How to contact me.
How to view the papers on this site.
Why the papers are long.
Software used for drawing commutative diagrams.
Software used for drawing graphics in geometry notes.
Mirrors of this website.
Book recommendations on algebraic combinatorics.
Book recommendations on elementary geometry.
Book recommendations on inequalities.
In September 2019, I am joining the Mathematics Department at Drexel University as an assistant professor. In 2016--2019, I have been a Dunham Jackson Assistant Professor at the University of Minnesota. I have defended my PhD at MIT, advised by Alexander Postnikov.
My work is in algebraic combinatorics and related subjects (symmetric functions, combinatorial Hopf algebras, Young tableaux, Lie algebras, invariant theory, etc.). For a quick glimpse at these areas, see, for instance, Bruce Sagan or Richard Stanley or Yufei Zhao or Jeremy Martin (sources I can all recommend).
More detailed information is found in my curriculum vitae, my research statement (longer version that goes into details on some finished projects), and my teaching statement. (They might not be 100% up-to-date.)
Before my academic life, I used to participate in high-school mathematical contests such as the IMO and various German competitions. In that time, I have written several notes and papers on contest-level elementary mathematics, particularly elementary geometry and algebraic inequalities; these too can be found on this website.
How to contact me
The best way to get in touch with me is via my email address: X@gmail.com, where X = darijgrinberg.
My old email addresses at MIT (Y@mit.edu and Y@math.mit.edu, where Y = darij) are no longer active since I have defended my thesis.
You can usually find me on MathOverflow and math.stackexchange, two forums I have spent much time in (and learned a lot from). Before that, I used to frequent Art of Problem Solving (particularly the Highschool and College parts); these days I only visit it on rare occasions (sorry).
I am not on facebook or other social networks.
How to view the papers on this site
The following instructions are written for Microsoft Windows users (specifically, Windows 7 and 10). Most Linux distributions are already capable of viewing these files "out of the box", and I have no experience with OS X.
PS files: Some (older) papers are available in PS (= PostScript) format only. (Those were written back when making a PDF was not as easy as it is now.) You can view PS files under Windows by installing the following two programs (in this order):
You won't ever have to run Ghostscript (it just needs to be
will open PS files in either GSView or SumatraPDF.
TEX files: A TEX file is a sourcecode for a paper. They are what the PDFs on this site are generated from. If you want (for some reason) to make some changes (e.g., alter the page layout) and regenerate the PDF file, you will need the LaTeX compiler and the packages that I have been using. These days you can get both by installing TeX Live.
Every file which
1. is hosted on this website -- i.e., its URL begins with "http://www.cip.ifi.lmu.de/~grinberg/" --, and
2. is not part of the teaching archive -- i.e., its URL does not begin with "http://www.cip.ifi.lmu.de/~grinberg/t/", and
3. is hyperlinked from an .html (or .htm) file on this website, and
4. the word "[Prop]" does not appear immediately in front of the hyperlink that links to it
is public domain (see the Creative Commons CC0 page for the precise meaning of this), i. e. you can use it in whatever way you want without asking me for it.
As for files that are part of the teaching archive, they are also public domain if I am their author.
[If you are wondering: The reason for condition 3. above is to exclude private files which are (usually temporarily) hosted on my website. Condition 4. is mainly meant to deal with joint papers.]
My posts on the MathLinks forum (a.k.a. Art of Problem Solving), the Hyacinthos newsgroup and in other internet resources are public domain as well,
with the exception of:
- quoted text (text inside "Quote" tags);
- attachments to posts (some of these are public domain, some aren't);
- posts on MathLinks beginning with the sentence "The author of this posting is:" (in fact, these are posts made by others which were lost during a hacker attack on MathLinks, and which I reposted because I happened to have backups of them).
Why the papers are long
Many (not all) of my writings are unusually detailed and, as a consequence, unusually long. This is a habit I picked up long ago; it has saved me from making numerous mistakes that would have otherwise slipped in through "obvious" statements and unwritten arguments "left to the reader". Furthermore, (I hope) the detailed proofs make the papers more accessible to inexperienced readers.
Unfortunately, there is a downside -- some of my proofs have so much detail that they take longer to read than to find. I would suggest treating theorems as exercises when reading -- i.e., first try proving it on your own, then skim the proof, then read it if the skim was not enough.
Software using for
drawing commutative diagrams
I just use the (rather standard) Xy-pic package. See a short introduction on wikibooks and J. S. Milne's tutorial.
If you find its syntax somewhat too spartan, you can try also try out Michael Barr's diagxy package, which provides a kind of UI for the \xymatrix command.
See also J. S. Milne's page for a list of other diagram packages.
Software used for drawing
graphics in geometry notes
I have been asked this several times. Unfortunately, I don't have a useful answer -- first, I haven't been writing about geometry for years now; second, the procedure I have been using was an unreliable hack that involved several pieces of fickle software. Sorry.
From what I have heard, Asymptote is a good tool for this; but I have no first-hand experience with it.
Mirrors of this website
This site is currently hosted on http://www.cip.ifi.lmu.de/~grinberg/ and mirrored on https://darijgrinberg.gitlab.io/.
(Formerly, this site also used to be hosted on http://de.geocities.com/darij_grinberg/ and on http://mit.edu/~darij/www/ .)
I also have a teaching website at http://www.math.umn.edu/~dgrinber/, but it will likely disappear soon as I am no longer at the UMN. You can find my teaching materials on the present website.
Book recommendations on algebraic combinatorics
I have listed some introductory sources in a math.stackexchange answer (which includes both books and online sources). An encyclopedic treatment of a major part of the subject is Richard Stanley's Enumerative Combinatorics, whose first tome is available for free from his website.
The OEIS is a wonderful electronic resource for integer sequences, useful not only in combinatorics.
Pascal's triangle from top to bottom has been a useful source for binomial coefficient identities (it is now hosted on archive.org's Wayback Machine). See the identities page in particular.
QEDMO stands for "QED Mathematical Olympiad". The QED is a society of high-school and undergraduate students (mostly) in southern Germany, organizing meetings that involve mathematical talks (as well as games and excursions). From 2005 on, some of these meetings have featured a math fight (a contest with two teams solving problems and debating the solutions in front of the blackboard) called the QEDMO. The difficulty of the problems ranges from school-level to hard IMO problems. I have taken part in the organization of some of the early QEDMOs, and created many of the problems.
Unfortunately, the QEDMO has no official site and archive. The problem statements of the first 5 QEDMOs can be downloaded. Some solutions appear on my solutions page, and others (in German) on my German page. Some problems appear on the AoPS forum.
Book recommendations on elementary geometry
This is from 2006; new things will have probably appeared by now, and some links are probably out of date.
See the links section of this website for a collection of online resources (including lecture notes and ebooks). In particular, there is a lot to be learnt from the Cut-the-Knot website and from Kedlaya's Geometry Unbound. Prasolov's Problems in Plane Geometry (translated by Leites) are invaluable to a problem solver who wants to get better in geometry.
As for books, my first choice would be
H. S. M. Coxeter, Samuel L. Greitzer, Geometry Revisited
(this one has been printed several times, and as far as I know, the editions only differ in the solutions to the exercises -- the newer ones have partly better solutions). The first three chapters of this book contain some real common knowledge on triangle geometry and radical axes theory. However, the "introduction into projective geometry" chapter is anything but an introduction into projective geometry, and the way inversion is introduced is far from complete.
The next textbook is
Nathan Altshiller-Court, College Geometry.
This has been reprinted in 2007 after having been elusive for a long time. It contains nearly all plane geometry you might need on contests.
Along with this book,
Roger A. Johnson, Advanced Euclidean Geometry
has also been reprinted. While I would not recommend this for olympiad training, it can be helpful for a more serious study since it contains a number of lesser known results.
Then, there is
Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry.
This one is really fun to read if you are into elementary geometry. Much of the content can help you in mathematical olympiads as well - though, for an IMO gold medaillist with focus on geometry, this one will be rather like a collection of simple exercises (which can still be of use).
I have now put up a page for errata in these three books (some other books may also be included).
Unfortunately, this is pretty much all books I can recommend - apart from the (also rather few) German and Russian ones I won't itemize.
Book recommendations on inequalities
I am afraid what follows is mostly of historical significance, as this subject is rather different now from what it was back when I wrote this.
Here is some random stuff I found useful. In fact, I have never learnt inequalities systematically; most of my basic knowledge comes from the German IMO training, and the rest is experience from solving MathLinks problems and reading others' solutions.
Thomas Mildorf has written nice notes on inequalities (2006).
Hojoo Lee is rather known for his "Topics in Inequalities".
Kiran Kedlaya has another text similar to the two above.
This MathLinks topic is partly of interest.
If you read German, Robert Geretschläger has his notes as well.
Now to actual books. There are several books by the (Romanian) GIL publishing house on olympiad-style inequalities. Two of them are in English. Please don't ask me how to order books published by GIL from outside Romania - apparently this has not been organized smoothly yet :( . Let me mention two books:
Vasile Cîrtoaje, Algebraic Inequalities - Old and New Methods, Gil: Zalau 2006.
This one is 480 pages long and features many interesting tactics and examples on solving inequalities. Unfortunately, you are not likely to enjoy all these 480 pages, because many of the modern methods for solving inequalities include applications of calculus and involved computations. However, a lot was done to keep these ugly parts at a minimum while keeping the whole power of the new methods.
The RCF ("right convex function"), LCF (guess what this means) and EV (equal variables) theorems as well as the AC (arithmetic compensation) and GC (geometric compensation) methods provide a means to solve >95% of olympiad inequalities using rather straightforward - not nice, but doable - computations. All of these methods are extensively presented with numerous examples. A short chapter underlines applications of the (underrated) generalized Popoviciu inequality. Finally, and - in my opinion - most importantly, a lot of exercises with solutions are given which don't require any strong new methods, but just creative ideas and clever manipulations.
Pham Kim Hung, Secrets in Inequalities (volume 1), Gil: Zalau 2007.
This one has 256 pages, and is remarkable for mostly avoiding computations. Numerous creative ideas can be found here - I was particularly surprised about some of the applications of the Chebyshev and rearrangement inequalities. Besides, a good introduction into the applications of convexity is given. I would recommend this book to olympiad participants who look for challenging problems and intelligent techniques without the aim to be able to kill every inequality.
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