Dersler

Ders 1

Dersi veren: Türkü Özlüm Çelik

Başlık (Title): From applied algebra and geometry

Özet (Abstract): Recent advances in both theoretical mathematics and efficient mathematical software have stimulated the use of algebra and geometry in tackling problems arising from the sciences. Conversely, developments on these topics inspire fundamental questions in algebra and geometry. This involves algebraic geometry, commutative algebra, combinatorics, polyhedral geometry, and more. This course aims to urge some interest along the lines. In particular, it will showcase examples from such studies. In this framework, the first part of the course will be a discussion about the methods to study some optimization problems. The second part targets making a rapid excursion into the study of integrable systems through the lens of computational algebraic geometry. Specifically, the emphasis will be on exploiting tools from the areas to investigate solutions of the differential equations. Also, there will be an exhibition of some fundamental questions emerging from this investigation.

AFTER THE COURSE:

Further reading on algebro-geometric solutions of the KP equation:

• Algebraic theory of the KP equations, Motohicho Mulase (1994)

• Theta functions and non-linear equations, Boris Dubrovin (1981)

• The Dubrovin threefold of an algebraic curve, Agostini, C., Sturmfels, (2020)

Some exemplary links for historical mathematical model collections in various universities:

• Leipzig University (not finished, ongoing project)

• Oxford University

• Karazin Kharkiv National University

• Martin-Luther-University Halle-Wittenberg

• University of Göttingen

• The Hebrew University of Jerusalem

• University of Groningen

• The University of Tokyo

...

Ders 2

Dersi veren: Nursel Erey

Başlık (Title): Kombinatoryal Değiştmeli Cebirden Seçme Konular/ Selected topics in combinatorial commutative algebra

Özet (Abstract): In this course, we will be interested in simplicial complexes, monomial ideals and some of their algebraic properties. We will see how one can use vertex decomposable or shellable simplicial complexes to investigate some problems about invariants of minimal free resolutions. In particular, I will present some open problems where this combinatorial method can be applied to powers of some monomial ideals.

Ders 3

Dersi veren: Özgür Esentepe

Başlık (Title): Explicit computations of cohomology annihilators for some hypersurfaces

Özet (Abstract): ​​The aim of these lectures is to introduce a research problem in algebra in 4 hours. In Lecture 1, there will be an introduction to homological algebra. We will start with some facts from linear algebra and continue to explore free resolutions. This lecture will be enough to explain the main problem in Lecture 2. We will devote the second lecture to the explanation of the problem and we will see several examples. In Lecture 3, we will focus on the hypersurface case. I will talk about a theorem due to Eisenbud and introduce matrix factorizations and maximal Cohen-Macaulay modules. In Lecture 4, we will see some methods to attack the problem. I am hoping to have 2 problem sessions with the students who are interested in these lectures. During the first session, I will introduce a computer algebra program: Singular and teach how to do some computations. During the second session, we will try to use what we have learned so far and discuss our findings.

After the course

UCGEN Research Group on Cohomology Annihilators

Ders 4

Dersi veren: Ilmar Gahramanov

Başlık (Title):

Özet (Abstract):

Ders 5

Dersi veren: Neslihan Güğümcü

Başlık (Title): Problems in Knot Theory

Özet (Abstract): ​​In this series of lectures, we will first review introductory notions of classical knot theory. Then we will study other knotted objects such as virtual knots, \Theta-curves, and specifically knotoids.

Knotoids are a part of geometric three-dimensional knot theory, and, as such, are related to open-ended embeddings of intervals in three dimensional space subject to some special ambient isotopy. This correspondence gives a more realistic understanding of the entanglement in physical structures such as open linear proteins. Another geometric aspect of knotoids is given by Turaev: He shows that there is a bijection between the set of \Theta-curves and the set of spherical knotoids. The theory of knotoids is naturally in relation also with virtual knot theory that studies embeddings of circles in thickened higher genus surfaces with a diagrammatic formulation using virtual knot diagrams with real and virtual crossings.

We will extract invariants of these objects using combinatorial and quantum topology techniques. We will also discuss some open problems in classical knot theory and also in the theories of virtual knots and knotoids.

Ders 6

Dersi veren: Ezgi Kantarcı Oğuz

Başlık (Title): Permütasyon istatistikleri ve polinomlar (Permutation statistics and corresponding polynomials)

Özet (Abstract): ​​Bu derste çok temel objeler olan permütasyonları inceleyeceğiz. Permütasyonlar üzerine tanımlı tersinim, düşüş, artış ve pik gibi bazı istatistiklerle ilgileneceğiz. Yakın zamanda, “n”nin belli bir düşüş kümesine sahip permütasyonlarının sayısının n üzerine düşüş polinomu adını verdiğimiz bir polinom verdiği gösterildi. Son on yılda bu polinomla ve benzer şekilde tanımlanmış pik polinomu ile birçok makale yazıldı. Vaktimiz elverirse, bu çalışmalarını bir kısmının üzerinden geçip ulaşılan sonuçlara ve hala açık olan sorulara göz atacağız.

(The objects we are interested in are basic and fundamental: permutations. We will look at some permutations statistics -inversions, ascents, descents, peaks- and corresponding generating polynomials. The number of permutations of n which have a given descent set has recently been shown to be a polynomial on n, and there has been some work studying the descent polynomial, and similarly defined peak polynomial in the last ten years. Time permitting, we will try to go over some recent work, see what is known and what is still open.)

Ders 7

Dersi veren: Mohan Ravichandran

Başlık (Title): The Wilf-Zeilberger method, Onsager’s theorem and open problems

Özet (Abstract): This course will discuss an approach due to Herbert Wilf and Doron Zeilberger (WZ) that can be used to automatically prove combinatorial identities using a computer. We will see applications of this method, notably to the planar Ising model, where we will use this approach to give a non-rigorous proof of Onsager's famous exact solution in the zero field case. I will then present four open problems in combinatorics and encourage you to form research groups to work on these.

Not: Dersler türkçe slaytlar ingilizce olacaktir.

Ders 8

Dersi veren: Sibel Şahin

Başlık (Title): Potansiyel Teorisi ve Holomorfik Fonksiyon Uzaylari / Potential theory and Holomorphic Function Spaces

Özet (Abstract): In this minicourse on Potential Theory and Holomorphic Function Spaces we will consider the following topics during lecture hours and in a final general seminar I’d like to discuss the contemporary research interests and some open problems in pluripotential theory and complex analysis.Topics planned to be covered during lectures:

• Harmonic functions and The Dirichlet Problem.

• Subharmonic functions, Potentials, Polar sets and the generalized Laplacian.

• Plurisubharmonic functions, Poistive currents and the complex Monge-Ampère equation. 􏰷

• Holomorphic function spaces (H^p)

References:

1. T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society Student Texts, 1995. P.L. Duren,

2. Theory of H^p-Spaces, Academic Press Inc., 1970. J.P. Demailly,

3. Complex Analytic and Differential Geometry, Institut Fourier open manuscript.

Ders 9

Dersi veren: Ayesha Asloob Qureshi

Başlık (Title): Combinatorics of monomial and binomial ideals

Özet (Abstract): Commutative algebra is the basic essential tool used in algebraic geometry and algebraic number theory. The very first major connection of combinatorics with commutative algebra appeared in 1975 in the work of Richard Stanley where he strengthened the proof of the Upper Bound Conjecture for simplicial spheres. Later on, due to wide applications of combinatorial techniques in commutative algebra, combinatorial commutative algebra emerged as a sub-branch of commutative al- gebra. The main goal of this lecture series is to briefly introduce several topics in combinatorial commutative algebra, particularly the combinatorial structure related to monomial and binomial ideals of a polynomial ring in several variables.

Lecture 1+2: First we will review some basic concepts in commutative algebra, such as, Z-grading, Zn-grading, complexes and resolutions, Hilbert series, operations on monomial ideals. Then we will review different combinatorial interpretations of monomial ideals. Some open problems related to monomial ideals will be also discussed.

Lecture 3+4: We will briefly discuss the ideal membership problem, Gro ̈bner basis and toric ideals. Then we will continue with the combinatorial interpretations of binomial ideals. In particular, we will discuss, binomial edge ideals, generalized binomial edge ideals and polyomino ideals. Some open problems related to these classes of binomial ideals will be also discussed.

Not: Dersler ingilizce olacaktir.

Ders notları için:

Ders 10

Dersi veren: Özge Ülkem

Başlık (Title): Elliptic Curves vs Drinfeld Modules

Özet (Abstract): ​​Bu derste eliptik eğriler ve Drinfeld modülleri ile ilgileneceğiz. Cebirsel sayılar teorisinin yapi taslarindan biri olan eliptik eğriler cesitli arastirma konularinda (modular formlar, kriptografi vb.) kullanilir. V. Drinfeld 70'li yıllarda eliptik eğrilerin fonksiyon cisimleri dünyasinda analoglari olarak Drinfeld modüllerini tanimladi ve böylece su an yeni bir arastirma alani olan fonksiyon cisimleri aritmetigi alani acildi. Bu ders kapsamında bu iki dünyadaki araştırmalardan bahsedip aralarindaki benzerlikler ve farklılıkları konuşacağız.

(In this lecture we will introduce elliptic curves and Drinfeld modules. Elliptic curves are a fundamental object in algebraic number theory. There are many different research areas that arise from elliptic curves such as modular curves, cryptography. In the 70's V. Drinfeld introduced analogues of elliptic curves in the function field setting, leading to the area of function field arithmetic. We will talk on some of the research areas within algebraic number theory and function field arithmetic. We will also discuss similarities and differences between them.)