Phd program

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

Prerequisites: -

Course Level: introductory PhD 

Brief introduction to the course:

The main theorems about the combinatorial structure of convex polytopes are presented concentrating on the numbers of faces in various dimensions.

The goals of the course:

To introduce the basic combinatorial properties of convex polytopes.

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):  

1. 
   Polytopes as convex hull of finite point sets or intersections of halfspaces.
   
2. 
   Faces of polytopes.
   
3. 
   Examples: Simplicial, simple, cyclic and neighbourly polytopes.
   
4. 
   Polarity for polytopes.
   
5. 
   The Balinski theorem.
   
6. 
   Discussion of the Steinitz theorem for three polytopes.
   
7. 
   Realizability using rational coordinates.
   
8. 
   Gale transform and polytopes with few vertices.
   
9. 
   The oriented matroid of a polytope
   
10. 
    Shelling.Euler-Poincaré formula
    
11. 
    h-vector of a simplicial polytope, Dehn-Sommerfield equations
    
12. 
    Upper bound theorem Stresses Lower bound theorem Weight algebra Sketch of the proof of the g-theorem.
    

Reference: G.M. Ziegler: Lectures on polytopes. Springer, 1995.

105) COMBINATORIAL 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:

Convexity, separation, Helly, Radon, Ham-sandwich theorems, Erdős-Szekeres theorem and its relatives, incidence problems, the crossing number of graphs, intersection patterns of convex sets, Caratheodory and Tverberg theorems, order types, Same Type Lemma, the k-set problem

The goals of the course:

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

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: convexity, linear and affine subspaces, separation

week 2: Radon' theorem, Helly's theorem, Ham-sandwich theorem

week 3: Erdős-Szekeres theorem, upper and lower bounds

week 4: Erdős-Szekeres-type theorems, Horton sets

week 5: Incidence problems

week 6: crossing numbers of graphs

week 7: Intersection patterns of convex sets, fractional Helly theorem, Caratheodory theorem

week 8: Tverberg theorem, order types, Same Type Lemma

week 9-10: The k-set problem, duality, k-level problem, upper and lower bounds

week 11-12: further topics, according to the interest of the students

Reference: J. Matousek: Lectures on Discrete Geometry, Springer, 200

106) GEOMETRY OF NUMBERS

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 basic properties Eucledian lattices and the Brunn-Minkowski Theory are presented, followed by the Minkowski theorems, basis reduction, and applications to Diophantine approximation, flatness theorem.

The goals of the course:

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

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):  

1. 
   Lattices, sublattices, bases, determinant of a lattice.
   
2. 
   Convex bodies, elements of the Brunn-Minkowski theory, duality, star bodies. Selection theorems of Blaschke and Mahler.
   
3. 
   The fundamental theorem of Minkowski, and its generalizations: theorems of Blichfeldt, van der Corput.
   
4. 
   Successive minima, Minkowski's second theorem.
   
5. 
   The Minkowski-Hlawka theorem.
   
6. 
   Reduction theory, Korkine-Zolotarev basis, LLL basis reduction.
   
7. 
   Connections to the theory of packings and coverings.
   
8. 
   Diophantine approximation: simultaneous, homogeneous, and inhomogeneous.
   
9. 
   Theorems of Dirichlet, Kronecker, Hermite, Khintchin
   
10. 
    Short vector problem, nearest lattice point problem Applications in combinatorial optimization.
    
11. 
    The flatness theorem.
    
12. 
    Covering minima Algorithmic questions, convex lattice polytopes.
    

References:

1. J.W.S Cassels: An introduction to the geometry of numbers, Springer, Berlin, 1972.

2. P.M. Gruber, C.G. Lekkerkerker: Geometry of numbers, North-Holland, 1987.

3. L. Lovász: An algorithmic theory of numbers, graphs, and convexity, CBMS-NSF regional conference series, 1986.

107) STOCHASTIC GEOMETRY

Course Coordinator: Karoly Boroczky

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

Prerequisites: -

Course Level: advanced PhD 

Brief introduction to the course:

The main theorems of Stochastic Geometry are presented among others about approximation by polynomials, and by the application related splines.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of Stochastic Geometry, and applications in various fields.  

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):  

1. 
   Space of lines, measures on the space of lines
   
2. 
   Spaces, groups, measures, intersection formulae
   
3. 
   Minkowski addition and projections
   
4. 
   Lines and flats through convex bodies, the Crofton formulae
   
5. 
   Valuations. Hadwiger’s charactherization of isometry invariant valuations.
   
6. 
   Random polytopes, approximation by random polytopes, expectation of the deviation in various measures
   
7. 
   Connections to floating bodies and affine surface area, extremal properties of balls and polytopes
   
8. 
   Random methods in geometry 1: the Erdos-Rogers theorem,
   
9. 
   Random methods in geometry 2: The Johnson-Lindenstrauss theorem, Dvoretzki's theorem,
   
10. 
    Random hyperplane arrangements.
    
11. 
    Applications in computational geometry
    
12. 
    Applications to isoperimetric deficit.
    

References:

1. L.A. Santalo, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Appl., Vol 1. Addison-Wiley, 1976.

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

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

108)BRUNN-MINKOWSKI THEORY

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 properties of convex bodies in Euclidean spaces are introduced. Inequalities like the Isoperimetric inequality and the Brunn-Minkowski inequality are discussed, and applications to analytic inequalities like the Sobolev inequality and the Prekopa-Leindler inequality are presented.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of the Brunn-Minkowski Theory.  

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):  

1. 
   Isoperimetric inequality in the plane, sharpening with the inradius.
   
2. 
   Distance function.
   
3. 
   Support properties, support function.
   
4. 
   Minkowski sum, Blaschke-Hausdorff distance.
   
5. 
   Blaschke selection theorem.
   
6. 
   Almost everywhere differentiability of convex functions.
   
7. 
   Cauchy surface formula.
   
8. 
   Steiner symmetrization, isoperimetric inequality via Steiner symmetrization.
   
9. 
   Mixed volumes.
   
10. 
    Brunn-Minkowski inequality. Minkowski's inequality for mixed volumes, isoperimetric inequality
    
11. 
    Alexandrov-Fenchel inequality
    
12. 
    Prekopa-Leindler inequality
    

References:

1. T. Bonnesen, W. Fenchel, Theory of convex bodies, BSC Associates, Moscow, Idaho, 1987.

2. R. Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge Univ. Press, Cambridge, 1993.

109) NON-EUCLIDEAN GEOMETRIES

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 non-Euclidean geometries, like Projective, Spherical and Hyperbolic geometry, are presented, and axiomatic aspects are discussed.

The goals of the course:

The main goal of the course is to introduce students to the main facts about non-Euclidean geometries.  

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):  

1. 
   Axiomatic foundation.
   
2. 
   Projective spaces over division rings, Desargues' and Pappus' theorem.
   
3. 
   The duality principle.
   
4. 
   Collineations, correlations, cross-ratio preserving transformations.
   
5. 
   Quadrics, classification of quadrics.
   
6. 
   Pascal's and Brianchon's theorems.
   
7. 
   Polarity induced by a quadric, pencils of quadrics, Poncelet's theorem.
   
8. 
   Models of the projective space, orientability.
   
9. 
   Spherical trigonometry.
   
10. 
    Hyperbolic geometry: the hyperboloid model
    
11. 
    Hyperbolic trigonometry, isometries.
    
12. 
    Other models of the hyperbolic space and the transition between them.
    

References:

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

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

110) DIFFERENTIAL GEOMETRY

Course Coordinator: Balázs Csikós

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

Prerequisites: -

Course Level: intermediatePhD 

Brief introduction to the course:

The main theorems of Differential Geometry are presented among others about curves, surfaces and the curvature tensor.

The goals of the course:

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

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):  

1. 
   Curves in R2.
   
2. 
   Hypersurfaces in R3. Theorema Egregium. Special surfaces.
   
3. 
   Differentiable manifolds, tangent budle, tensor bundles;
   
4. 
   Lie algebra of vector fields, distributions and Frobenius' theorem;
   
5. 
   Covariant derivation, the Levi-Civita connection of a Riemannian manifold,
   
6. 
   Parallel transport, holonomy groups;
   
7. 
   Curvature tensor, symmetries of the curvature tensor,
   
8. 
   Decomposition of the curvature tensor;
   
9. 
   Geodesic curves, the exponential map,
   
10. 
    Gauss Lemma, Jacobi fields, the Gauss-Bonnet theorem;
    
11. 
    Differential forms, de Rham cohomology, integration on manifolds,
    
12. 
    Stokes' theorem.
    

References:

1. M.P. do Carmo: Differential Geometry of Curves and Surfaces Prentice-Hall, Englewood Cliffs, NJ, 1976.

2. W. Klingenberg: A course in differential geometry, Springer, 1978.

3. W.M. Boothby: An introduction to differentiable manifolds and Riemannian geometry, Second Edition, Academic Press, 1986.

111) HYPERBOLIC MANIFOLDS

Course Coordinator:Gabor Moussong

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

Prerequisites: -

Course Level: advanced PhD 

Brief introduction to the course:

The main theorems about the structure and construction of Hyperbolic Manifolds are presented like Discrete groups of isometries of hyperbolic space, Margulis’ lemma, Thurston’s geometrization conjecture, and an overview of Perelman’s proof.

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 Hyperbolic Manifolds.  

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):  

1. 
   Hyperbolic space. Overview of the projective, quadratic form, and conformal models.
   
2. 
   Isometries and groups of isometries.
   
3. 
   Hyperbolic manifolds. Hyperbolic structures, developing and holonomy, completeness.
   
4. 
   Discrete groups of isometries of hyperbolic space. The case of dimension two.
   
5. 
   Constructing hyperbolic manifolds. Fundamental polyhedra and the Poincaré theorems. Some arithmetic constructions.
   
6. 
   Mostow Rigidity. Extending quasi-isometries.
   
7. 
   The Gromov-Thurston proof of the rigidity theorem for closed hyperbolic manifolds.
   
8. 
   Structure of hyperbolic manifolds.
   
9. 
   Margulis' Lemma and the thick-thin decomposition of complete hyperbolic manifolds of finite volume.
   
10. 
    Thurston's hyperbolic surgery theorem. The space of hyperbolic manifolds.
    
11. 
    Properties of the volume function. Dehn surgery on three-manifolds and Thurston's theorem.
    
12. 
    The geometrization conjecture, and discussion of Perelman’s proof. Topology of three-manifolds: geometric structures and the role of hyperbolic geometry in Thurston's theory.
    

References:

1. R. Benedetti, C.~Petronio, Lectures on Hyperbolic Geometry, Springer, 1992

2. J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, 1994

112) MODERN SET THEORY

Course coordinator: István Juhász

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

Prerequisites:-

Course level: intermediate PhD

Brief introduction to the course:

The basics of both axiomatic and combinatorial set theory will be presented.

The goals of the course:

One goal is to present the most important results in set theory. Another goal is to get the students acquainted with consistency and independence results.

The learning outcomes of the course:

The students will learn not just the axiomatic development of set theory but the more general significance of the axiomatic method in mathematics.

More detailed display of contents:

Week 1-2 The cumulative hierarchy and the ZFC axiom system

Week 3-4 Axiomatic exposition of set-theory

Week 5-6 Absoluteness and reflection

Week 7 Models of set-theory, relative consistency

Week 8 Constructible sets, consistency of AC and GCH

Week 9 Combinatorial set-theory and combinatorial principles

Week 10-11 Large cardinals

Week 12 Basic forcing

References:

1. András Hajnal, Peter Hamburger: Set Theory, Cambridge University Press,1999.

2. Thomas Jech: Set Theory, Spinger-Verlag, 1997.

3. Kenneth Kunen: Set theory. An introduction to Independence Proofs,   Elsevier,1999
113) INTRODUCTION TO FORCING
Course coordinator: Laszlo Csirmaz
Prerequisites: Modern Set Theory

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

Course Level: advanced PhD 

Objective of the course:
The course is an introduction to this important branch of modern set theory: a general method to prove that a statement is independent of the usual axioms of set theory. We take a tour to investigate different models of set theory, the constructible universe (Godel's L), and the permutation models. We also touch the role of regularity, and the anti-well-founded axiom of Aczel. Both approaches to forcing is considered: via complete Boolean algebras and via partially ordered sets, we also show the equivalnce of the two methods. The course ends with an introduction of iterated forcing, and constructing models without the axiom of choice.
Learning outcomes of the course:
At the end of the course, the students

• 
  will be able to understand the structure of models of set theory,
  
• 
  can apply the forcing argument to create different models of set theory,
  
• 
  can construct models where certain set theoretical statements are satisfied,
  
• 
  understand the permutation model and the role of the axiom of choice,
  
• 
  create models of the negation of axiom of choice, and the negation of continuum hypothesis,
  
• 
  will understand the main properties of the constructible universe.
  

Detailed contents of the course:

1. 
   Axioms of set theory; models, collapsing, reflection principle
   
2. 
   Godel's operations, the Godel-Bernays axiomatization, absoluteness, the constructible universe
   
3. 
   Statements true in V=L: axiom of choice, generalized continuum hypothesis, diamond principle, Existence of Kurepa trees
   
4. 
   Partially ordered sets, complete Boolean algebras, topological equivalence, dense sets, filters, Rasiowa-Sikorski theorem
   
5. 
   Antichains and kappa-completeness. Transitive epsilon models, consistency of non-existence of such models. The method of forcing
   
6. 
   The M-generic model and its properties; names and interpretation; evaluation of formulas as elements of a Boole algebra, the notion of forcing
   
7. 
   The M-generic model is a model of ZFC; basic properties of forcing; M[G] is constructed from M and G
   
8. 
   Forcing constructions: continuum hypothesis and its negation, the role of kappa-completeness and kappa-antichain condition: preserving and collapsing cardinals
   
9. 
   Permutation models, models with urelements, permutation model with urelements where the axiom of choice fails
   
10. 
    Creating generic model where the AC fails; ordinal-definable elements; models where all ultrafilters on omega are trivial; Ajtai's construction of a Hilbert space where every linear operator is bounded
    
11. 
    Iterated forcing, iterating with finite support, Martin's axiom
    
12. 
    Hajnal-Baumgartner result which proves a ZFC result through forcing and absoluteness.
    

References:

1. 
   T. Jech: Set Theory
   
2. 
   K.Kunen: Set Theory
   

Assessment:
Students can a) either choose one of the topics 6—12 above and make presentation from it or b) take an oral exam after the course.

114) DESCRIPTIVE SET THEORY

Course Coordinator: Istvan Juhasz

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

Prerequisites: Modern Set Theory

Course Level: advanced PhD 

Brief introduction to the course:

The main theorems of Descriptive Set Theory are presented, explaining the relation to real analysis.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of Descriptive Set Theory.  

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.