Phd program

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References:
A. Pillay, An Introduction to Stability Theory, Clarendon Press, Oxford, 1983 and 2002.

S. Shelah, Classification Theory,Elsevier, 2002.

W. Hodges, Model Theory, Oxford Univ. Pres., 1997.

95) LOGICAL SYSTEMS (AND UNIVERSAL LOGIC)

Course Coordinator: Andreka Hajnal

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Introduction to Mathematical logic.

Course Level: advanced PhD

Brief introduction to the course:

Establishing a meta-theory for investigating logical systems (logics for short), the concept of a general logic, some distinguished properties of logics.

The goals of the course:

The main goal of the course is to introduce students to the main concepts of the logical systems.

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents (week-by-week):

Week 1 Filter-property (syntactical) substitution property.

Week 2-3 Semantical substitution property. Structurality.

Week 4-5 Algebraizability. Algebraization of logics. Linden-baum-Tarski algebras.

Week 6-7 Characterization theorems for completeness, soundness and their algebraic counterparts.

Week 8-9 Concepts of compactness and their algebraic counterparts; definability properties and their algebraic counterparts; properties and their algebraic counterparts; omitting types properties and their algebraic counterparts.

Week 10 Applications, examples; propositional logic; (multi-)modal logical systems; dynamic logics (logics of actions, logics of programs etc.).

Week 11 Connections with abstract model theory, Beziau’s Universal Logic, Institutions theory.

Week 12 Elements of Abstract Model Theory (AMT); absolute logics; Abstract Algebraic Logic (AAL); Lindstrom's theorem in AMT versus that in AAL.

References:

1. J. Barwise and S. Feferman, editors, Model-Theoretic Logics, Springer-Verlag, Berlin, 1985.

2. W.J. Blok and D.L. Pigozzi: Algebraizable Logics, Memoirs AMS, 77, 1989.

3. L. Henkin, J.D. Monk, and A. Tarski: Cylindric Algebras, North-Holland, Amsterdam, 1985.

4. H. Andreka, I. Nemeti, and I. Sain: Algebraic Logic. In: Handbook of Philosophical Logic, 2, Kluwer, 2001. Pp.133-247.

96) LOGIC AND RELATIVITY 1

Course Coordinator:Istvan Nemeti

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Algebraic logic, the notion of a first order theory and its models.

Course Level: advanced PhD

Brief introduction to the course:

Axiomatization of the theory of relativity.

The goals of the course:

The main goal of the course is to introduce students to the main concepts of the axiomatization of the theory of relativity.

The learning outcomes of the course:

More detailed display of contents (week-by-week):

Week 1-2 Axiomatizing special relativity purely in first order logic. (Arguments from abstract model theory against using higher order logic for such an axiomatization.)

Week 3-4 Proving some of the main results, i.e. "paradigmatic effects", of special relativity from the above axioms. (E.g. twin paradox, time dilation, no FTL observer etc.)

Week 5-6 Which axiom is responsible for which "paradigmatic effect"

Week 7-8 Proving the paradigmatic effects in weaker/more general axiom systems (for relativity).

Week 9-10 Applications of definability theory of logic to the question of definability of "theoretical concepts" from "observational ones" in relativity. Duality with relativistic geometries.

Week 11-12 Extending the theory to accelerated observers. Acceleration and gravity. Black holes, rotating (Kerr) black holes. Schwarzschild coordinates, Eddington-Finkelstein coordinates, Penrose diagrams. Causal loops (closed time-like curves).Connections with the Church-Turing thesis.

References:

1. Andréka, H., Madarász, J., Németi, I.., Andai, A. Sain, I., Sági, G., Tıke, Cs.: Logical analysis of relativity theory. Parts I-IV. Lecture Notes. www.mathinst.hu/pub/algebraic-logic.

2. d'Inverno, R.: Introducing Einstein's Relativity. Clarendon Press, Oxford, 1992.

3. Goldblatt, R.: Orthogonality and spacetime geometry. Springer-Verlag, 1987.

97) LOGIC AND RELATIVITY 2

Course Coordinator:Istvan Nemeti

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Logic and relativity 1

Course Level: advanced PhD

Brief introduction to the course:

Among others, the course provides a logical/conceptual
analysis of relativity theories (both special and general, with a hint
at cosmological perspectives, too). We build up (and analyse)
relativity theories as theories in first order logic (FOL).

The goals of the course:

The main goal of the course is to introduce students to the advanced concepts of the axiomatization of the theory of relativity.

The learning outcomes of the course:

More detailed display of contents (week-by-week):

Week 1: Recalling FOL, model theory, definability theory.
Week 2: The theory SpecRel for special relativity.
Week 3. Analysing SpecRel, its variants.
Week 4: Interpreting SpecRel in an operational theory (Ax’s signalling theory).
Week 5: E=mc2
Week 6: The theory AccRel for the theory of accelerated observers (a
theory between special relativity and general relativity).
Week 7: Preparations for formalizing general relativity in FOL.
Week 8: The theory GenRel for general relativity.
Week 9: Spacetime of a black hole. Schwarzshild geometry.
Week 10: Einstein’s equation.
Week 11: A glimpse of cosmology.
Week 12: Accelerated expansion of the Universe.

References:

1. Rindler, W., Relativity. Special, General and Cosmological. Oxford
University Press, 2001.
2. d’Inverno, R., Introducing Einstein’s Relativity. Oxford University
Press, 1992.
3. Székely, G., First-order logic investigation of relativity theory
with an emphasis on accelerated observers. PhD Thesis, ELTE TTK, 2009.

98) FRONTIERS OF ALGEBRAIC LOGIC 1

Course Coordinator: Andréka Hajnal

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Algebraic logic, basic universal algebra

Course Level: advanced PhD

Brief introduction to the course:

Classical problems in Algebraic Logic are discussed, like the finitization problem, its connections with finite model theory, or parallels and differences between algebraic logic and (new trends in) the modal logic tradition, or Tarskian representation theorems and duality theories in algebraic logic and their generalizations (e.g. in axiomatic geometry and relativity theory).

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of the theory of Algebraic Logic.

The learning outcomes of the course:

More detailed display of contents (week-by-week):

1. Short overview of the process of algebraization of a logic.

2. Re-thinking the role of algebraic logic in logic. Theories as algebras,

interpretations between theories as homomorphisms.

3. Parallels and differences between algebraic logic and (new trends in) the

modal logic tradition.

4. Connections and differences between the algebraic logic based approach and

abstract model theory (e.g. in connection with the Lindström type theorems).

5. Tarskian representation theorems and duality theories in algebraic logic

and their generalizations (e.g. in axiomatic geometry and relativity theory).

6. The finitization problem, its connections with finite model theory.
7. On the finitization problem of first order logic (FOL). FOL without equality

versus FOL with equality.

8. The solution for FOL without equality. A meta-theorem: reducing the

algebraic logic problem to a semigroup problem.

9. On the proof of the meta theorem: adopting the neat embedding theorem for

our situation. Ultraproduct representation.

10. Finite schema axiomatization of generalized weak set G-algebras.
11. Finite axiomatization of our finite schema.
12. On the solution of the semigroup problem.

References:

1. Andréka, H., Németi, I., Sain, I.: Algebraic Logic. Chapter in Handbook of

Philosophical Logic, second edition. Kluwer.

2. van Benthem, J.: Exploring Logical Dynamics. Studies in Logic, Language

and Information, CSLI Publications, 1996.

3. Henkin, L. Monk, J. D. Tarski, A. Andréka, H. Németi, I.: Cylindric Set

Algebras. Lecture Notes in Mathematics Vol 883, Springer-Verlag, Berlin, 1981.

4. Henkin, L. Monk, J. D. Tarski, A.: Cylindric Algebras Part II.

North-Holland, Amsterdam, 1985.

5. Nemeti, I.: Algebraization of Quantifier Logics, an Introductory

Overview. Preprint version available from the home page of the Renyi

institure. Shorter version appeared in Stuia Logica 50, No 3/4, 485-570,

1991.
6. Sain, I.: On the search for a finitizable algebraization of first order

logic. Logic Journal of the IGPL, 8(4):495-589, 2000.

99) FRONTIERS OF ALGEBRAIC LOGIC 2

Course Coordinator: Andréka Hajnal

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Frontiers of algebraic logic 1

Course Level: advanced PhD

Brief introduction to the course:

Advanced topics in Algebraic Logic are discussed, like the solution of the finitization problem for classical first order logic, andfinite schematization problem.

The goals of the course:

The main goal of the course is to introduce students to the advanced topics and methods of the theory of Algebraic Logic.

The learning outcomes of the course:

More detailed display of contents (week-by-week):

1-2. Brief overview of the finitization problem.

3-4. Ideas on representation theory.
5. Connection between logic and algebra.
6. A solution of the finitization problem for classical first order logic

without equality, in algebraic form.

7. Application of the algebraic solution to logic.
8. Completeness, compactness, strong completeness.
9. The case of first order logic with equality.
10-11. The finite schematization problem.
12. Open problems

References:

1. Henkin, L. Monk, J. D. Tarski, A. Andréka, H. Németi, I.: Cylindric Set

Algebras. Lecture Notes in Mathematics Vol 883, Springer-Verlag, Berlin, 1981.

2. Henkin, L. Monk, J. D. Tarski, A.: Cylindric Algebras Part II.

North-Holland, Amsterdam, 1985.

3. Nemeti, I.: Algebraization of Quantifier Logics, an Introductory

Overview. Preprint version available from the home page of the Renyi

institure. Shorter version appeared in Stuia Logica 50, No 3/4, 485-570,

1991.
4. Sain, I.: On the search for a finitizable algebraization of first order

logic. Logic Journal of the IGPL, 8(4):495-589, 2000.

100) LOGIC OF PROGRAMS

Course coordinator: Laszlo Csirmaz

No. of Credits: 3 and no. of ETCS credits: 6

Prerequisities: Introduction to Logic

Course Level: advanced PhD

Objective of the course:

Logic of programs stemmed from the requirement of automatically proving that a computer program behaves as expected. Format methods are of great importance as they get rid of the subjective factor, checking by examples,and still letting the chance of a computer bug somewhere. When an algorithm is proved to be correct formally, it will always perform according to its specification. The course takes a route around this fascinating topic. How programs are modeled, what does it mean that a program is correct, what are the methods to prove correctness, and when are these methods sufficient.

The course requires knowledge of mathematical methods, especially Mathematical Logic and Universal Algebra; and perspective students should have some programming experience as well.

The course discusses how programs can be proved correct, what are the methods, their limitations. We state and prove characterizations for some of the most well-known methods. Temporal logic is a useful tool for stating and proving such theorems. The limitations of present-day methods for complex programs containing recursive definitions or arrays are touched as well.

Learning outcomes of the course:

At the end of the course students will know an overview of the recent results and problems, will be able to start their own research in the topics. Students will be able to judge the usefulness and feasibility of formal methods in different areas, including formal verification of security protocols. The course concludes with an oral exam or by preparing a short research paper.

Detailed contents of the course:

Computability, computations, structures, models
Programs, program scheme, chart and straight-line programs.
Interpreting program runs in structures, Herbrandt universe, partial and total correctness. Formalizing statements
Methods proving program termination
Calculus of annotated programs
Example: an “evidently wrong” sorting program is, in fact, correct
Floyd-Hoare method, correctness and completeness
Programs with strange time scale: parallel execution, eventuality and liveness; Dijsktra conditional statementsDynamic logic: soundness and completeness
Temporal logic of programs: modalities and expressive power
Recursive program schemes, operators, fixed-point theorems
Weak higher order structures, weak program runs, characterizations of the Floyd-Hoare method.
BAN logic, compositional logic for proving security properties of protocols.

Reference: T. Gergely, L Ury, First-order programming theories, Springer, 1991, Z.Manna, Mathematical theory of computation, Courier Dover Publications, 2003

101) CONVEX GEOMETRY

Course Coordinator: Karoly Boroczky

No. of Credits: 3, and no. of ECTS credits: 6.

Prerequisites:-

Course Level: introductory PhD

Brief introduction to the course:

The main notions and theorems of the Brunn-Minkowski Theory are presented like relatives of the isoperimetric inequality.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of Convex Geometry.

The learning outcomes of the course:

More detailed display of contents (week-by-week):

1. Affine spaces.

2. Euclidean space, structure of the isometry group, canonical form of isometries, Cartan's theorem.

3. Spherical trigonometry.

4-5. Fundamental theorems on convex sets (Caratheodory, Radon, Helly, Krein-Milman, Straszewicz etc.).

6-7. Convex polytopes, Euler's formula, classification of regular polytopes,

8. Cauchy's rigidity theorem, flexible polytopes.

9. Hausdorff metric, Blaschke's selection theorem,

10. Cauchy's formula, the Steiner-Minkowski formula, quermassintegrals

11-12. symmetrizations, isoperimetric and isodiametral inequalities.

References:

1. M. Berger, Geometry I-II, Springer-Verlag, New York, 1987.

2. K.W. Gruenberg and A.J. Weir, Linear Geometry, Springer, 1977.

102) FINITE PACKING AND COVERING BY CONVEX BODIES

Course Coordinator: Karoly Boroczky

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: -

Course Level: introductory PhD

Brief introduction to the course:

The main theorems of the Theory of Finite Packing and Covering by convex bodies are presented, among others about bin packing and covering.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of the Theory of Finite Packing and Covering.

The learning outcomes of the course:

More detailed display of contents (week-by-week):

Planar arrangements: Translative packings of a centrally symmetric convex domain, the Oler inequality.
Translative coverings by a centrally symmetric convex domain, the Fejes Tóth inequality.
The optimal packing of equal Euclidean circles (G. Wegner).
Density inside r-convex domains for arrangements of equal circles in the hyperbolic plane.
The extremal perimeter for packings and coverings by congruent convex domains.
The maximal perimeter for coverings by equal Euclidean circles.
The Hadwiger number in the plane. Clouds in the plane.
Higher dimensional arrangements: Optimal arrangements of balls in the sphericalspace.
The Sausage Conjecture and Theorem for Euclidean ball packings.
Extremal mean width for packings by congruent convex bodies
The Hadwiger number in high dimensions. Clouds in high dimensions.
Parametric density for translative arrangements. The asymptotic (Wulff) shape for translative lattice packings.

References:

J.H Conway, N.J.A. Sloane: Sphere packings, Lattices and Groups, Springer, 1999.
K. Boroczky: Finite Packing and Covering, Cambridge, 2004.

103) PACKING AND COVERING

Course Coordinator: Karoly Boroczky

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: -

Course Level: introductory PhD

Brief introduction to the course:

The main theorems of Packings and Coverings by convex bodies in the Euclidean space, and by balls in the spherical and hyperbolic spaces concentrating on density.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of the Theory of Packing and Covering.

The learning outcomes of the course:

More detailed display of contents (week-by-week):

Theorem of Groemer concerning the existence of densest packings and thinnest coverings. Dirichlet cells, Delone triangles.
Theorems of Thue and Kershner concerning densest circle packings and thinnest circle coverings. Packing and covering of incongruent circles. Theorems of Dowker, generalized Dirichlet cells. Packing and covering of congruent convex discs: theorems of C.A. Rogers and L.Fejes Tóth.
The moment theorem. Isoperimetric problems for packings and coverings. Existence of dense packings and thin coverings in the plane: p-hexagons, extensive parallelograms, theorems of W. Kuperberg, D. Ismailescu, G. Kuperberg and W. Kuperberg. The theorem of E. Sas.
Multiple packing and covering of circles.
The problem of Thammes; packing and covering of caps on the 2-sphere. The moment theorem on S2, volume estimates for polytopes containing the unit ball.
Theorem of Lindelöff, isoperimetric problem for polytopes. Packing and covering in the hyperbolic plane.
Packing of balls in Ed the method of Blichfeldt, Rogers' simplex bound. Covering with balls in Ed the simplex bound of Coxeter, Few and Rogers.
Packing in Sd, the linear programming bound. Theorem of Kabatjanskii and Levenstein.
Existence of dense lattice packings of symmetric convex bodies: the theorem of Minkowski-Hlawka.
Packing of convex bodies, difference body, the theorem of Rogers and Shephard concerning the volume of the difference body.
Construction of dense packings via codes.
The theorem of Rogers concerning the existence of thin coverings with convex bodies. Approximation of convex bodies by generalized cylinders, existence of thin lattice coverings with convex bodies

References:

1. L. Fejes Tóth: Regular figures, Pergamon Press, 1964.

2. J. Pach and P.K. Agarwal: Combinatorial geometry, Academic Press, 1995.

3. C.A. Rogers: Packing and covering, Cambridge University Press, 1964.

104) CONVEX POLYTOPES

Course Coordinator:Karoly Boroczky

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