Přednášky v podzimním semestru 2015 |

Přednášky se konají na Ústavu matematiky a statistiky,budova č.8 v areálu Přírodovědecké fakulty, Kotlářská 2, Brno. ## středa 4. listopadu 2015, v 16:00 v M1
## Abstract:Contemporary statistics is often faced with data sets consisting of data units that are complex objects, such as trajectories, curves, surfaces or images that can be seen as realizations of stochastic processes. Functional data analysis is a branch of statistics that deals with collections of such random variables in function spaces. I will present two areas of my recent research experience in this field. Both problems originate from applied settings and lead to the development of novel statistical methods involving ill-posed inverse problems and the asymptotic analysis of their regularized solutions. The first one is motivated by a problem from molecular biology, namely the study of the mechanical properties of DNA molecules. I will present procedures for comparing covariance operators using spectrally truncated approximations of the Hilbert--Schmidt distance of the empirical covariance operators. The second part comes from a public health study of heart activity patterns. A complete inferential framework is developed for a new type of data, partially observed functional data (fragments of temporal heart rate profiles in this example), with a focus on prediction of principal components and function completion via regularized conditional expectation. ## pátek 6. listopadu 2015, ve 14:00 v M1
## Abstract:
Quadratic forms have long played a central role in number theory - for example, the investigation of which primes are of the form \(x^2+ny^2\) led to the development of a number of crucial tools in algebraic number theory. In the talk, we will be mostly interested in universal forms, i.e., positive definite quadratic forms which represent all natural numbers - a classical example is the sum of four squares \(x^2 + y^2 + z^2 + w^2\). ## čtvrtek 12. listopadu 2015, ve 14:00 v M5
## Abstract:The work deals with the existence of solutions of an integro-differential equation arising in population dynamics in the case of anomalous diffusion involving the negative Laplace operator raised to a certain ractional power. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for non-Fredholm elliptic operators in unbounded domains along with the Sobolev inequality for a fractional Laplacian are being used. ## středa 25. listopadu 2015, v 17:00 v M1
## Abstract:The subject of Cauchy-Riemann Geometry (shortly: CR-geometry), founded in the research of Henri Poincare, is remarkable in that it lies on the border of several mathematical disciplines, among which we emphasize Complex Analysis and Geometry, Differential Geometry, and Partial Differential Equations. Very recently, in my research, I have discovered a new face of CR-geometry. This is a novel approach of interpreting objects arising in CR-geometry (called CR-manifolds) as certain Dynamical Systems, and vice versa. It turns out that geometric properties of CR-manifolds are in one-to-one correspondence with that of the associated dynamical systems. In this way, we obtain a certain vocabulary between the two theories. The latter approach has enabled us recently to solve a number of long-standing problems in CR-geometry related to mappings of CR-manifolds with degeneracies of the CR-structure. We call this method the CR (Cauchy-Riemann manifolds) - DS (Dynamical Systems) technique. In this talk, I will outline the CR - DS technique, and describe its recent applications to Complex Geometry and Dynamics. ## pátek 27. listopadu 2015, ve 14:00 v M1
## Abstract:
In this talk we survey, first, the recent development in computational homotopy theory and, second, the consequent applications in computer science and computational geometry. The basic and important example of the former is the computational complexity of the ## středa 2. prosince 2015, v 17:00 v M1
## Abstract:Quantum mechanics is a fundamental theory to investigate the structure of matter, from the small scale of atoms to the large scale of stars. In principle, the full quantum theory is linear, but it is very hard to compute explicitly when the number of particles becomes large. Therefore, physical properties of large systems are often addressed using mathematical tools from functional analysis, spectral theory and partial differential equations. I will discuss several mathematical problems in many-body quantum mechanics, from the structure of atoms in the periodic table to the Bose-Einstein condensation and superfluidity of cold gases. ## pátek 4. prosince 2015, ve 14:00 v M1
## Abstract:
According to Einstein's theory of general relativity, the model for our universe is a four-dimensional spacetime, whose points represent events. Distances between two events are measured by means of a metric, and there are non-zero vectors, said to be null, with zero length. In particular, light rays in our universe are geometrically described by null geodesics. |

Aktualizováno Úterý, 01 Prosinec 2015 14:01 |

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