# Master's Degree Projects

## Supervisor: Alexander Vasiliev

Room 513

e-mail: alexander.vasiliev at uib.no

*Master's Degree Projects in* Analysis

*Analysis**Master's Degree Projects in*

### Moduli of families of curves and extremal problems for conformal maps (2-3 students)

**Abstract**. The project is aimed at thorough study of moduli of free families of homotopy classes of curves and their application to concrete extremal problems for conformal maps.

**Description**. The project includes topics from Differential Geometry, Real and Complex Analysis, Differential Equations. Students are supposed to learn Riemann surfaces and conformal invariants on them. Free families of homotopy classes of curves are to be defined. Modulus of a family of curves is a quantitative description of so-called “length-area” principle and possesses some properties of measure. Extremal families of curves is of particular importance. They are geodesics in certain Riemannian metric on the underlying Riemann surface generated by a quadratic differential. The method of moduli goes back to Ahlfors’ extremal length method and represents a powerful tool to solve extremal problems for conformal maps being combined with symmetrization and polarization. Students will be offered some concrete extremal problems:

- Distortion problems for conformal maps;
- Boundary distortion;
- Distortion of the modulus and quasiconformal maps.

**References**:

- A.Vasil'ev:
*Moduli of families of curves for conformal and quasiconformal mappings.*- ISBN 3-540-43846-7, Lecture Notes in Mathematics, vol. 1788, Springer-Verlag, Berlin- New York, 2002. - L.Ahlfors:
*Lectures on quasiconformal**maps*.- University Lecture Series, vol. 38, AMS, 2006. - Ch. Pommerenke:
*Boundary behavior of conformal maps*.- Springer-Verlag, New York, 1992.

### Dynamical systems in the unit disk and boundary behavior of conformal maps (1-2 students)

**Abstract**. The project is devoted to the Denjoy-Wolff theory of the dynamical systems in the unit disk. Students will study it and apply to fine problems of boundary behavior of univalent dynamical systems.

**Description**. The project includes topics from Dynamical Systems, Real and Complex Analysis, Ordinary Differential Equations. Conformal maps defined on some canonical domain (the unit disk in most of the cases) being analytic inside this domain show quite irregular behavior near the boundary. Many properties of boundary limits of conformal maps are known, in particular, angular limits have been studied deeply starting from the famous Beurling-Ahlfors paper in 1950. It is known that the angular limit exists at almost all points of the unit circle. However generally, very little can be said about the existence of the boundary derivatives. Moreover, one may observe the twisting effect at the boundary. For self-maps of the disk Julia-Wolff’s lemma implies that the angular derivative exists at almost all points where the angular limit exists and is equal to 1. Boundary derivatives are closely connected with reduced moduli of triangles and digons. The project is aimed to the study of mutual behavior of angular and usual internal derivatives applying the moduli technique as well as other extremal problems are expected to be solved.

**References**:

- Ch.Pommerenke,
*Boundary behaviour of conformal maps*, Springer-Verlag New-York, 1992. - Ch.Pommerenke, A.Vasil'ev:
*Angular derivatives of bounded univalent functions and extremal partitions of the unit disk.*- Pacific. J. Math.**206**(2002), no.2, 425-450.
t;
- A.Vasil'ev:
*Moduli of families of curves for conformal and quasiconformal mappings.*- Lecture Notes in Mathematics, vol. 1788, Springer-Verlag, Berlin- New York, 2002, 212 pp.

### Classical actions for evolution dynamics of conformal maps (1-2 students)

**Abstract**: The project is devoted to applications of String Theory to some classical problems of growth evolution in the complex plane.

**Description**: The project includes topics from Mathematical Physics, Differential Geometry, Real and Complex Analysis, Hamiltonian Systems. Classical actions have been proved to be one of the principal subjects of the research in Field Theory in general, and in Gauge String Theory in particular, where the Liouville action plays an important role in the definition of the two-dimensional gravity. This leads to so called Quantum Geometry of Riemann Surfaces. Takhtajan and Zograf made a rigorous mathematical treatment of Polyakov’s discovery that first-quantized bosonic string propagation can be described as a theory of free bosons coupled with the two-dimensional quantum gravity. It is not surprising that several classical mechanical processes possess some quantum features (KdV and the soliton theory, e.g.). Some two-dimensional evolution processes in Fluid Mechanics can be described by means of conformal maps. The project is aimed at study of Liouville and similar classical actions first of all for the Laplacian Growth and then, for general evolution families of conformal maps described by the Löwner-Kufarev equation and its modifications for quasiconformal and quasiconformally extendable maps. This study is closely connected with the Kirillov definition of infinite-dimensional Lie algebras (Virasoro, in particular), and with their links to the Teichmüller theory. It is expected to obtain also variations of the actions and these variations are expected to clear up that the actions represent potentials of the Weil-Petersson metric on the Teichmüller spaces. In the project we expect to discover other connections between evolution equations for conformal maps and problems of Quantum Field Theory.

**References**:

- A.M.Polyakov,
*Quantum geometry of bosonic strings*, Phys.Lett. B**103**(1981), no. 3, 207-210. - L.A.Takhtajan,
*Topics in quantum geometry of Riemann surfaces: two-dimensional quantum gravity*, Proc. Internat. School Phys. Enrico Fermi, 127, IOS, Amsterdam, 1996. - L.D.Faddeev and O.A.Yakubovsky,
*Lectures in Quantum Mechanics for students-mathematicians*, 1980. - P.G.Zorgraf, L.A.Takhtajan,
*Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler metrics on*M(0,n), Trans. Amer. Math. Soc.**355**(2003), no. 5, 1857-1867. - A.Vasil'ev,
*On a parametric method for conformal maps with quasiconformal extensions*.- Publ. de l'Institut Math. (Nouvelle Ser.), Belgrad**75(89)**(2004), 9-24. - V.I. Arnold,
*Mathematical methods of classical mechanics,*Springer-Verlag New-York, 1989.

*Master Degree Projects in Mathematics Methodology*

### Exceptional sets in first-year Calculus

**Abstract:** The project addresses the problem of construction of a function by its derivative.

**Description**: A well-known theorem in the first-year Calculus states that a continuous function in the interval [a,b] differentiable in this interval is constant if and only if its derivative is zero. The result remains true if we assume differentiability in [a,b]\E, where E is a finite number of points in [a,b]. How big can the set E be? The project includes some bibliography work.

**References:**

- N.Bourbaki,
*Functions of a real variable. Elementary theory.*Translated from the 1976 French original Elements of Mathematics (Berlin).*Springer-Verlag, Berlin,*2004.

## Supervisor: Irina Markina

Room 535

e-mail: irina.markina at uib.no

*Master's Degree Projects in* Analysis

*Analysis*

*Master's Degree Projects in*

### The local and global sub-Riemannian structures of spheres and Heisenberg groups

**Description.** The first task is to understand how the global sub-Riemannian structure of one-dimensional Heisenberg group is related to the sub-Riemannian structure of 3-dimensional sphere. This question can be generalized to the relation between n-dimensional Heisenberg group and odd-dimensional spheres. It is also interesting to study the relation between other Carnot groups related to division algebras and sub-Riemannian structures of corresponding spheres.

**References:**

- A. Bellaïche, J.J. Risler.
*Sub-Riemannian geometry*. Progress in Mathematics, Birkhäuser, 1996

### Existence of a quasiconformal flow of sub-Riemannian spheres

**Description.** It is known that Carnot groups, in general, is quite rigid and do not admit quasiconformal maps. Nevertheless, the Heisenberg group has infinite-dimensional flow of quasiconformal maps. Question: does the 3-dimensional sphere admit an infinite-dimensional flow of quasiconformal maps? Describe this flow and study quasiconformal maps on sphere. This question can be asked about any odd-dimensional sphere.

**References:**

- A. Korányi, H. M. Reimann.
*Foundation for the theory of quasiconformal mappings on the Heisenberg group*. Advanced in Math. (1995),**111,**1—87.

### Module of family of measures on nilpotent groups

**Description.** What is the correct notion of a family of measures on the nilpotent groups? Does there exist any relation between the module of a family of curves (horizontal curves) connecting two compacts in a domain and the family of the surfaces separating these compacts? It would be interesting to construct simple domains on other sub-Riemannian manifolds for which the module of a family of curves can be calculated exactly.

**References:**

- B. Fuglede. Extremal length and functional completion. Acta Math.
**98**1957 171--219.

*Master Degree Projects in Mathematics Methodology*

*Master Degree Projects in Mathematics Methodology*

### Variety of integrals

**Description.** It is known that in analysis there exist different definitions of integrals. Why we need several of them? The project is intended to describe different constructions of integrals and explain their historical and mathematical reasons.

**References:**

- D. Bressoud.
*A radical approach to Lebesgue’s Theory of integration*. Cambridge University Press, 2008.

### What are the exclusive sets?

**Description.** The notion of “small”, “exclusive”, “polar”, ”thin” set is used to denote a set that we can neglect in one or another sense. Typical examples are sets of zero Lebesgue measure and zero capacity. Are there other notions of “measures” to measure a negligible set? Where are they used and where can they be applied?

**References:**

- L.C. Evans, R.F.Gariepy.
*Measure theory and fine properties of functions*. Studies in Advanced mathematics.*CRC Press, 1*992.

## Supervisor: Arne Stray

Room 515

e-mail: arne.stray at mi.uib.no

*Master's Degree Projects in* Analysis

*Analysis**Master's Degree Projects in*

### Approximation in function spaces

**Abstract.** The project concerns approximation with smooth harmonic or analytic functions in various norms.

**Description. **The project includes topics from complex analysis, functional analysis and harmonic analysis on Euclidean spaces. The interplay between growth properties and boundary values for analytic functions in the unit disc are important. Function spaces of particular interest are Hardy spaces, Bergman spaces and the Dirichlet space.**References:**

- J.B.Garnett.
*Bounded analytic functions*, Springer Verlag 2007

### Pick – Nevanlinna interpolation.

**Abstract. **The project is aimed at studying the solution sets of general Pick-Nevanlinna problems using Nevanlinna's parametrization formula.

**Description. **Around 1930 R. Nevanlinna found a parametrization of all solutions to certain interpolation problems. This is a chapter in classical function theory that more recently arouse interest because its connections to functional analysis and operator theory. The project concerns a detailed study of Nevanlinna's formula and

connections with approximation problems for bounded analytic functions.**References:**

- J.B.Garnett.
*Bounded analytic functions*, Springer Verlag 2007