Tere-tere! Hello Hello!
Kursuse eesmärk on tuua 3. aasta bakalaureuseüliõpilasteni ja ka magistrantide/ doktorantideni erinevaid teemasid, objekte, valdkondi millega tegeleb tänapäeva matemaatiline uurimustöö, ning seeläbi avardada õpilaste silmaringi ja tekitada huvi. Kursus koosneb mitmest erinevast minikursusest, igaüks vähemalt 3h, ja ka mõnest eraldiseistvast loengust. Need leiavad aset umbes tihedusega üks nädal kuus. Allpool leiate kahe esimese minikursuse lühikirjeldused. Ülejäänud teemad ja kirjeldused lisanduvad semestri jooksul kursuse veebilehele ja Moodle' keskkonda. Umbes nädal enne ilmub mõnikord ka ettevalmistavat materjali nii siia kui Moodle' keskkonda.
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The aim of the course is to introduce third-year undergraduate students, as well as Master's and doctoral students, to a range of topics, objects, and areas that are part of contemporary mathematical research. In doing so, the course seeks to broaden students' mathematical horizons and stimulate their interest in current research. The course consists of several separate mini-courses, each lasting at least three hours, as well as a number of standalone lectures. These will take place approximately one week per month. Here are the descriptions of the first two topics, more will be communicated as the semester progresses on the course website or Moodle. Look out also for preparatory materials both here and on Moodle!
Topic 1 (week of 28th of September, Prof. F. Richter): A dynamical approach to number theory
Abstract: What can the study of orbits in dynamical systems tell us about seemingly static, discrete objects such as the prime numbers, integer solutions to equations, or set partitions? Far more than one might expect. Ergodic theory, which studies the long-term statistical behavior of such orbits, has found remarkable applications in combinatorics and number theory. In this short lecture series, I will introduce the main ideas behind this connection, explaining how arithmetic problems can be recast in dynamical terms and how this perspective has led to breakthroughs beyond the reach of other methods.
Preparatory material (some basic definitions and results in measure theory): Attach:MeasureTheoreticPreliminaries.pdf
Topic 2 (week of 19th of October, Prof. D. Wyss): P-adic numbers and some of their applications
For every prime numberp the field of rational numbers Q admits a completion called the p-adic numbers Q_p and these are the only completions of Q other than the real numbers R. I will present an overview of the properties and applications of these numbers to various areas of mathematics, in particular I will try to explain the monodromy conjecture of Igusa.
The aim of the course is to introduce third-year undergraduate students, as well as Master's and doctoral students, to a range of topics, objects, and areas that are part of contemporary mathematical research. In doing so, the course seeks to broaden students' mathematical horizons and stimulate their interest in current research.