Jan Draisma
Professor of Mathematics
Contact
Publications
Programs
Talks
Teaching
Organisational
Recreational Maths
Links
News
Spring 2023: I'm organising the SMG Spring Meeting. Please
attend!
Fall 2022: I'll be on sabbatical at the IAS as a participant in the
Special
Year on Dynamics, Additive Number Theory and Algebraic Geometry.
January 2022: I have succeeded Bernd Sturmfels as Editor-in-Chief of
the SIAM Journal on
Applied Algebra and Geometry.
Dec 2020: my former Ph.D. student Arthur Bik as won an SNF
Postdoc.Mobility grant, with which he will spend a year at
MPI MIS (Leipzig) and a year at Texas A&M (College
Station). Congrats to Arthur!
Nov 2020: my Ph.D. student Alejandro Vargas won an SNF Early
Postdoc.Mobility grant, with which he will spend time in Frankfurt
and Nantes. Congrats to Alejandro!
Feb 2020:
We have created a movie visualising parts of
my joint paper with Alejandro
Vargas on the gonality of metric graphs.
Feb 2020: Shreehari Bodas, a Bachelor's student from IIT Bombay, has
successfully applied for a ThinkSwiss
2020 Research Scholarship to visit my group for two months this
summer---unfortunately, due to the Covid-19 pandemic, his visit did
not take place.
Apr 2019: The current,
online version of the programme for SIAM AG 19 is online.
Mar 2019: My paper on
topological Noetherianity of polynomial functors was accepted for
publication in the Journal of the AMS.
July 2018: My former student Rob
Eggermont has won a Veni grant.
June 2018: The website for
MEGA 2019 is up and running.
Apr 2018: The website for
SIAM AG 19 is up and running.
Dec 2017: My student Arthur Bik won the
DIAMANT prize for best Ph.D. talk at the DIAMANT symposium;
see this picture.
Nov 2017: MEGA (Effective Methods in Algebraic Geometry) now has a
permanent
website collecting information about past and future MEGAs!
May 2017: Christandl, Jensen, and Zuiddam give very simple and
elegant examples of strict submultiplicativity of tensor rank under
the tensor product in this
preprint!
Jan 2017: Paul Breiding has succeeded at evaluating our formula for the average ED degree of the cone over Veronese embeddings. See Paul's impressive preprint and our paper for the formula.
Jan 2017: Andries Brouwer and Dan Christensen refuted the Gale-Neyman
conjecture that in chomp on the n-dimensional hypercube {0,1}^n with poisoned
block at the origin, the all-one vector is a winning first move. They show this is not
true for n=7. See this page.
Aug 2016: I have moved to the Bern math institute, succeeding Christine Riedtmann.
Mar 2016: Anna Seigal
wrote a blog
about the pictures on the cover of our new SIAM journal.
Feb 2016: My Vici proposal entitled Stabilisation in Algebra and
Geometry was successful!
Feb 2016: The website for the journal is up, and paper submission will open on 23 March.
Jan 2016: Here is the cover art for the new SIAM Journal on Applied Algebra and Geometry---the website will be up soon!
Dec 2015: Our proposal for a new SIAM Journal on Applied Algebra and
Geometry was approved by SIAM. We aim to attract the best papers in this
broad area. Submission will open soon, but please contact me if you have
fantastic work that could fit!
June 2015: the Mittag-Leffler proposal by Anders Jensen, Hannah Markwig, Benjamin Nill, and myself for a semester on Tropical geometry, amoebas, and polytopes was accepted for Spring 2018!
May 2015: Rob Eggermont got his Ph.D. with the distinction cum laude; congratulations! See this news item.
April 2015: My article on Plücker varieties with Rob Eggermont was accepted by Crelle's journal.
April 2015: I have joined the editorial boards of Linear and Multilinear Algebra, Nieuw Archief voor Wiskunde and, last year, of Experimental Mathematics. Where appropriate, please consider submitting your work to one of these journals.
March 2015: I wrote a SIAM News article loosely based on lectures by Peter Bürgisser and others during the Simons Institute semester on Algorithms and Complexity in Algebraic Geometry.
January 2015: I was appointed parttime full professor at the VU Amsterdam; see
this news item.
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.
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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.
Clicking on the picture below opens 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. S/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?
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