appoaches, priors such as total-variation ot meancurvature motion and the Euler-Lagrange equations associated to the functional will be
described and illustrated for denoising, motion estimation, inpainting, segmentation, etc.
Title of the course: Mathematical foundation of signal and image processing
Instructor: Asst. Prof. Leila Muresan
Institution: ENS (IBENS)
Dates: 23 – 29 July 2012
Prerequisites: Level: Undergraduate.
Abstract: The lecture is an introduction to mathematical tools for image processing. It
covers topics such as: discrete and continuous Fourier transform, wavelet transform,
deconvolution, Wiener filter, Richardson-Lucy algorithm, etc. Throughout the course,
practical illustration of the methods is provided, with emphasis on applications to
microscopy images.
Başlık: Oyun, Olasılık ve Strateji
Tarih: 23 Temmuz – 5 Ağustos 2012
İçerik: Çeşitli matematiksel oyunlar vasıtasıyla, olasılık ve oyunlar kuramına bir nevi
giriş yapacağız. Ayrıca kombinatoriyal hesaplarla da haşır neşir olacağız. Oyun kazanma
stratejileri belirleyeceğiz. Zaman kalırsa sonsuz oyunlar konusunu işleyeceğiz.
Title of the course: Number Sequences
Instructor: MSc. Maximilian Jaroschek
Institution: RISC, Hagenberg
Dates: 30 July - 5 August 2012
Prerequisites: Basic Algebra
Level: Advanced undergrad.
Abstract: In the course we will deal with algebraic theories concerning number
sequences like the Fibonacci numbers or the harmonic numbers. We'll identify different
types of sequences and investiagte their properties as well as their interplay with
recurrences, sums and generating functions. Furthermore, we'll discuss approches for
solving summation problems often arising in combinatorial applications.
Title of the course: Introduction to Tropical Geometry
Instructor: MSc. Hamid Rahkooy
Institution: RISC (Research Institute for Symbolic Computation)
Prerequisites: Preliminary knowledge on Groups and Rings.
Level: Advanced Undergraduate
Abstract: The course will be an invitation to a recently growing field, Tropical Geometry,
based on draft of a book by Maclagan & Sturmfels. "In tropical algebra, the sum of two
numbers is their minimum and the product of two number is their product. This algebraic
structure is known as the tropical semiring or as the min-plus algebra... The origins of
algebraic geometry lie in the study of zero sets of systems of multivariate polynomials.
These objects are algebraic varieties, and they include familiar examples such as plane
curves and surfaces in three-dimensional space. It makes perfect sense to define
polynomials and rational functions over the tropical semiring. The functions they define
are piecewise- linear. Also, algebraic varieties can be defined in the tropical setting. They
are now subsets of R^n that are composed of convex polyhedra. Thus, tropical algebraic
geometry is a piecewise-linear version of algebraic geometry."
Title of the course: Automated Theorem Proving
Instructor: Asst. Prof. Madalina Erascu
Institution: RISC, Linz
Prerequisites: No prerequisites, the lecture is self-contained
Level: Undergraduate, graduate
Abstract: The course is an introduction to logic for Computer Science and Mathematics
students. We plan to discuss the following:
- The principles of mathematical logic and its role in human activity;
- Propositional logic, first-order predicate logic, higher-order logic, proof systems:
correctness, completeness;
- Application of logic in program verification.
Title of the course: An Introduction to Social Choice Theory
Instructor: Prof. Remzi Sanver
Dates: 30 July – 5 August 2012
Prerequisites: Basic concepts of logic.
Level: Graduate, advanced undergraduate
Abstract: Collective decision making in societies can be modeled as an aggregation of ntuples of binary relations into a single one. The course aims to introduce the basic
concepts and results of this well-elaborated literature.
Title of the course: Oyunlar Kuramı - Game Theory
Instructor: Asst. Prof. Uğur Özdemir
Institution: İstanbul Bilgi Üniversitesi
Prerequisites: English
Level: Abstract:
1- Normal Form Games and Rationalizability
2- Static Games of Imperfect Information (Bayesian Games)
3- Dynamic Games of Perfect Information
4- Dynamic Games of Imperfect Information
5- Repeated Games and Reputation
6- Mechanism Design and Agency Theory
Textbook:
Fudenberg, D. and Tirole, J. (1991). Game Theory. MIT Press
Osborne, M. and Rubinstein, A. (1994). A Course in Game Theory. MIT Press
Title of the course: Eşleşme - Matching
Instructor: Prof. Ahmet Alkan, Prof. İpek Sanver
Institution: Sabancı University, İstanbul Bilgi University
Prerequisites: Level: Graduate, advanced undergraduate
Abstract: Day 1 (3 hours). Classical results in matching theory: Stable matchings and
strategic questions
The (non-)existence of stable matchings in several types of matching problems
The structure of the set of stable matchings in “marriage” problems
Decomposition lemma
The manipulability results and the blocking lemma of Hwang
Day 2 (2 hours). A characterization of the core of “marriage” problems
The core of a game
Consistency and Converse Consistency
Elevator Lemma
Bracing Lemma
Day 3 (2 hours). Matching with Money: Assignment Game
Existence and Structure of Stable Matchings
Multiobject Auction
Strategyproofness
Day 4 (3 hours). One-to-Many Extensions
Substitutability and Existence
Multiobject Auction and Matroids
Generalized Deferred Acceptance and Tarski Fixed Point Theorem
Day 5 (3 hours). Many-to-Many Schedule and Random Matching
Substitutability and Revelaed Preference Lattice
Size Monotonicity
Lattice and Other Properties
Day 6 (2 hours). Selected Topics from Current Theory
Title of the course: Matching Markets: A Market Design Approach
Instructor: Prof. Tayfun Sönmez
Institution: Boston College
Prerequisites: -
Level: Graduate
Abstract: The mini-course will provide an overview of some recent research and policy
work on matching markets. The focus of the course is the evolution of the literature both
from a theoretical and also practical perspective. Topics include two-sided matching,
house allocation, school choice, kidney exchange, matching with contracts, and cadet
branching.
Textbook: https://www2.bc.edu/~sonmezt/
Title of the course: Cooperative Game Theory
Instructor: Assoc. Prof. Özgür Kıbrıs
Institution: Sabancı University
Prerequisites: Level: Graduate, advanced undergraduate
Abstract: This course will cover transferable utility games and nontransferable utility
games (focusing on bargaining games). It will discuss well-known axioms for solution
concepts and present characterizations of well-known solutions concepts, such as
Shapley value, Nash bargaining rule, Kalai-Smorodinsky rule, Egalitarian rule, Utilitarian
rule, based on these axioms.
Title of the course: Decision Theory for Risk and Uncertainty
Instructor: MSc. Umut Keskin
Institution: Erasmus School of Economics
Prerequisites: Basic utility theory, elementary probability and calculus
Level: Graduate, advanced undergraduate, beginning undergraduate
Abstract: We will have a rigorous analysis of different decision models used in
economics: Expected value, expected utility, choquet integral, prospect theory. For
motivating these models, some of the famous paradoxes in decision theory will be
discussed: St. Petersburg Paradox, Allais Paradox, Ellsberg Paradox etc.
Title of the course: Classical Groups
Instructor: Asst. Prof. Şükrü Yalçınkaya
Prerequisites: Linear algebra and group theory.
Level: Advanced undergraduate or graduate.
Abstract: Review of affine and projective geometries. Linear groups. The 7-point plane.
Bilinear and hermitian forms. Symplectic, unitary and orthogonal groups.
Title of the course: Concrete Group Theory I
Instructor: Prof. Alexandre Borovik
Institution: Manchester University
Prerequisites: Linear algebra, basics of general algebra. It will be expected that
students know basic facts about groups, rings, fields.
Level: Junior undergraduate
Abstract: The group PGL(2,k) as the group of fractional-linear transformations of
projective line, and its structure. Cross ratio as invariant of action. The case of finite
fields and some applications. The real orthogonal group SO(3,R) and "calculus of
reflections". Finite subgroups of SO(3,R). Finite subgroups of PSL(2,R).
Title of the course: Concrete Group Theory II
Dates: 6 – 12 August 2012
Level: Senior undergraduate
Abstract: Linear fractional transformations on the Riemann sphere. Stereographic
projection. Moebius geometry. Models of the Lobachevsky plane. The Mercator projection
as uniformisation of logarithm.
Title of the course: Concrete Group Theory III
Level: Postgraduate
Abstract: Linear algebraic groups over dual numbers and their Lie algebras. The vectors
in dimension 3 with cross product as the Lie algebra of SO(3,R). The quaternion theory,
SU(2,C) as the simply connected cover of SO(3,R). PGL(2,k) as an isotropic orthogonal
group in dimension 3. PSL(2,C) as the connected component of the Minkovsky group.
Title of the course: Special Relativity and its applications
Instructor: Asst. Prof. Tonguç Rador
Institution: Boğaziçi University
Prerequisites: Very basic knowledge of Newton's three laws.
Level: Undergraduate, graduate.
Abstract: The course will include a review of Galilean relativity and an in depth study of
inertial and non-inertial observers followed by the derivation of Lorentz transformations.
The rest of the course can be summarized as: Applications of Lorentz transformations:
Simultaneity ,length contraction and time dilation issues. Derivation of Special Relativistic
generalization of Newton's laws. Special relativistic kinematics and some exactly solvable
systems Approach to Maxwell's equations from the Lorentz group and some exactly
solvable systems. This is a tentative outline. Complete lecture notes will be available,
hopefully, before the course starts.
Title of the course: Introduction to Representation Theory
Instructor: Assoc. Prof. Adrien Deloro
Institution: Paris VI University
Prerequisites: Students taking this class should know what a group action is and be
familiar with Cayley's theorem on embedding into symmetric groups. Classical notions
from linear algebra (eigenvalues, etc.) will be used as well.
Abstract:
Week 1: Elementary representation theory.
Syllabus: permutation representations, Burnside's formula, Representations, Maschke's
Theorem, Schur's Lemma, Characters, Orthogonality relations.
Recommended prerequisites: Groups, group actions, elementary linear algebra
(vector spaces, linear maps, and matrices).
Week 2: Basic representation theory.
Syllabus: Representations, Complete reducibility, Character Theory, Character Tables.
Recommended prerequisites: Abstract linear algebra, Rings and modules, bilinear
algebra.
Week 3: Topics in representation theory.
Syllabus: Real and quaternionic representations, Induced representations, Frobenius
reciprocity, Mackey Criterion, Frobenius complements.
Recommended prerequisites: Tensor Algebra, Character Theory, Sylow theory.
Başlık: Sayma ve Kombinatoriks
Tarih: 6 - 19 Ağustos 2012
İçerik: Güvercin Yuvası İlkesi. Tümevarım. Birkaç temel sayılar kuramnı teoremi. Kombinasyon
hesapları ve binom katsayıları. Catalan sayıları. Stirling sayıları. Ramsey sayıları.
Kaynakça: Ali Nesin, Sayma, Nesin Yayıncılık 2012.
Başlık: Hızlandırılmış Mekanik
Eğitmen: Yard. Doç. Dr. Tonguç Rador
İçerik: Lise bilgilerinin üstüne temel klasik mekanik.
Başlık: Sayılar Kuramı
Eğitmen: Yard. Doç. Dr. Zeynep Özkurt
Kurum: Çukurova Üniversitesi
İçerik: Tamsayılarda bölünebilme ve bölme algoritması, Euclid algoritması, Çarpanlara
ayırma, Kongruans denklemleri, Çin-kalan teoremi, Fermat teoremi ve Lagrange teoremi.
Title of the course: Ultraproducts and their consequences
Instructor: Assoc. Prof. David Pierce
Institution: Mimar Sinan GSÜ
Prerequisites: Some knowledge of algebra, including the theorem that a quotient of a
ring by an ideal is a field if and only if the ideal is maximal.
Level: Advanced undergraduate and graduate
Abstract: An ultraproduct is a kind of average of infinitely many structures. The
construction is usually traced to a 1955 paper of Jerzy Los; however, the idea of an
ultraproduct can be found in Kurt Goedel's 1930 proof (from his doctoral dissertation) of
the Completeness Theorem for first-order logic. Non-standard analysis, developed in the
1960s by Abraham Robinson, can be seen as taking place in an ultraproduct of the
ordered field of real numbers: more precisely, in an ultrapower. Indeed, for each integer,
the "average" real number is greater than that integer; therefore an ultrapower of the
ordered field of real numbers is an ordered field with infinite elements and therefore
infinitesimal elements. Perhaps the first textbook of model theory is Bell and Slomson's
“Models and Ultraproducts” of 1969: the title suggests the usefulness of ultraproducts in
the development various model-theoretic ideas. Our course will investigate ultraproducts,
starting from one of the simplest interesting examples: the quotient of the cartesian
product of an infinite collection of fields by a maximal ideal that has nontrivial projection
onto each coordinate. No particular knowledge of logic is assumed.
Title of the course: Groups and Geometry
Instructor: Prof. Ayşe Berkman
Institution: Mimar Sinan GSÜ
Prerequisites: Some familiarity with groups (such as subgroups, homomorphisms,
Lagrange’s Theorem) and with linear algebra (such as vector spaces, linear
transformations).
Level: Advanced undergraduate
Abstract: Isometry groups, symmetry groups, frieze groups and plane crystallographic
groups.
Title of the course: Nonstandard Calculus
Instructor: MSc. Haydar Göral, MSc. Sinem Odabaşı
Institution: Lyon University
Prerequisites: Calculus
Level: Advanced undergraduate, beginning undergraduate
Abstract: Some set theory and filters, construction of hyperreals, infinitesimals,
differentiation and integration without using the limit concept like in the times of Newton
and Leibniz.
Başlık: Lambda-calculus
Eğitmen: MSc. Chris Stephenson
Kurum: İstanbul Bilgi University
Tarihler: 20 – 26 August 2012
Önkoşul: Basic mathematical skills.
Seviye: Beginning, advanced undergraduate.
İçerik: λ-calculus 80 sene önce icat edilmiş. Tamamen soyut matematiksel bir yapı.
Ancak giderek λ-calculus ya da en az “lambda” kelimesi “Pratik” dünyada da moda olmuş.
Python'da, Java'da, C#'da artık “ λ” var. Dersta λ-caclulus'un hem pratik ve teorik
yanlarına bakılacak. Uygulamada λ-calculus değerlendiren programlar yazacağız.
Program
Neden λ-calculus? Tarih ve önemi. Gödel, Church and Turing.
Sembollar ve ikame sorunları.
Bağlı ve serbest semboller
De Bruijn sayıları
Schönfinkelling
Church-Rosser niteliği ve pratik önemi.
λ-calculus Church-Rosser'dir.
Church sayılar, mantık, aritmetik.
Normal form
Özyineleme ve Y combinatörü
Açgöz ve tembel değerlendirilme
Gerçek programlama dilleri ve λ-calculus.
λ-calculus gibi bir dilde λ-calculus değelendiren bir program yazmak
Church-Turing tezi.
Title of the course: Model Theory
Instructor: MSc. Uğur Efem
Institution: Oxford University
Prerequisites: A solid course in abstract algebra, a basic understanding of order
structures and graphs will be necessary, also some knowledge in logic would be useful
but not necessary.
Level: Graduate and advanced undergraduate
Abstract: The course is aimed to present a very brief introduction to model theory for
those who are interested in the topic and/or working outside of model theory (and
perhaps logic). The course will be as follows:
Part 1 – Around Compactness Theorem: Languages, structures, a bit of definability,
Compactness Theorem and its applications mostly in model theory and algebra.
Part 2 : Up and Down, Back and Forth, Fraise Games.
Başlık: Matrisler
Eğitmen: Yard. Doç. Dr. Zeynep Özkurt
İçerik: Temel kavramlar ve bazı özel tipteki matrisler. Laplaca açılımı. Matrislerin
denkliği. Bir matrisin tersi. Determinant fonksiyonu. Matrislerle lineer denklem
sistemlerinin çözümü.
Başlık: Uygulamalı Doğrusal Cebir
Eğitmen: Yard. Doç. Dr. Ayhan Dil
Kurum: Akdeniz Üniversitesi
Tarih: 20 – 26 Ağustos 2012
İçerik: Doğrusal denklem sistemleri ve matrisler, matris cebiri ve determinantlar.
Başlık: Otomata ve Biçimsel Diller
Eğitmen: MSc. Gabriela Aslı Rino Nesin
Kurum: Leicester University
İçerik: Bu ders sonlu otomataları, regüler ifadeler ve gramerleri, ve Pompalama
Önsavı'nı kapsayacaktır. Eğer zamanımız olursa başka dil sınıflarından da bahsedeceğiz,
öreğin lineer, tek sayaçlı veya içerik bağımsız diller. Daha ileri seviyedeki öğrenciler için
bir grubun cebirsel yapısı ile kelime probleminin bulunduğu sınıf arasındaki bağlantıdan
kısaca bahsedilecektir.
Title of the course: Generators and Relations (Group Theory)
Instructor: Prof. Ali Nesin
Institution: İstanbul Bilgi Üniversitesi
Dates: 20 - 26 August 2012
Prerequisites: Group Theory
Level: Undergraduate and graduate
Abstract: Finitely presented groups. Small cancellation theory. Trees and groups.
Grigorchuk-Gupta-Sidki theorem. Free groups and their subgroups. ReidemeisterSchreier method.
Title of the course: Quadratic Reciprocity Law and more
Prerequisites: A solid background in ring theory
Level: Advanced graduate and undergraduate
Abstract: Dirichler characters. Gauss sum. Conductor. Quadratic Reciprocity Law.
Examples. The ring »p of p-adic integers and the field »p of p-adic numbers. Hensel’s
Lemma.
Title of the course: Analytic Number Theory
Instructor: MSc. Doğacan Sertbaş
Institution: Bonn University
Prerequisites: Calculus, Basic Complex Analysis
Level: Advanced undergraduate & Undergraduate
Abstract: Arithmetic Functions, Dirichlet convolution, Möbius Inversion Formula, Big-O
and Little-O notations, Abel & Euler Summation Formula.
Title of the course: Topics in Classical Number Theory
Instructor: Asst. Prof. Ayhan Dil
Institution: Akdeniz University
Prerequisites: Calculus
Level: Beginning Undergraduate
Abstract: Primes and Their Distribution, The Theory of Congruences, Arithmetical
Functions, Numbers of Speical Forms.
Title of the course: Distributions and their applications
Instructor: MSc. Arif Mardin
Institution: Dates: 27 August – 2 September 2012
Prerequisites: Familiarity with the basics of Lebesgue integration, Fourier transforms
and Hilbert spaces would be helpful.
Level: Graduate, Advanced undergraduate
Abstract: The aim of these lectures is to present the basic theory of distributions and
some of their applications.
Kaynakça: References: (the list is arranged in increasing order of difficulty)
1) "The Theory of Distributions: A Nontechnical Introduction", J.I.Richards & H.K.Youn,
Cambridge University Press, 1995;
2) "Distribution Theory and Transform Analysis", A.H.Zemanian, Dover Publications,
1987;
3) "A Guide to Distribution Theory and Transform Analysis", R.S.Strichartz, World
Scientific, 2003;
4) "Distributions: Theory and Applications", J.J.Duistermaat & J.A.C.Kolk, Birkhauser,
2010;
5) "Mathematics for the Physical Sciences", L.Schwartz, Dover Publications, 2008;
6) "Theorie des Distributions", L.Schwartz, Hermann, 1966;
7) "The Analysis of Partial Differential Equations. Volume1: Distribution Theory and
Fourier Analysis" (2nd edn.), L.Hörmander, Springer Verlag, 1990.
Title of the course: Riesz Spaces
Instructor: Assoc. Prof. Mert Çağlar
Institution: Kültür University
Dates: 27 August – 2 September 2012
Prerequisites: Acquaintance, at the graduate level, with the basics of General Topology,
Real Analysis, and Functional Analysis.
Level: Graduate
Abstract: The course will be a self-contained introduction to the theory of topological
Riesz spaces. Having introduced the lattice structure of Riesz spaces, namely the ideals,
bands, Riesz subspaces, order completeness and projection properties, Freudenhal’s
Spectral Theorem, order bounded operators and the order duals of Riesz spaces, locally
solid topologies on Riesz spaces will be examined. Of all locally convex-solid ones,
emphasis will be on the Lebesgue and Fatou topologies.
Textbook:
C. D. Aliprantis & O. Burkinshaw, Locally Solid Riesz Spaces with Applications to
Economics, Second Edition, American Mathematical Society, Mathematical Surveys and
Monographs, Vol. 105, Providence, RI, 2003
SUGGESTED READING:
C.D. Aliprantis & O. Burkinshaw, Positive Operators, Springer, The Netherlands, 2006
D.H. Fremlin, Topological Riesz Spaces and Measure Theory, Cambridge University Press,
1974
W. A. J. Luxemburg & A.C. Zaanen, Riesz Spaces, Vol. 1, North-Holland Publishing Co.,
Amsterdam, 1971
A. L. Peressini, Ordered Topological Vector Spaces, Harper & Row Publishers, New York
,1967
B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Trans. by Leo F.
Boron with the editorial collaboration of Adriaan C. Zaanen and Kiyoshi Iséki, WoltersNoordhoff, Ltd., Groningen, The Netherlands, 1967
A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, Berlin,
Heidelberg, 1997
Title of the course: Positive Operators
Instructor: Asst. Prof. Tunç Mısırlıoğlu
Institution: İstanbul Kültür University
Prerequisites: It requires only a basic knowledge of classical analysis in undergraduate
level: measure theory, Hilbert and Banach spaces, elementary functional analysis.
Abstract: We will first present an introduction to Banach spaces and operator theory.
Also, we will introduce Banach lattices and positive operators. We then review the basic
structural properties of operators with special emphasis on positive operators such as
bounded below operators and compact and weakly compact positive operators. After, we
will deal with AM- and AL-spaces and demonstrate their importance in operator theory.
We will also study the center of a Banach lattice. Finally, we will present detailed
accounts regarding the following classes of operators: finite-rank operators,
multiplication operators, lattice and algebraic homomorphisms, Fredholm operators, and
strictly singular operators.
Title of the course: Number Theory and Polyhedral Geometry
Instructor: MSc. Zafeirakis Zafeirakopoulos
Institution: Johannes Kepler University
Dates: 27 August – 2 September
Prerequisites: Basic knowledge of algebra and geometry.
Abstract: The course will provide a basic introduction to polyhedral geometry, number
theory and algorithms. Next we will explore the connections between them concentrating
to the problem of linear Diophantine systems of equalities and inequalities.
Day 1:
* Introduction to Linear Diophantine Systems (LDS)
* Number Theory and Geometry of (LDS)
Day 2:
* Inroduction to Algorithms
* Complexity and Hardness of the problem
Day 3:
* Number Theory for LDS
* Partition Analysis for LDS
Day 4:
* Polyhedral Geometry for LDS
Day 5:
* The Polyhedral Geometry of Partition Analysis
* Generalizations
Day 6 (if possible):
* Applications to Algebra
* Applications to Combinatorics
Başlık: Türev
Tarih: 27 Ağustos – 2 Eylül 2012
İçerik: Reel sayılarda limit kavramı sezgisel olarak anlatıldıktan sonra önce türev, sonra
da integral konusu işlenecek. Türevin ve integralin geometrik anlamları irdelenecek,
türevle ilgili geometrik ve fiziksel sorular çözülecek ve integral almanın çeşitli yöntemleri
açıklanacak.
Kaynakça: Matematik Dünyasi dergisi 2010, 2011 ve 2012 sayıları.
Eğitmen: Yard. Doç. Dr. Özlem Beyraslan
koordinatlar.
Başlık: Eğimden İntegrale Kısa Bir Yolculuk
Eğitmen: Prof. Dr. Haluk Oral
Kurum: Bahçeşehir Üniversitesi
İçerik: Doğrunun eğimi ve denklemi. Limit kavramının sezdirilmesi. Eğrilerde eğim
olmadığı için yapılan teğet eğimi tanımı. Doğrusal yaklaşım. Belirsiz integral. Sonsuz
troplam ve belirli integral. Calculuş'un temel teoremi. İki katlı integral.
Title of the course: Cones and Duality
Instructor: Prof. Eduard Emelyanov
Institution: ODTÜ
Prerequisites: Basic functional analysis
Abstract: The main aim of this course is an introduction to basic concepts of the theory
of ordered vector spaces which plays a prominent role not only in functional analysis but
also in engineering, economics, and other fields that uses optimization techniques. We
consider only real vectors spaces. First four lectures are devoted to basic properties of
cones and positive linear operators. Then, in the next five lectures we give an inroduction
to the theory of ordered topological vector spaces. The last five lectures are devoted to
Krein operators, K-lattices, and Riesz-Kantorovich formulas.
Kaynakça: Aliprantis, C.D. Tourky, R. Cones and duality, Graduate Studies in Math.
Vol.84 American Math. Soc. 2007
Title of the course: The Transseries Field
Instructor: Asst. Prof. Salih Durhan
Institution: ODTÜ, Kuzey Kıbrıs
Prerequisites: Previous exposure to basic concepts in model theory (not much more
than compactness theorem), differential algebra, ordered fields and valued fields is
helpful but none of these are particularly required.
Level: Advanced Undergraduate, Graduate
Abstract: The transseries field is an ordered differential field and behaves like a
universal domain for real differential algebra. In this course we will carefully define this
field, explore its basic properties, define certain important automorphisms and study the
natural valuation it carries. This object is relatively new so open problems and further
research topics will be discussed through the course.
Title of the course: Introduction to Modal and Temporal Logic
Instructor: Dr. Guido Sciavicco
Institution: University of Udine
Dates: 3 - 9 September 2012
Prerequisites:
Level:
Abstract: In this course, intended for a audition with mathematical, computer science,
or phylosophy background, with a basic knowledge of propositional and first-order logics,
I intend to present modal logics as a subject. We will start from basic definitions and
intended semantics, progressively going towards a precise notion of truth in modal logic,
and, on the side, of formal deduction.
After that, we will define the notion of Hilbert calculus, and prove its soundness and
completeness. We will introduce the notion of modal frame definability, with a short
detour on relative completeness in those cases. Then, we will start introducing the subset
of modal temporal logics, syntax, semantics, and decidability/undecidability of them.
Finally, we will consider the special case of modal temporal logics of intervals, and recent
results on them. Exercises can be proposed if anyone wants to use this as a PhD course.
Please see the attached slides for further detail.
Title of the course: Topics in Analysis
Instructor: Prof. Dr. Ali Nesin
Dates: 3 – 9 September 2012
Prerequisites: Basic knowledge of integrals.
Level: Advanced undergraduate and graduate.
Abstract: Liouville sayıları. Wallis formülü. n! için Stirling formülü, Γ fonksiyonu.
Title of the course: Riesz Uzaylarinda İki Temel Temsil Teoremi.
Instructor: Prof. Zafer Ercan
Institution: Abant İzzet Beysal Üniversitesi
Prerequisites: Level: Abstract: (E, +, .) vektör uzayı ve, ≤, E üzerinde kısmi sıralama olmak üzere (E, ≤) ikilisi
bir lattice ve, x, y ∈ E ve x ≤ y olduğunda her z ∈ E ve 0 ≤ α ∈ R icin x + z ≤ y + z ve αx
≤ αy eşitsizlikleri sağlanıyor ise, (E, +, ., ≤) dörtlüsüne Riesz uzayı (vektor lattice) denir.
Kakutani-Krein Temsil Teoremi: 0 < e ∈ E elemanı şu özelliği sağlasın: Verilen her x ∈ E
için |x| ≤ αe olacak biçimde α ∈ R vardır. Bu özellikteki e ∈ E elemanına "sıra birimi"
(order unit) denir. Archimedean özelliği olan bir Riesz uzayı E'nin sıra birimi var ise, E,
C(K) biçimindeki bir Riesz uzayının sıra yoğun bir Riesz altuzayına Riesz uzay eşyapılıdır.
Burada geçen K (topolojik eşyapılılık olarak tek olan) bir kompact Hausdorff uzayı, C(K)
ise, K'da tanımlı gerçel değerli sürekli fonksiyonlar Riesz uzayını göstermektedir. Bahsi
geçen K'ye "E'nin Kakutani Krein uzayı" denir. Eğer E düzgün tam (bir Riesz uzayında bir
çesit her Cauchy dizisinin yakınsaması) ise E ile C(K) Riesz uzayı olarak eşyapılıdır.
Maeda-Ogasawara Temsil Teoremi: Her açık kümenin kapanışı da açık olan kompakt
Hausdorff uzayına "Stonean" denir. S bir Stonean uzay olmak üzere f: S → [-∞,∞]
fonksiyonu sürekli ve f^{-1}R, S'de yoğun ise f'ye "genelleştilmiş sürekli fonksiyon"
denir. Genelleştirilmiş sürekli fonsiyonların uzayı C^{∞}(S) ile gösterilir. Uygun cebirsel
işlemler ve noktasal sıralamaya göre C^{∞}(S) bir Riesz uzayıdır (ve kanıtlanacaktır).
Verilen her Archimedean Riesz uzayı E için, E, bir Stonean uzayı S için, C^{∞}(S)'nin bir
sıra yoğun Riesz altuzayına Riesz uzay eşyaplıdırve kanıtı verilecektir. Bu Teoreme
Maeda-Ogasawara Temsil Teoremi denir. Bahsi geçen S, topolojik eşyapılı olarak tektir.
Tekliğin bilinen kanıtı yanında yeni bir kanıtı da verilecektir.
Zamanın yetmesi durumunda, C^{∞}(S)'nin sıralanmış halka oduğu gösterilerek
C^{∞}(S)'nin bazı halka özelliklerinden bahsedilecektir.
Title of the course: Cohomology of SL_2(Z) and Number Theory
Instructor: Asst. Prof. Haluk Şengün
Institution: University of Warwick
Prerequisites: Basic group theory, basic topology, basic field theory
Level: Graduate, advanced undergraduate, enthusiastic beginning undergraduate
Abstract: The modular group SL(2,Z) lies in the intersection of many exciting areas of
mathematics. In this lecture series, we will study fundamental aspects of SL(2,Z) which
are important for number theory, modular forms and arithmetic geometry. Our aim is to
provide the students with an understanding and appreciation of the “Big Picture”.
Title of the course: Elliptic Curves, Mordell-Weil Groups, Modular Forms and Modular
Curves.
Instructor: Asst. Prof. Erhan Gürel
Institution: ODTÜ, Kıbrıs
Prerequisites: Algebra course and some knowledge of Algebraic Geometry would be
better.
Abstract: Basic aspects of Arithmetic will be introduced in the first week. According to
people’s interests some elementary level papers will be discussed in the second week.
Texbook:
http://www.math.ncc.metu.edu.tr/content/courses/previous/2011spring/math526/course.php
Title of the course: Galois Theory (a field guide to algebra)
Instructor: Assoc. Prof. Martin Hils, Asst. Prof. Özlem Beyraslan
Institution: Paris VII, Boğaziçi Üniversitesi
Prerequisites: Basic algebra
Level: Undergraduate, graduate.
Abstract:
1. Ruler compass constructions
2. Field extensions
3. Some classical imposibilities
4. Symmetric functions
5. Algebraically closed fields, algebraic closure
6. Puiseux Theorem
7. Automorphism group of an extension
8. The Galis group as a permutation group
9. Discriminant, resolvent polynomials
10. Equations with degrees up to 4
11. How (not) to compute Galois groups
Textbook: Antoine Chambert-Loir, "A Field Guide to Algebra", Undergraduate Texts in
Mathematics, Springer, 2005.
Başlık: Cebir ve Analiz Problemleri Çözme Saati
Eğitmen: MSc. Uğur Doğan
Kurum: Koç Üniversitesi
Tarih: 3 - 16 Eylül 2012
İçerik: Öğrenciler asistan eşliğinde problem çözeceklerdir.
Başlık: Sayılar Kuramı ve Aritmetik Problemleri Çözme Saati
Eğitmen: Ms. Türkü Özlüm Çelik
İçerik: Öğrenciler asistan eşliğinde problem çözeceklerdir.
Başlık: Olimpiyatlarda Cebir ve Analiz
İçerik: Daha çok uluslararası matematik olimpiyatlarında çıkmış sorular çözülecek ve bu
soruların teorik altyapısı gösterilecektir.
Başlık: Olimpiyatlarda Sayılar Kuramı ve Aritmetik
İçerik: Bölünebilme, asallar, aritmetiğin temel teoremi, obeb, okek, Euclid Algoritması,
Bezout Teoremi, modüler aritmetik, çarpımsal fonksiyonlar, Fermat sayıları, Mersenne
sayıları, mükemmel sayılar.
Title of the course: Group Cohomology
Instructor: Prof. Piotr Kowalski
Institution: Wroclaw University
Prerequisites: Basic group and field theory.
Abstract: Definition of group cohomology using cochains (most basic way). Interpration
in low dimensions: principal homogenous spaces, group extensions. Basic Galois
cohomology: Hilbert's Theorem 90, normal basis theorem.
Title of the course: Character Theory with Applications
Instructor: Asst. Prof. Pınar Uğurlu
Prerequisites: Basic Group Theory and Linear Algebra
Abstract: Representations and characters of groups, basic definitions, examples, some
character tables, applications to groups. Burnside’s Theorem.
Title of the course: Quadratic Forms and Classical Groups
Prerequisites: Linear algebra and basic group theory
Level: Advanced undergraduate and graduate
Abstract: Nondegenerate quadratic forms.Some classification theorems. The study of
the classical groups associated.
Title of the course: Clifford cebirleri, spin gruplari ve trialite
Instructor: Prof. Şahin Koçak
Institution: Anadolu University
Prerequisites:
Level: Advanced undergraduate or graduate
Abstract: Normed algebras, Cross products of octonions, Bonan form, Clifford cebirleri,
Periodisite teoremleri, Spin grupları, Spin(7), Trialite.
Yüksek Lisans Cebir (in Turkish)
Başlık: Grup Teorisi I
Önkoşul: Seviye: İçerik:
Gruplar, bölüm grupları, temel izomorfizma teoremleri, alterne, simetrik ve dihedral
gruplar, direkt çarpımlar, otomorfizma grupları and yarı direkt çarpımlar, serbest gruplar,
üreteçler ve bağıntılar.
Başlık: Grup Teorisi II
Önkoşul: Seviye: İçerik:
Grup etkimesi, Sylow teoremleri, sıfırgüçlü (nilpotant) ve çözülebilir (solvable) gruplar,
normal ve altnormal seriler.
Başlık: Halkalar Teorisi
Önkoşul: Seviye: Lisansüstü ya da ileri seviyede lisans
İçerik: Halkalar, halka homomorfizmaları, idealler, polinom halkaları, bölüm halkası,
komütatif halkalarda asallar ve indirgenemezler ve indirgenemezlere ayırma. Yerelleşme.
Esas ideal bölgesi, Öklid bölgeleri, tek çarpanlama bölgesi, polinomlar, biçimsel kuvvet
serileri.
Başlık: Halkalar ve Modüller
Önkoşul: Bir önceki ders
Seviye: Lisansüstü ya da ileri seviyede lisans
İçerik:
30-07-2012 haftası: Hilbert taban teoremi, sıralı halkalar ve cisimler, Jacobson ideali,
yarıbasit halkalar.
06-08-2012 haftası: Modüller, homomorfiler, tam diziler, serbest modüller, vektör
uzayları, üreteçler ve taban, matrisler ve matris halkaları.
Başlık: Lineer Cebir
Önkoşul: Halka ve modül bilgisi
İçerik: Tek üreteçli ideal halkası üzerine modüller, Jordan formu. Sonlu eleman
tarafından üretilmiş abelyen gruplara uygulama. Bölünebilir abelyen gruplar. Kuadratik
formlar.
Başlık: Cisimler Kuramı
Önkoşul: Grup teorisi ve halkalar kuramı bilgisi
İçerik: Cisimler, cisim genişlemesi, halka genişlemesi, tam kapanış, ayrıştırma (splitting)
cisimleri, Galois kuramının temel teoremi, cebirsel kapanış, bir polinomun Galois grubu,
sonlu cisimler.
Başlık: Tansörler
Önkoşul: Modüller kuramı bilgisi.
İçerik: Modül tensörleri. Örnek ve uygulamalar. Tensör cebiri. Simetrik ve alterne cebir.
Determinantın doğru tanımı.
Yüksek Lisans Real Analysis (in Turkish)
Başlık: Ölçüm Kuramı
Eğitmen: Yard. Doç. Dr. Kemal Ilgar Eroğlu
Önkoşul: Seviye: Lisansüstü ya da ileri seviyede lisans
İçerik: Sigma cebirleri, ölçüm kavramı, dış ölçüm, Caratheodory teoremi, Lebesgue
ölçümü, Borel ölçümleri.
Başlık: İntegral
Önkoşul: Ölçüm kuramı bilgisi
İçerik: Integrasyon: Ölçülebilir fonksiyonlar, Lebesgue integrali, Fatou Önsavı ve
Monoton Yakınsaklık Teoremi
Başlık: Karmaşık İntegral
Eğitmen: Prof. Dr. Yusuf Ünlü
Kurum: Yeditepe Ü.
Önkoşul: Ölçüm kuramı ve integral
İçerik: Karmaşık fonksiyonların integrali ve “Dominated Convergence Theorem”,
yakınsaklık teoremleri, Egoroff teoremi. Analize uygulamalar.
Başlık: Çarpım Ölçümü, Fubini, Lebesgue
Eğitmen: Prof. Dr. Şafak Alpay
Kurum: ODTÜ
Önkoşul: Bu programdaki önceki derslerin içerikleri bilinmeli
İçerik: Çarpım ölçümleri, Fubini-Tonelli teoremi, »n üstünde Lebesgue ölçümü ve
integrali. Uygulamalar.
Başlık: Ölçümleri Ayrıştırmak
Eğitmen: Prof. Dr. Şafak Alpay
Kurum: ODTÜ
İçerik: Ölçümlerin ayrıştırılması: işaretli ölçümler, Hahn ayrıştırma teoremi, LebesgueRadon-Nikodym teoremi, karmaşık ölçümler. Uygulamalar.
Başlık: L^p Uzayları
Eğitmen: Prof. Dr. Zafer Ercan
Kurum: Abant İzzet Baysal Üniversitesi
İçerik: L^p uzayları: Hölder ve Minkowski's eşitlikleri, L^p’nin düali , konvolusyonlar.
Uygulamalar.
Başlık: Radon ölçümleri, Riesz Tasvir Teoremi
İçerik: Radon ölçümleri: C_c(X) üzerine pozitif lineer fonksiyonlar. Riesz temsil teoremi.
Uygulamalar.
Başlık: Reel Analiz Problemleri
Önkoşul: Reel Analiz
İçerik: Örnekler, problemler, alıştırmalar, sınavlar.
Yüksek Lisans Karmaşık Analiz (in Turkish)
Başlık: Karmaşık Analizin Temelleri
Eğitmen: Prof. Dr. Mehmet Sait Eroğlu
Kurum: Tarih: 9 – 15 Temmuz 2012
Önkoşul: -
İçerik: Karmaşık Düzlemde Analiz: Karmasik düzlemin topolojisi, karmaşık (fonksiyon)
dizileri ve serileri, özellikle kuvvet serileri, sürekli ve karmaşık türevlenebilme, gerçel
türevlenebilmeyle ilişkisi, eğrisel integraller ve temel fonksiyonlar: eksponensiyel,
logaritmik, üssel ve trigonometrik fonksiyonlar.
Başlık: İntegral I
Eğitmen: Prof. Dr. Mehmet Sait Eroğlu
Kurum: Tarih: 16 – 22 Temmuz 2012
İçerik: Cauchy İntegral Teoremi: Kapalı eğrilerin dönme sayıları, en genel biçimiyle
Cauchy integral teoremi (ve sıfıra homolog kapalı eğriler için kanıtı), Cauchy integral
formülü, kalan teoremi. Bu teoremlerin ilk sonuçları.
Başlık: İntegral II
Eğitmen: Prof. Dr. Naime Ekici
İçerik: Integral I dersinde kalınan yerden devam edilecek ve uygulamalar verilecektir.
Başlık: Argüman İlkesi
Eğitmen: Dr. Uğur Gül
Kurum: Hacettepe Üniversitesi
İçerik: Argüman İlkesi: kapalı bir eğrinin indeksi, Cauchy teoreminin genel biçimi,
kalanlar (residue) teoremi, argüman ilkesi, Rouché teoremi. Uygulamalar.
Başlık: Maksimum Modulus İlkesi
Eğitmen: Dr. Uğur Gül
İçerik: Maksimum Modulus İlkesi, Schwarz önsavı, birim dairenin üstüne holomorfik
dönüşümleri, Mobius dönüşümleri. Uygulamalar.
Başlık: Meromorfik Fonksiyonlar
Eğitmen: Doç. Dr. Ali Özgür Kişisel
Kurum: ODTÜ, Kuzey Kıbrıs
İçerik: Analitik fonksiyonların sıfırları ve kutupları. Runge teoremi, meromorphik
fonksiyonlar, sonsuz çarpımlar, Weierstrass çarpanlama teoremi. Uygulamalar.
Başlık: Analitik Devamlılık
Eğitmen: Yard. Doç. Dr. Mohan Ravichandran
Kurum: Sabancı Üniversitesi
İçerik: Bir eğri üzerinde analitik devamlılık, analitik devamlılık, monodromy teoremi.
Uygulamalar.
Başlık: Kompleks Analiz Yeterlilik Sınavı Çözümü
Eğitmen: MSc. Nazlı Doğan
Kurum: Kültür Üniversitesi
Önkoşul: Lisans kompleks analiz bilgisi bilinmeli.
İçerik: Katılımcılarla her gün bir adet kompleks analiz yeterlilik sınavı ve çözümleri
üzerine tartışılacaktır.
Başlık: Riemann Mapping Teoremi
Eğitmen: Prof. Dr. Doğan Dönmez
İçerik: Normal aileler, Riemann mapping teoremi. Uygulamalar.
Yüksek Lisans Geometri (in Turkish)
Başlık: Diferansiyel Çokyüzlüler (Manifold)
Önkoşul: Temel analiz bilgisi
İçerik: Differansiye edilebilir fonksiyonlar, Differansiye edilebilir çokkatlılar, bir
fonksiyonun derecesi, immersiyonlar ve sübmersiyonlar, alt-çokkatlılar ve gömmeler.
Başlık: Vektör Alanları, Teğet Uzay
Önkoşul: Diferansiyel çokyüzlüler
İçerik: Bir çokkatlıya bir noktada teğet uzay. Bir fonksiyonun diferansiyeli. Vector
alanları. Vectör alanların Lie çarpımı.
Başlık: Çokkatlılar Üzerine Tensörler ve Tensör Alanları
Eğitmen: Doç. Dr. Murat Limoncu
Kurum: Anadolu Üniversitesi
İçerik: Kotanjant vektörler ve alanlar. Tensör ve tensör alanları. Tensör çarpımı.
Tensörlerin ve tensör alanlarının cebirsel yapısı. Tensör bileşenleri ve büzüşme
operatörleri. Diferansiye edilebilen bir fonksiyonun geriçekişi (pullback) Vektör uzayları
üzerine simetrik çifte doğrusal formlar. Çokkatlılar üzerine Rieman metriği. Riemanyen
metrik tensörün varlığı. Çokkatlılar üzerine p-formlar. Dış türev. Yönlendirilebilir
çokkatlılar. Hacim elemanı.
Başlık: İntegral Alma
Eğitmen: Doç. Dr. Ferit Öztürk
İçerik: Çok katlılar üzerine integrasyon, sınırı olan çokkatlılar, çokkatlının sınırını
yönlendirmek. Stokes teoremi. Uygulamalar.
Yüksek Lisans Topoloji (in Turkish)
Başlık: Temel Topoloji
Eğitmen: Prof. Dr. Hatice Tuna Yalvaç
Önkoşul: Seviye: İçerik: Topolojik uzaylar, taban and alttaban, altuzay topolojisi, sürekli fonksiyonlar,
çarpım topolojsi, metrik uzaylar ve metrik topolojisi, bölüm topolojisi.
Başlık: Tıkızlık
Önkoşul: Önceki dersi almış ya da biliyor olmak
Seviye: Önceki dersin içeriği aşağı yukarı bilinmeli.
İçerik: Tıkız uzaylar, Heine-Borel teoremi, Tychonoff teoremi, değişik tıkızlık kavramları.
Metrik uzaylarda tıkızlık. Yerel tıkızlık. Tıkızlaştırma.
Başlık: Bağıntılılık
İçerik: Bağıntılı uzaylar, yol bağıntılı uzaylar, bağıntı bileşenleri, yerel bağıntılılık,
aradeğer teoremi ve diğer uygulamalar.
Başlık: Ayrışma Özellikleri
İçerik: T0 uzaylar, Hausdorff, düzgün ve normal uzaylar, Uryshon önsavı. Metrikleşme.
Başlık: İleri Topoloji
Eğitmen: Doç. Dr. Ali Özkurt
İçerik: Tietze genişleme teoremi, sayılabilirlik özellikleri, Lindelöf, ayrıştırılabilir,
sayılabilir tıkız uzaylar.
Başlık: Topoloji Problemleri (İngilizce)
Eğitmen: Prof. Eduard Emelyanov
Kurum: ODTÜ
Önkoşul: Topoloji
İçerik: Örnekler, problemler, alıştırmalar, sınavlar.

Title of the course - Nesin Matematik Köyü

Transkript

Benzer belgeler

Title of the course: Character Theory with Applications Instructor

Prof. Dr. Ali Nesin Kurum: stanbul Bilgi Üniversitesi Tarih: 4