Serdecznie zapraszamy wszystkich studentów studiów stacjonarnych naszego wydziału na wykłady online, które w dniach 7.04.2021 - 10.06.2021 r. poprowadzi Prof. David Grimm z Universidad de Santiago de Chile.
Tematem wykładów będzie: COMMUTATIVE ALGEBRA
Wykłady odbędą się w ramach projektu ZINTEGROWANY PROGRAM ROZWOJU Uniwersytetu Śląskiego w Katowicach.
Prof. Przemysław Koprowski współpracujący z wykładowcą w następujący sposób przedstawia jego sylwetkę:
Professor David Grimm obtained his diploma (equivalente of master) and doctoral degree (equivalente of PhD) at the Universität Konstanz, Germany in 2006 and 2011, respectively. Between 2011 and 2015, he has worked as a postdoctoral researcher and teacher at the Universität Ulm (Germany) and the Ecole Polytechnique Federale de Lausanne (Switzerland). Since 2015, he is working assistant professor at the Universidad de Santiago de Chile, where he teaches both on graduate and undergraduate level, and is currently responsible for coordinating Algebra 1 for first year engineering students.
His main research interest is focused on applications of valuation theory and arithmetic geometry to problems from real algebra and quadratic forms.
Most recent publications:
Hereditarily non-pythagorean fields: https://arxiv.org/abs/2001.00618 (with David Leep, Journal of Algebra, vol. 574, 2021)
Non-split conics in the reduction of an arithmetic curve https://arxiv.org/abs/2005.11855 (with Karim J. Becher, preprint 2021).
WAŻNE! Osoby zainteresowane udziałem w wykładach obowiązuje PRZESŁANIE DOKUMENTÓW APLIKACYJNYCH.
Dokumenty, które należy wypełnić można pobrać ze strony internetowej: https://drive.google.com/drive/folders/1jdJA-yzeaXr9Ot-6UEFnw-9EQsbE_LwY?usp=sharing
Prosimy przesłać skany, bądź zdjęcia wszystkich wypełnionych dokumentów na adres: koordynatora projektu na kierunku Matematyka: jolanta.sobera@us.edu.pl w terminie do niedzieli 28 marca br.
W tytule wiadomości prosimy wpisać: “Wykładowca z zagranicy – Commutative algebra”
Ze względu na bieżącą sytuację, prosimy w Załączniku 1b miejsce na podpis i pieczęć pracownika dziekanatu pozostawić puste.
W załączniku 5b prosimy umieścić znak X tylko w wierszu: Zajęcia prowadzone przez wykładowców zagranicznych, a w kolumnie: SZCZEGÓŁOWA NAZWA WYBRANEGO WSPARCIA – wpisać: Commutative algebra.
Wykłady online będą się odbywać użyciem platformy MS Teams (szczegółowy harmonogram zostanie uzgodniony z zainteresowanymi osobami) :
Prerequisits
Some prior knowledge on rings and ideals is stongly recommended (for example principal ideal domains,
unique factorization domains, ideals, prime ideals and maximal ideals should be known concepts). The
aspects of field theory that enter this course will be discussed when needed (e.g. algebraical or transcendental
extensions). In particular, no knowledge of Galois theory is assumed.
Teaching
Due to the probable late start in the semester, the course is planned to happen in a compact form,
preferably In total we will have 30 teaching hours.
At least 15 hours will be direct teaching by the professor (wyk lad). The remaining 15 or less hours
will depend on choices of the students (see evaluation options). Some (or possibly all of them) will go
into presentations of some topics as seminar talks by students (konsultacje), and some will be reserved for
discussing handed in home works (for students who will chose this evaluation option) or to guide the student
in the preparation of the topic (konsultacje).
The course will be primarily based on the book by Atiyah-Macdonald (see bibliography), and here is a
list of topics that students may choose to present (if they opt for the seminar option):
Content
Commutative Algebra studies properties of distinct abstract types of rings that appear naturally in various
areas of mathematics, predominantly in number theory and algebraic geometry. In this course we will exhibit
some of those properties in the abstract context and also have a peak at examples from the aforementioned
less abstract areas.
We start by introducing basic concepts of ideals and modules over rings, such as radicals o ideales or
localizations and the tensor product. The concept of finitely generated modules will be applied to define
integral extensions and integral closures of domains, as well as extension behaviour of prime ideales under
integral extension. We then introduce the concept of Noetherian and Artinian properties both of rings and
modules, and see the prior as a natural abstraction of some essential properties of polynomial rings or more
generally coordinate rings of varieties. We also introduce the corresponding abstract notion of dimension
for such rings, and identify Artinian rings as the zero dimensional Noetherian rings. Then we focus on one
dimensional normal Notherian (local) domains, namely Dedekind (or discrete valuation) domains.
If time permits, we will introduce various different notions of dimension of (local) Noetherian rings which
are motivated by different settings of geometry, and we show the dimension theorem which basically states
that they all coincide. As a byproduct, we show Krull’s Height theorem (which shows that the dimension
of an algebraic set at a given point is equal to the minimal set of equations needed to define the set in a
neighbourhood), and we show equality of analytic and algebraic dimension at a smooth point of an algebraic
set defined over C via the "geometric case" of Cohen’s structure theorem.