SoSe 2017

Talk : Introduction to root systems, part III

This is the follow-up to my talk on root systems given in Klaus's seminar last semester (20.02.17), held on the 08.05.17, starting at 14:15. See webpage for more information.

Toric Varieties Seminar

Klaus Altmann's weekly seminar.
Mo 14:15-15:45, Arnimallee 3, Raum 119
See webpage for more information or potential modifications to the schedule.

WiSe 2016/17

Talk : Introduction to root systems in Lie algebras, part I & II

The talks I will give are to be held in Klaus Altmann's seminar on 13.02.017 and 20.02.17, starting at 14:15. See webpage for more information.

Toric Varieties Seminar

Klaus Altmann's weekly seminar.
Mo 14:15-15:45, Arnimallee 3, Raum 119
See webpage for more information or potential modifications to the schedule.

Curriculum Vitae

Available here.


No papers available yet.

WiSe 16/17

No courses taught this semester.

The following is my personal database of moderately to completely cleaned up sets of notes. If you wish to comment/contribute/notice errors, feel free to do so via e-mail. I will keep previous versions so that a reader used to a particular version (knowing some cross-references by heart or something like that) can still have access to them.

Commutative Algebra

Version 13.06.2017 available here.
Version 02.06.2017 available here.

This set of notes is meant to support my future set of notes on algebraic geometry. It is not complete (I don't think I'll ever consider it complete), so one should think of it as being a document which is "alive" in some sense. It currently discusses a wide variety of topics, such as : algebras, ideals, prime ideals, modules, tensor products, Hom functors, noetherian/artinian rings, the theory of associated primes, field theory (algebraic and transcendental), primary decomposition (I would be happy if someone knew a proof of the characterization of Lasker rings claimed in Atiyah-Macdonald! I couldn't complete it myself), graded rings and modules, dimension theory and the Nullstellensatz.
I would love to add a section on Dedekind domains and develop some of the standard algebraic number theory in there, as well as a section on homological algebra concerning Koszul complexes and so on. Sadly, this will have to wait!

Non-commutative Algebra

Version 02.06.2017 available here.

This document was the result of my read of Bourbaki's Chapter VIII on non-commutative algebra. I went through almost the entire chapter and added a lot of references to justify the arguments which were not always present in the original document (it links to my commutative algebra notes for some crucial steps, which are particular hard to follow using the cross-references in Bourbaki). A possible advantage for those who consider reading the original~: my notes are in English, whereas the original is in French.
I consider this set of notes complete for the moment, but it would be nice one day to know some Brauer theory and possibly non-commutative homological algebra. This will also have to wait!

Representation Theory of Finite Groups

Version 09.01.2017 available here.

This is my personal introduction to representation theory. Currently, the only completed section is the one concerning the standard properties of representations and characters over $\mathbb C$. I plan on adding a second chapter over arbitrary fields and another chapter dedicated to the fascinating theory of representations of the permutation groups $S_n$ somewhere in the future.

Algebraic Groups and Lie Algebras

Version 09.01.2017 available here. These notes are not complete.

This will become my personal introduction to algebraic groups. Since they are closely related to their Lie algebras which is the tangent space at the identity, I began by writing an extensive section concerning Lie algebras with a focus in characteristic zero since I am interested in the reductive case. The notes will be further developed in the near future.

Introduction to Representation Theory

BMS Friday "What is...?" Seminar
October 16th 2015

A "What is...?" seminar is a ~20-25-minutes talk that answers a concise mathematical question in a manner accessible to students and mathematicians not familiar with the field. Get to know the many different parts of mathematics in a relaxed and open environment. We offer you the opportunity to meet other graduate students from various fields while enhancing your general knowledge of mathematics. Our speakers are welcome to experiment in their talks and enjoy the friendly feedback from the audience.

This "What is...?" seminar is meant to be a rough introduction to representation theory to be able to follow Geordie Williamson's talk. I sketch the basic concepts : definition of a representation, Maschke's theorem, Schur's Lemma and character theory. Since Geordie will mention modular representation theory I will give an example in characteristic p, where Maschke's theorem fails.

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Patrick Da Silva

Freie Universität Berlin
Fachbereich Mathematik und Informatik
Mathematisches Institut
Arnimallee 3, Büro 113
14195 Berlin
E-mail address:
StackExchange profile: Patrick Da Silva

Welcome to my academic homepage. I am currently a Phase II PhD student at the Berlin Mathematical School (BMS) under the doctoral supervision of Prof. Dr. Klaus Altmann and working on toric varieties.

My mathematical interests are mainly focused on algebraic geometry, but I am also interested in related/unrelated fields such as geometric invariant theory, representation theory, algebraic and analytic number theory, differential geometry, and "on the week-ends", probability theory and functional analysis.