# Mathematics Department Colloquium: Autumn 2013 schedule

Monday, November 18, 1530 - 1630. |

SPEAKER: Tatyana Shulman, University of Copenhagen.

TITLE: Mathematical aspects of zero-error quantum information theory.

ABSTRACT: In quantum information theory for mathematical description of quantum channels one uses the notion of completely positive maps on matrix spaces. In the first part of the talk we will discuss basic facts about these maps and their connection with quantum channels. In the second part of the talk we will focus on some mathematical problems related with superactivation of zero-error capacities of quantum channels. This is a joint work with M. Shirokov.

Monday, October 14, 1530 - 1630. |

SPEAKER: Pekka Pankka, University of Helsinki.

TITLE: Sharpness of Picard's theorem in space.

ABSTRACT: One of the classical theorems in complex analysis is

Picard's theorem stating that a non-constant entire holomorphic map

from the complex plane to the Riemann sphere omits at most two points.

In the late 1960's and early 1970's, results of Reshetnyak and

Martio-Rickman-Väisälä showed that mappings of bounded distortion,

also called as quasiregular mappings, can be viewed as a counterpart

for holomorphic mappings in conformal(/quasiconformal) geometry. One

of the natural goals from the very beginning in this theory was a

Picard type theorem.

In 1980, Rickman showed that a non-constant quasiregular mapping from

the Euclidean n-space to the n-sphere omits only finitely many points,

where the number depends only on the dimension and distortion. It was

however expected that this result would hold in the same strong form

as in the plane, i.e. the number of omitted points would be two in all

dimensions. In 1984, Rickman showed by a surprising construction that

given any finite set in the 3-sphere there exists a quasiregular

mapping from the Euclidean 3-space into the 3-sphere omitting exactly

that set.

In this talk, I will discuss joint work with David Drasin on the

sharpness of Rickman's Picard theorem in all dimensions. Especially, I

will discuss the role of bilipschitz geometry in the proof which leads

to a stronger stament on the metric properties of the map and is a

crucial ingredient in dimensions n > 3.

Picard's theorem stating that a non-constant entire holomorphic map

from the complex plane to the Riemann sphere omits at most two points.

In the late 1960's and early 1970's, results of Reshetnyak and

Martio-Rickman-Väisälä showed that mappings of bounded distortion,

also called as quasiregular mappings, can be viewed as a counterpart

for holomorphic mappings in conformal(/quasiconformal) geometry. One

of the natural goals from the very beginning in this theory was a

Picard type theorem.

In 1980, Rickman showed that a non-constant quasiregular mapping from

the Euclidean n-space to the n-sphere omits only finitely many points,

where the number depends only on the dimension and distortion. It was

however expected that this result would hold in the same strong form

as in the plane, i.e. the number of omitted points would be two in all

dimensions. In 1984, Rickman showed by a surprising construction that

given any finite set in the 3-sphere there exists a quasiregular

mapping from the Euclidean 3-space into the 3-sphere omitting exactly

that set.

In this talk, I will discuss joint work with David Drasin on the

sharpness of Rickman's Picard theorem in all dimensions. Especially, I

will discuss the role of bilipschitz geometry in the proof which leads

to a stronger stament on the metric properties of the map and is a

crucial ingredient in dimensions n > 3.