Jan Draisma

Associate Professor

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October 2014: Rekha Thomas wrote a beautiful article for SIAM News on the Euclidean distance degree.

Fall 2014: The MEGA 2015 website is in place, and submission of papers, extended abstracts, computations, and posters is open!

October 2014: both my last-year's Master students have started Ph.D.'s: Jasmijn Baaijens in the CWI life sciences group, under the supervision of Alexander Schönhuth; and Guus Bollen in Discrete Maths here at TU/e, under the supervision of Hans Cuypers, Rudi Pendavingh, and myself. The best of luck to both!

September 2014: The conference website for the SIAM Conference on Applied Algebraic Geometry is up and running. Check it out, and consider organising a minisymposium! The conference has also been accepted as a ICIAM 2015 satellite.

July 2014: my Department nominated me for a TU/e Education Award, as their candidate in the category Best Master's Program Lecturer.

Spring 2014: three members of the SIAM (AG)^2 were elected SIAM fellows: Jean Lasserre, Peter Olver, and Bernd Sturmfels. Congratulations!

24 February 2014: I wrote a guest post on Rota's basis conjecture for The Matroid Union.

9 January 2014: I won the GEWIS teaching award 2013/2014 for Mathematics. Needless to say, I'm very, very proud of this!

December 2013: I was elected Chair of the SIAM activity group on Algebraic Geometry. This group brings together researchers who use algebraic geometry in industrial and applied mathematics. If you are an algebraic geometer interested in applications, or if you have a maths/statistics/engineering/CS/... problem that you think might benefit from algebraic techniques, please check out this activity group (or contact me). It's this interplay that makes the group such a success!

24 October 2013: my paper with Eggermont on the existence of poly-time membership tests for a wide class of phylogenetic models has just been accepted for J. Eur. Math. Soc.

1 May 2013: my paper with Kuhnt and Zwiernik on groups acting on Gaussian graphical models has just been accepted for Annals of Statistics.

Spring 2013: Robert Krone from GeorgiaTech is visiting for three months. Together with Anton Leykin and Rob Eggermont we aim to prove finiteness-up-to-symmetry results for certain infinite-dimensional toric varieties.

15 October 2012: the website of the CIME/CIRM course Combinatorial Algebraic Geometry, taking place from 10-15 June 2013 in Levico Terme, is up. Check it out!

1 September 2012: Emil Horobeţ from Babes-Bolyai university joins the group on the project Tensors of Bounded Rank.

23 January 2012: Tensors of Bounded Rank, an NWO free competition Ph.D. project proposal together with Monique Laurent and Siep Weiland has been awarded funding!

23 January 2012: Piotr Zwiernik is starting a Post-doc in the Vidi project.

1 September 2011: Rob Eggermont started his Ph.D. in the Vidi project.

2 March 2011: a manuscript with Johan P. de Jong on his Bachelor's project has been accepted for publication in the EMS Newsletter. See this page.

6 October 2010: A Vidi grant! See this page or this page or this page.


































Recreational Mathematics

The Casas-Alvero conjecture

The Casas-Alvero conjecture states that a univariate, complex polynomial f of positive degree n that has a non-trivial gcd (i.e., a root in common) with each of its derivatives f', f'', ..., f(n-1) must be of the form a(x-b)n. The conjecture is known to hold if n=q pe with q in {1,2,3,4} and p a prime larger than q, except possibly for q=4 and p=7. For q=1,2 this is proved here, and for q=3,4 the statement can be proved in a similar fashion. Below is an applet, written by my Bachelor student Johan P. de Jong, with which you can get a feeling for why the conjecture might be true---if you try to make roots of f coincide with roots of its derivatives, then the roots seem to have the tendency to "collapse" into a single point. Left-clicking adds a root or increases the multiplicity of an existing root, right-clicking deletes a root or decreases the multiplicity of an existing root, and the roots of derivatives of f are depicted in ever lighter shades of grey. The applet uses Michael Thomas Flanagan's Java library for complex polynomials.

Tree chomp

Chomp is a game that can be played on any partially ordered set with a a smallest element. Two players alternatingly remove an element of the set, together with all larger elements. The player who is forced to remove the minimum loses the game. Classical chomp is played on a chocolate bar, and very hard to analyse. See this page for mathematics and links and an applet. Another special of chomp case is graph chomp. A yet more special case, which should be analysable by good high school students, is tree chomp, for which I wrote an applet once. He who takes the last vertex wins the game; can you find a strategy?

A poem about mathematics

Arrows

A mathematician's most common mistake
takes products to sums, and sub-things to quotients;
semisimplicity can only fake
the lurking, neglected emotions.
But don't despair!
A one-way road back is there, though narrow:
immediate re-reversal of the arrow.

A three-gap proof??

Every once in a while you read a mathematical statement that you just have to try and prove yourself. This happened to me when browsing through the latest issue of Nieuw archief voor de wiskunde. In this article, in which Krzysztof Apt passionately explains why everyone should post their papers on the arxiv, he also recalls the "Three-Gap Theorem", which states the following: let b be a positive real number, let a be a non-negative real number smaller than b, and let n be a non-negative integer. Remove from the circle R/bZ (R=real numbers, Z=integers) the points (cosets of) ka for k=0,...,n. Then the lengths of the connected components of what remains attain at most three distinct values. Put more operationally: stepping around a circle of length b with constant step size a, and deleting each point where you step, you cut the circle into smaller and smaller pieces. The theorem says that, after every step, the pieces take only three different lengths.

Very unscientifically, I haven't done any research as to whether this is a famous theorem---I just had to think it over myself. Would you say that what follows is a proof?

Let L(n,a,b) be the set of lengths. So for instance L(0,a,b)=L(n,0,b)={b} and if a is in the open interval (0,b) we have L(1,a,b)={a,b-a}. For fixed positive n, let P(n) be the statement that the union of L(n,a,b) and L(n-1,a,b) has cardinality at most 3 for all a,b. Yes, we are going to prove something stronger! We will prove P(n) by induction on n; suppose that P(m) holds for all m smaller than n, and fix a,b as above. If a=0 then L(n,a,b)=0 and we are done. So we may assume that a is strictly between 0 and b. Write b=q*a+r with q a positive integer and r in the half-open interval (0,a], and let s:=a-r, in the half-open interval [0,a). Then if n=1,..,q we have L(n,a,b)={a,b-na} and, indeed, the union of L(n,a,b) and L(n-1,a,b) has cardinality at most three.

So suppose that n>q. Then I claim that this union equals the union of L(m,s,a) and L(m-1,s,a) for some m at least 1 and at most n-q. Since P(m) holds by assumption, we are then done. To get a feeling why the claim is true, look at the situation after q steps: the circle of length b has been cut into q pieces of length a and an additional piece of length r. Now concentrate first on those steps among the ones that follow that hit the first piece of length a; roughly 1 in every q steps does so, and it hits that piece at the positions s, 2s, 3s, ... (all modulo a). So in fact you are stepping along a circle of length a with constant step size s.

But what about the remaining pieces of size a? They are all, one-by-one, following the pattern in the first piece of size a, until all have been "updated" and you step back onto the first---or, indeed, onto the piece of size r; what's with that? Well, it's easy to see that that piece behaves exactly like the r-tail of the a-pieces: when it is hit, the a-pieces are consecutively hit at the corresponding spot in their r-tail. In rounds when the r-bit is not hit, the a-pieces are consecutively hit in their s-heads.

We conclude that after every step hitting the q-th a-piece the set of lengths is exactly L(m,s,a) for some m. One step earlier it is L(m-1,s,a) union L(m,s,a) because the q-th a-piece has not yet been updated, and one step later L(m,s,a) union L(m+1,s,a), etc. This proves the claim, and hence the theorem.

This process can be formalised, of course. But perhaps more interesting is: are there higher-dimensional analogues on tori or other compact Lie groups? Several years after I put this item online, a version of this on tori was proved by Henk Don; see this paper.

A more efficient way to pack (a kind of) Tangram

You probably know the classical puzzle of tangram. Somehow we obtained a variant of the game, containing two more small triangles; see the pictures below. It came packed in a square box, assembled as on the left. Note that if the short side of the smallest triangle is 1, then the sides of the square filled by the puzzle, after moving the pieces tightly together, have length 3, so that the square has area 9. But my father-in-law discovered that this is not quite the most efficient way to pack the puzzle. Indeed, consider the rectangle on the right. Its horizontal side has length 2, and its vertical side has length 1+sqrt(2)+2 (the lengths of the pieces on the left border, from top to bottom), which is approximately 4.414. So the area of this rectangle is approximately 2*4.414=8.83, which is smaller than 9! Funny, right? What's wrong here?