Historically, finite groups appeared in mathematics together with an action on some object, for example, as the groups of symmetries of polygons or polytopes. We are going to study, in some sense, the easiest possible actions of finite groups, namely, the linear actions on vector spaces. Such actions are called representations of finite groups. We will only consider representations in complex vector spaces.

Preliminary program:

Generalities on linear representations, irreducible representations and operation on representations. Character theory, Schur’s lemma, decompositions of representations. Subgroups and induced representations. Explicit examples. Representations of the symmetric group.

W. Fulton, J. Harris, Representation Theory - A First Course, GTM 129

Lineare Algebra I

 Overview  

 The Classification of Finite Simple Groups was one of the biggest achievements in 20th century group theory. By Jordan Hölder theorem, every finite group has a composition series consisting of finite simple groups. So, finite simple groups can be seen as building blocks for all finite groups. According to the classification theorem, a finite non-abelian simple group is either an alternating group, or a simple group of Lie type or one of the 26 sporadic groups which are not belong to any infinite family. In this seminar, the theory of simple groups of Lie type as groups automorphisms of simple Lie algebras will be discussed from the basics. The classical simple groups will be introduced and then root systems and Weyl groups, Chevalley groups and their structure will be investigated. After dealing with automorphisms of Chevalley groups, at the end of the seminar, we will introduce some geometric structures associated with simple groups of Lie type.

Textbook: Simple Groups of Lie Type, Roger Carter, Pure and Applied Mathematics Volume XXVIII, 1972

 Seminar plan:  A detailed plan of the talks will be available after the first meeting. 1st meeting : The Classical Simple Groups will be introduced (K. Ersoy)

2nd meeting : Weyl groups and root system. Abstract root systems, fundamental systems, length function, definitions by generators and relations and chamber systems.

3rd meeting: Simple Lie algebras, subalgebras, Cartan decomposition, roots of a simple Lie algebra, Dynkin diagrams, descriptions of simple Lie algebras, existence and isomorphism theorems

4th meeting : Chevalley groups, Chevalley basis, structure constants, the exponential map, algebras over arbitrary fields, the groups A_1(K)

5th meeting : Unipotent subgroups, Chevalley commutator formula.

6th meeting: Root SL(2,K) subgroups, their homomorphisms

The rest of the plan will be announced after first meeting.

 Overview  

 The Classification of Finite Simple Groups was one of the biggest achievements in 20th century group theory. By Jordan Hölder theorem, every finite group has a composition series consisting of finite simple groups. So, finite simple groups can be seen as building blocks for all finite groups. According to the classification theorem, a finite non-abelian simple group is either an alternating group, or a simple group of Lie type or one of the 26 sporadic groups which are not belong to any infinite family. In this seminar, the theory of simple groups of Lie type as groups automorphisms of simple Lie algebras will be discussed from the basics. The classical simple groups will be introduced and then root systems and Weyl groups, Chevalley groups and their structure will be investigated. After dealing with automorphisms of Chevalley groups, at the end of the seminar, we will introduce some geometric structures associated with simple groups of Lie type.

Textbook: Simple Groups of Lie Type, Roger Carter, Pure and Applied Mathematics Volume XXVIII, 1972

 Seminar plan:  A detailed plan of the talks will be available after the first meeting. 1st meeting : The Classical Simple Groups will be introduced (K. Ersoy)

2nd meeting : Weyl groups and root system. Abstract root systems, fundamental systems, length function, definitions by generators and relations and chamber systems.

3rd meeting: Simple Lie algebras, subalgebras, Cartan decomposition, roots of a simple Lie algebra, Dynkin diagrams, descriptions of simple Lie algebras, existence and isomorphism theorems

4th meeting : Chevalley groups, Chevalley basis, structure constants, the exponential map, algebras over arbitrary fields, the groups A_1(K)

5th meeting : Unipotent subgroups, Chevalley commutator formula.

6th meeting: Root SL(2,K) subgroups, their homomorphisms

The rest of the plan will be announced after first meeting.

Basics of non-standard analysis (according to Nelson)

In the analysis, calculating with in?nitesimalen sizes is formalized by the Grenzwertbegri?. Robinson has extended the body of real numbers so that it contains in?nitesimale elements, i.e. positive numbers that are smaller than all positive real numbers. Nelson later observed that the axioms of set theory can be expanded so that the "usual body" of the real numbers "standard" and "non-standard elements". Among the non-standard elements ?nden are again those that are positive and smaller than all positive standard elements.

The proseminar will discuss how the basic statements of the analysis can be reformulated and proven with the help of the in?nitesimalen elements. The proseminar is divided into the following sections:

I. Logical basis of NSA

II. Analysis I in the language of the NSA

III. advanced applications of NSA

IV. A look at Robinson's NSA

Homepage Prof. Schmitt

[1]    D. Landers, L. Rogge, Nichtstandard Analysis, Springer-Lehrbuch, Springer-Verlag, Berlin, 1994, x+485 S.
[2]    A.M. Robert, Nonstandard analysis, revised reprint of the 1988 translation, Dover Publications, Inc., Mineola, NY, 2003, xx+156 S.
[3]    H.-P. Tuschik, H. Wolter, Mathematische Logik—kurzgefaßt. Grundlagen, Modelltheorie, Entscheidbarkeit, Mengenlehre., 2. Au?., Spektrum Hochschultaschenbuch, Spektrum Akademischer Verlag, Heidelberg, 2002, viii+214 S.

Unbedingt zur Seminarvorbereitung lesen:

M. Lehn: Wie halte ich einen Seminarvortrag?

 Overview  

 The Classification of Finite Simple Groups was one of the biggest achievements in 20th century group theory. By Jordan Hölder theorem, every finite group has a composition series consisting of finite simple groups. So, finite simple groups can be seen as building blocks for all finite groups. According to the classification theorem, a finite non-abelian simple group is either an alternating group, or a simple group of Lie type or one of the 26 sporadic groups which are not belong to any infinite family. In this seminar, the theory of simple groups of Lie type as groups automorphisms of simple Lie algebras will be discussed from the basics. The classical simple groups will be introduced and then root systems and Weyl groups, Chevalley groups and their structure will be investigated. After dealing with automorphisms of Chevalley groups, at the end of the seminar, we will introduce some geometric structures associated with simple groups of Lie type.

Textbook: Simple Groups of Lie Type, Roger Carter, Pure and Applied Mathematics Volume XXVIII, 1972

 Seminar plan:  A detailed plan of the talks will be available after the first meeting. 1st meeting : The Classical Simple Groups will be introduced (K. Ersoy)

2nd meeting : Weyl groups and root system. Abstract root systems, fundamental systems, length function, definitions by generators and relations and chamber systems.

3rd meeting: Simple Lie algebras, subalgebras, Cartan decomposition, roots of a simple Lie algebra, Dynkin diagrams, descriptions of simple Lie algebras, existence and isomorphism theorems

4th meeting : Chevalley groups, Chevalley basis, structure constants, the exponential map, algebras over arbitrary fields, the groups A_1(K)

5th meeting : Unipotent subgroups, Chevalley commutator formula.

6th meeting: Root SL(2,K) subgroups, their homomorphisms

The rest of the plan will be announced after first meeting.

 Overview  

 The Classification of Finite Simple Groups was one of the biggest achievements in 20th century group theory. By Jordan Hölder theorem, every finite group has a composition series consisting of finite simple groups. So, finite simple groups can be seen as building blocks for all finite groups. According to the classification theorem, a finite non-abelian simple group is either an alternating group, or a simple group of Lie type or one of the 26 sporadic groups which are not belong to any infinite family. In this seminar, the theory of simple groups of Lie type as groups automorphisms of simple Lie algebras will be discussed from the basics. The classical simple groups will be introduced and then root systems and Weyl groups, Chevalley groups and their structure will be investigated. After dealing with automorphisms of Chevalley groups, at the end of the seminar, we will introduce some geometric structures associated with simple groups of Lie type.

Textbook: Simple Groups of Lie Type, Roger Carter, Pure and Applied Mathematics Volume XXVIII, 1972

 Seminar plan:  A detailed plan of the talks will be available after the first meeting. 1st meeting : The Classical Simple Groups will be introduced (K. Ersoy)

2nd meeting : Weyl groups and root system. Abstract root systems, fundamental systems, length function, definitions by generators and relations and chamber systems.

3rd meeting: Simple Lie algebras, subalgebras, Cartan decomposition, roots of a simple Lie algebra, Dynkin diagrams, descriptions of simple Lie algebras, existence and isomorphism theorems

4th meeting : Chevalley groups, Chevalley basis, structure constants, the exponential map, algebras over arbitrary fields, the groups A_1(K)

5th meeting : Unipotent subgroups, Chevalley commutator formula.

6th meeting: Root SL(2,K) subgroups, their homomorphisms

The rest of the plan will be announced after first meeting.

Basics of non-standard analysis (according to Nelson)

In the analysis, calculating with in?nitesimalen sizes is formalized by the Grenzwertbegri?. Robinson has extended the body of real numbers so that it contains in?nitesimale elements, i.e. positive numbers that are smaller than all positive real numbers. Nelson later observed that the axioms of set theory can be expanded so that the "usual body" of the real numbers "standard" and "non-standard elements". Among the non-standard elements ?nden are again those that are positive and smaller than all positive standard elements.

The proseminar will discuss how the basic statements of the analysis can be reformulated and proven with the help of the in?nitesimalen elements. The proseminar is divided into the following sections:

I. Logical basis of NSA

II. Analysis I in the language of the NSA

III. advanced applications of NSA

IV. A look at Robinson's NSA

Homepage Prof. Schmitt

[1]    D. Landers, L. Rogge, Nichtstandard Analysis, Springer-Lehrbuch, Springer-Verlag, Berlin, 1994, x+485 S.
[2]    A.M. Robert, Nonstandard analysis, revised reprint of the 1988 translation, Dover Publications, Inc., Mineola, NY, 2003, xx+156 S.
[3]    H.-P. Tuschik, H. Wolter, Mathematische Logik—kurzgefaßt. Grundlagen, Modelltheorie, Entscheidbarkeit, Mengenlehre., 2. Au?., Spektrum Hochschultaschenbuch, Spektrum Akademischer Verlag, Heidelberg, 2002, viii+214 S.

Unbedingt zur Seminarvorbereitung lesen:

M. Lehn: Wie halte ich einen Seminarvortrag?

Historically, finite groups appeared in mathematics together with an action on some object, for example, as the groups of symmetries of polygons or polytopes. We are going to study, in some sense, the easiest possible actions of finite groups, namely, the linear actions on vector spaces. Such actions are called representations of finite groups. We will only consider representations in complex vector spaces.

Preliminary program:

Generalities on linear representations, irreducible representations and operation on representations. Character theory, Schur’s lemma, decompositions of representations. Subgroups and induced representations. Explicit examples. Representations of the symmetric group.

W. Fulton, J. Harris, Representation Theory - A First Course, GTM 129

Lineare Algebra I

 Overview  

 The Classification of Finite Simple Groups was one of the biggest achievements in 20th century group theory. By Jordan Hölder theorem, every finite group has a composition series consisting of finite simple groups. So, finite simple groups can be seen as building blocks for all finite groups. According to the classification theorem, a finite non-abelian simple group is either an alternating group, or a simple group of Lie type or one of the 26 sporadic groups which are not belong to any infinite family. In this seminar, the theory of simple groups of Lie type as groups automorphisms of simple Lie algebras will be discussed from the basics. The classical simple groups will be introduced and then root systems and Weyl groups, Chevalley groups and their structure will be investigated. After dealing with automorphisms of Chevalley groups, at the end of the seminar, we will introduce some geometric structures associated with simple groups of Lie type.

Textbook: Simple Groups of Lie Type, Roger Carter, Pure and Applied Mathematics Volume XXVIII, 1972

 Seminar plan:  A detailed plan of the talks will be available after the first meeting. 1st meeting : The Classical Simple Groups will be introduced (K. Ersoy)

2nd meeting : Weyl groups and root system. Abstract root systems, fundamental systems, length function, definitions by generators and relations and chamber systems.

3rd meeting: Simple Lie algebras, subalgebras, Cartan decomposition, roots of a simple Lie algebra, Dynkin diagrams, descriptions of simple Lie algebras, existence and isomorphism theorems

4th meeting : Chevalley groups, Chevalley basis, structure constants, the exponential map, algebras over arbitrary fields, the groups A_1(K)

5th meeting : Unipotent subgroups, Chevalley commutator formula.

6th meeting: Root SL(2,K) subgroups, their homomorphisms

The rest of the plan will be announced after first meeting.

 Overview  

 The Classification of Finite Simple Groups was one of the biggest achievements in 20th century group theory. By Jordan Hölder theorem, every finite group has a composition series consisting of finite simple groups. So, finite simple groups can be seen as building blocks for all finite groups. According to the classification theorem, a finite non-abelian simple group is either an alternating group, or a simple group of Lie type or one of the 26 sporadic groups which are not belong to any infinite family. In this seminar, the theory of simple groups of Lie type as groups automorphisms of simple Lie algebras will be discussed from the basics. The classical simple groups will be introduced and then root systems and Weyl groups, Chevalley groups and their structure will be investigated. After dealing with automorphisms of Chevalley groups, at the end of the seminar, we will introduce some geometric structures associated with simple groups of Lie type.

Textbook: Simple Groups of Lie Type, Roger Carter, Pure and Applied Mathematics Volume XXVIII, 1972

 Seminar plan:  A detailed plan of the talks will be available after the first meeting. 1st meeting : The Classical Simple Groups will be introduced (K. Ersoy)

2nd meeting : Weyl groups and root system. Abstract root systems, fundamental systems, length function, definitions by generators and relations and chamber systems.

3rd meeting: Simple Lie algebras, subalgebras, Cartan decomposition, roots of a simple Lie algebra, Dynkin diagrams, descriptions of simple Lie algebras, existence and isomorphism theorems

4th meeting : Chevalley groups, Chevalley basis, structure constants, the exponential map, algebras over arbitrary fields, the groups A_1(K)

5th meeting : Unipotent subgroups, Chevalley commutator formula.

6th meeting: Root SL(2,K) subgroups, their homomorphisms

The rest of the plan will be announced after first meeting.

Basics of non-standard analysis (according to Nelson)

In the analysis, calculating with in?nitesimalen sizes is formalized by the Grenzwertbegri?. Robinson has extended the body of real numbers so that it contains in?nitesimale elements, i.e. positive numbers that are smaller than all positive real numbers. Nelson later observed that the axioms of set theory can be expanded so that the "usual body" of the real numbers "standard" and "non-standard elements". Among the non-standard elements ?nden are again those that are positive and smaller than all positive standard elements.

The proseminar will discuss how the basic statements of the analysis can be reformulated and proven with the help of the in?nitesimalen elements. The proseminar is divided into the following sections:

I. Logical basis of NSA

II. Analysis I in the language of the NSA

III. advanced applications of NSA

IV. A look at Robinson's NSA

Homepage Prof. Schmitt

[1]    D. Landers, L. Rogge, Nichtstandard Analysis, Springer-Lehrbuch, Springer-Verlag, Berlin, 1994, x+485 S.
[2]    A.M. Robert, Nonstandard analysis, revised reprint of the 1988 translation, Dover Publications, Inc., Mineola, NY, 2003, xx+156 S.
[3]    H.-P. Tuschik, H. Wolter, Mathematische Logik—kurzgefaßt. Grundlagen, Modelltheorie, Entscheidbarkeit, Mengenlehre., 2. Au?., Spektrum Hochschultaschenbuch, Spektrum Akademischer Verlag, Heidelberg, 2002, viii+214 S.

Unbedingt zur Seminarvorbereitung lesen:

M. Lehn: Wie halte ich einen Seminarvortrag?

Historically, finite groups appeared in mathematics together with an action on some object, for example, as the groups of symmetries of polygons or polytopes. We are going to study, in some sense, the easiest possible actions of finite groups, namely, the linear actions on vector spaces. Such actions are called representations of finite groups. We will only consider representations in complex vector spaces.

Preliminary program:

Generalities on linear representations, irreducible representations and operation on representations. Character theory, Schur’s lemma, decompositions of representations. Subgroups and induced representations. Explicit examples. Representations of the symmetric group.

W. Fulton, J. Harris, Representation Theory - A First Course, GTM 129

Lineare Algebra I

 Overview  

 The Classification of Finite Simple Groups was one of the biggest achievements in 20th century group theory. By Jordan Hölder theorem, every finite group has a composition series consisting of finite simple groups. So, finite simple groups can be seen as building blocks for all finite groups. According to the classification theorem, a finite non-abelian simple group is either an alternating group, or a simple group of Lie type or one of the 26 sporadic groups which are not belong to any infinite family. In this seminar, the theory of simple groups of Lie type as groups automorphisms of simple Lie algebras will be discussed from the basics. The classical simple groups will be introduced and then root systems and Weyl groups, Chevalley groups and their structure will be investigated. After dealing with automorphisms of Chevalley groups, at the end of the seminar, we will introduce some geometric structures associated with simple groups of Lie type.

Textbook: Simple Groups of Lie Type, Roger Carter, Pure and Applied Mathematics Volume XXVIII, 1972

 Seminar plan:  A detailed plan of the talks will be available after the first meeting. 1st meeting : The Classical Simple Groups will be introduced (K. Ersoy)

2nd meeting : Weyl groups and root system. Abstract root systems, fundamental systems, length function, definitions by generators and relations and chamber systems.

3rd meeting: Simple Lie algebras, subalgebras, Cartan decomposition, roots of a simple Lie algebra, Dynkin diagrams, descriptions of simple Lie algebras, existence and isomorphism theorems

4th meeting : Chevalley groups, Chevalley basis, structure constants, the exponential map, algebras over arbitrary fields, the groups A_1(K)

5th meeting : Unipotent subgroups, Chevalley commutator formula.

6th meeting: Root SL(2,K) subgroups, their homomorphisms

The rest of the plan will be announced after first meeting.

 Overview  

 The Classification of Finite Simple Groups was one of the biggest achievements in 20th century group theory. By Jordan Hölder theorem, every finite group has a composition series consisting of finite simple groups. So, finite simple groups can be seen as building blocks for all finite groups. According to the classification theorem, a finite non-abelian simple group is either an alternating group, or a simple group of Lie type or one of the 26 sporadic groups which are not belong to any infinite family. In this seminar, the theory of simple groups of Lie type as groups automorphisms of simple Lie algebras will be discussed from the basics. The classical simple groups will be introduced and then root systems and Weyl groups, Chevalley groups and their structure will be investigated. After dealing with automorphisms of Chevalley groups, at the end of the seminar, we will introduce some geometric structures associated with simple groups of Lie type.

Textbook: Simple Groups of Lie Type, Roger Carter, Pure and Applied Mathematics Volume XXVIII, 1972

 Seminar plan:  A detailed plan of the talks will be available after the first meeting. 1st meeting : The Classical Simple Groups will be introduced (K. Ersoy)

2nd meeting : Weyl groups and root system. Abstract root systems, fundamental systems, length function, definitions by generators and relations and chamber systems.

3rd meeting: Simple Lie algebras, subalgebras, Cartan decomposition, roots of a simple Lie algebra, Dynkin diagrams, descriptions of simple Lie algebras, existence and isomorphism theorems

4th meeting : Chevalley groups, Chevalley basis, structure constants, the exponential map, algebras over arbitrary fields, the groups A_1(K)

5th meeting : Unipotent subgroups, Chevalley commutator formula.

6th meeting: Root SL(2,K) subgroups, their homomorphisms

The rest of the plan will be announced after first meeting.