David E Speyer

This is my old webpage. The most current version is http://www-math.mit.edu/~speyer/. Please update your links accordingly.

Contact Information:


E-mail: speyer@post.harvard.edu or speyer@umich.edu
The former is a forwarding address that currently forwards to the latter. The former will always be correct, but has problems with large files. The latter is my current e-mail address at the University of Michigan. My Berkeley account should be going away any day now, so please do not send mail to that address.

Office: 1846 East Hall

Phone (Cell): (734)-255-8610
Phone (Office): (734)-936-4824

Mail (Work):
David E Speyer
Department of Mathematics
2074 East Hall
530 Church Street
Ann Arbor, MI
48109-1043 USA
Mail (Home):
David E Speyer
1312 Wright Street
Ann Arbor, MI
48105-1654 USA
Map


My Name:
My last name is pronounced "spire", like the top of a church. But I'll answer to "Sp-vowel-r!" or to "David!". For the curious, it is derived from the city of Speyer, Germany, although my friend Aaron tells me it could also be Yiddish for "pistol".

About Me:

I am a visiting scholar in the department of mathematics at the Univeristy of Michigan. I am funded by a Five Year Clay Research Fellowship.

In this term (Winter 2007), I will be teaching Combinatorics of Perfect Matchings.

I recently graduated from UC Berkeley, where I was a student of Bernd Sturmfels. My thesis is on tropical geometry, an approach to turning algebraic geometry problems into polyhedral geometry. Before coming to Berkeley, I was an undergraduate at Harvard. While there, I worked for Jim Propp's research group REACH and wrote an undergraduate thesis on the Eichler-Shimura correspondence under William Stein. I spent the rest of my time doing technical theater, hanging out with science fiction fans and working on Les Phys -- the physics musical! I spent four years as a counselor at PROMYS, a number theory program for high school students, and highly endorse it either as a place to go learn or to go work. I spent my own high school summers at MOP , which I found great but works better for some people than for others. I went to High School at Choate Rosemary Hall and to Middle School at Talcott Mountain Academy. If you are a young nerd in Connecticut, looking for a middle school or after school program, I highly recommend Talcott.

I enjoy algebraic problems with a combinatorial flavor. If you started out with a well motivated algebraic question but wound up with lots of little complicated pictures, I probably want to hear about it. Some topics I can usually be relied upon to think about: tropical geometry, cluster algebras, flag manifolds and other geometry of Lie Groups, interesting degenerations of algebraic structures, exact results and asymptopics of perfect matchings ( eg Arctic Circle phenomena). I also know a reasonable amount of Number Theory and enjoy talking about it, although as yet this is not a research interest.

If you want to know more about me, you can read my cv, see what I told the NSF I would work on or what I would have told them had there been no page limit. The second version is has some modifications reflecting later developments on these projects. Alternatively, you can see what you can figure out from my friends' websites.

I was part of a group of 60 or so people who wrote the 2006 MIT Mystery Hunt. All of our puzzles are now archived online, along with a web script that will check your answers and reasonably complete solutions. I wrote Connect Four (along with Noah Snyder), Tea for Two and Two for Tea (with Aaron Dinkin), Just a Jump to Left, The Scrambler, Wry, Ergo, Dead (with Noah Snyder), All Work and No Play Makes Jack a Dull Boy (with Noah Snyder) and Drop Everything. I also wrote the Moscow Meta (puzzle here, solution here), the Buenos Aires Ante and the McMurdo Station Ante. Ante puzzles are concealed among all of the other data for the round they are hidden in and their answers are the identity of a SPIES agent and the location of the next round. Our hunt was shorter than we expected, but most of the people I talked to seemed to have had a lot of fun solving it. And I have plenty of puzzle ideas left for next time (knock wood)...

Ezra Miller and I are organizing the Postdoc Algebraic Geometry seminar for Fall 2006. We are going to be learning Gelfand, Kapranov and Zelevinsky's material — discriminants, hypergeometric functions, secondary polytopes and so forth. We need lots of (primarily local) speakers, so please contact me if you would like to give a talk.

In my best news, I am now married to Lark-Aeryn Speyer, perviously known as Erin Ruth Larkspur, the lovely lady pictured above. We will be adding photos of the wedding to this page.

Papers:

All of my papers that are in a reasonably polished state can be found here on the arXiv . (With the exception of my first paper, "Every tree is 3-equitable" with Zsuzsanna Szansiszlo and my thesis, which will be broken into several papers for publication.) The arXiv version is often not completely final -- some papers have minor improvements in the published version.
Here is a complete annotated listing of my papers as of September 6, 2006.

  • Cambrian Fans with Nathan Reading

    Let W be a Coxeter group of finite type and c a Coxeter element. In a series of papers (see 1, 2 and 3), Reading introduced an equivalence relation on W called the c-Cambrian congruence whose equivalence classes are in natural bjection with clusters of the corresponding Cluster Algebra. Combining cones of the Coxeter fan corresponding to equivalent elements of W yields a coarsening of the Coxeter fan which we term the c-Cambrian fan. In this paper, we show that the c-Cambrian fan is a simplicial fan whose combinatorics matches the cluster complex. We dispose of almost all of the conjectures remaining from Reading's earlier papers and establish several connections between the Cambrian fan and Cluster Algebras — in particular, the g-vectors and quasi-Cartan companions occur naturally in the Cambrian setting. Our proofs depend on carefully checking the compatibility of a large number of bijections when the Coxeter element c is changed in a manner related to reversing a source in a quiver to a sink. Thankfully, now that these compatibilities have been checked, they will be available for future use.

    Nathan and I have been experimenting with enlarging the ideas of this paper to infinite Coxeter groups, and hope to write another paper on this topic.


  • A Matroid Invariant via the K-theory of the Grassmannian

    Let x be a point in the grassmannian G(d,n) and let T be the n-1 dimensional torus which acts on G(d,n). Take the closure of the T-orbit through x; the class of the structure sheaf of thish subvariety in the K-theory of G(d,n) depends only on the matroid of x. By some standard operations in K-theory, I associate a polynomial to x which behaves nicely under every standard matoid operation. Using this invariant, I prove the f-vector conjecture from Tropical Linear Spaces when all of the matrods involved are realizable in characteristic zero.

    I still don't have a great combinatorial interpretation of this polynomial -- it imposes very strong restrictions on decompositions of matroid polytopes into smaller matroid polytopes. If anyone recognizes what this guy is, please let me know!

  • A Kleiman-Beritini Theorem for sheaf tensor products with Ezra Miller

    Let X be a variety with a transitive action by an algebraic group G and let E and F be coherent sheaves on X. We prove that, for elements g in a dense open subset of G, the sheaf Tori(E, g F) vanishes for all i > 0. This says that, when performing intersections in K-theory, we may take a generic translate and then forget about higher Tor's. This is like the Kleiman-Bertini theorem, which says the same for intersections in Chow theory in characteristic zero.

  • Engagement Announcement with Erin Larkspur
    New Britain Herald December 3, 2005 p. C8

    Erin and I announce the beginning of a collaboration.

  • Cyclically Orientable Graphs

    In Cluster Algebras II, Fomin and Zelevinsky classified cluster algebras of finite type. Their classification did not yield an effective way of deterimning whether a given cluster algebra was of finite type. In Cluster Algebras of Finite Type and Symmetrizable Matrices, Barot, Geiss and Zelevinsky give an algorithm for performing this test, one step of which involves testing whether a graph is "cyclically orientable"; i.e., whether it has an orientation in which every cycle which occurs as an induced subgraph is cyclically oriented. In this paper, I give a simple and rapid algorithm for solving this graph theoretic problem and show that all cyclically orientable graphs are essentially built by gluing together cycles along single edges.

    Shortly after writing this, I learned that my main results had been obtained independently and several months earlier by Vladimir Gurvich of Rutgers, see his preprint Cyclically Orientable Graphs. With Gurvich's gracious agreement, I am posting my note so that people will be aware of the results; I completely acknowledge that he has several months of priority.

  • Computing Tropical Varieties with Tristram Bogart, Anders Jensen, Bernd Sturmfels and Rekha Thomas
    Accepted to appear in a special issue of the Journal for Symbolic Computation.

    We describe an algorithm for computing tropical varieties that is roughly a thousand times faster on high codimension examples than the naive approach via Groebner fans. There is some non-trivial math in the proof of correctness -- we show that the tropicalization of a prime variety is connected in codimension one. We have implemented our algorithm as an extension to Gfan; it is included with Gfan 0.2.

  • Tropical Geometry

    This is my dissertation, which attempts to do the ground work to establish tropicalization as a major tool of algebraic geometry. There are four major sections (plus a historical introduction.) The first section tries to develop general tools, including establishing the equivalence of several notions of tropicalization and describing the tropical degeneration and compactification -- these are schemes assosciated to a subvariety of a torus over a nonarchimedean field. The combinatorics of these schemes are indexed by a polyhedral complex whose underlying point set is the tropicalization. I have done more work here then is in the written version; I hope to eventually publish a paper on this subject, possibly in collaberation with Paul Hacking. The second section and third section respectively cover the material in my papers The Tropical Grassmannian and Tropical Linear Spaces below, rewritten to emphasize their connections to the other material of the dissertation. The final section studies the probleming of recognizing which graphs embedded in R^n occur as tropicalizations of curves embedded in the torus. It turns out that Mumford's techniques of nonarchimedean uniformization are admirably suited to this problem. The curve material, with a few technical hypotheses removed, will probably appear in a seperate publication.

    A few minor changes have been made to this file as compared to the original dissertation.

  • Tropical Linear Spaces

    I define tropical analogues of the notion of "linear space" and "Plucker coordinates" and basic constructions for working with them. This paper is an exhaustive introduction that tells almost everything I have figured out. The most interesting aspect of the paper is the f-vector conjecture -- I conjecture what the maximal possible f-vector of a tropical linear space should be and provide a great deal of evidence for this claim. Out of all the conjectures I have made, this is the one that frustrates me the most; I would really like to get it before I graduate.

    Although it can be read independently, this paper is naturally a sequel to my paper The Tropical Grassmannian below.

  • A Broken Circuit Ring with Nick Proudfoot
    Accepted, Beiträge zur Algebra und Geometrie


    Given a linear subspace of affine space, we study the ring of rational functions on the linear space generated by the reciprocals of the coordinate functions. This ring has been studied previously by Terao and others. We find a universal Groebner basis and show that the ring degenerates to the Stanley-Reisner ring of the broken circuit complex.

  • Tropical Mathematics with Bernd Sturmfels

    An elementary introduction to tropical mathematics, expanding on my co-author's Clay Public Lecture at Park City Math Institute 2004 (IAS/PCMI)

  • An arctic circle theorem for groves with Kyle Petersen
    Presented at Formal Power Series and Algebraic Combinatorics 2004.
    Journal of Combinatorial Theory: Series A 111 Issue 1 (2005), p. 137-164

    Proves that a randomly chosen grove (introduced in my paper with Gabriel Carroll below) is "frozen" outside a certain circle. This is analogous to results on random tilings of Aztec Diamonds and random Alternating Sign Matrices.

  • The tropical totally positive Grassmannian with Lauren Williams
    Journal of Algebraic Combinatorics 22 no. 2 (2005), p. 189-210

    We study the tropical analogue of the totally positive cell in the Grassmannian, introduced by Lusztig and studied in detail by Postnikov and others. We discover a tight connection to the combinatorics of cluster algebras and conjecture a general connection between the cluster complex of a cluster algebra and its totally positive tropicalization.

  • Horn's Problem, Vinnikov Curves and Hives
    Duke Journal of Mathematics 127 no. 3 (2005), p. 395-428

    Horn's Problem asks to characterize the possible eigenvalues of a triples of Hermitian matrices with sum 0. Allen Knutson and Terry Tao gave an answer in terms of combinatorial objects called honeycombs which look like tropical curves. I explain this phenomenon by showing that Horn's problem is equivalent to studying the possible intersections of plane curves with prescribed topology with the coordinate axes and then showing that the tropical version of this criterion recovers the results of Knutson and Tao.

    Note: The above linked paper does not fully spell out the link between honeycombs and eigenvalues; the chain of logic is as follows: by appendix I of the above linked paper, honeycombs are equivalent to Berenstein-Zelevinsky patterns, which compute tensor product multiplicities, which are related to eigenvalue computations by the Kirwan-Ness theorem. Allen Knutson has a good survey paper explaining the last step.

  • Reconstructing Trees from Subtree Weights with Lior Pachter
    Applied Mathematical Letters, 17 (2004), p. 615-621

    In computational phylogenetics, the problem of reconstructing a metric tree from the distances between its leaves frequently arises. We study the similar problem of reconstructing a tree from the total lengths of the subtrees spanned by k of its leaves.

  • The Tropical Grassmannian with Bernd Sturmfels
    Advances in Geometry, 4 (2004), no. 3, p. 389-411

    We study the tropicalization of the Grassmannian in its standard Plucker emebedding. We show that its points parameterize tropicalizations of linear spaces, give a complete description of the case of G(2,n) and do some computations of larger cases.

    I have done a good deal more work on the properties and classification of tropicalizations of linear spaces, see my paper Tropical Linear Spaces above.

  • The Cube Recurrence with Gabriel Carroll
    Electronic Journal of Combinatorics, 11 (2004) #R73

    This paper is similar to the octahedron recurrence paper below, but with applications to Propp's cube recurrence, a peculiar recurrence that has Laurentness and positivity properties similar to the octahedron recurrence but has no known relation to cluster algebras. The relevant combinatorial objects are no longer perfect matchings but "groves", certain highly symmetric forests that deserve further study. This paper was primarily written in Propp's REACH program.

  • Perfect Matchings and the Octahedron Recurrence
    Journal of Algebraic Combinatorics, to appear

    The octahedron recurrence is a certain recurrence whose entries are indexed by a three dimensional lattice; the recurrence grows from a two dimensional surface of initial conditions. It follows from Fomin and Zelevinski's results on Cluster Algebras that all of the terms of the recurrence are Laurent polynomials in the initial values. I show that every term in these polynomials has coefficient 1 by establishing a bijection between these monomials and the perfect matchings of certain graphs. Special cases include formulas for Somos-4 and Somos-5 and for the number of perfect matchings of many families of graphs. This paper is based on research done in Propp's REACH program.

  • Every tree is 3-equitable (link may require academic access) with Zsuzsanna Szansiszlo
    Discrete Mathematics, 220 (2000) 283-289

    Let G be a graph whose vertices are labelled with the numbers 0, 1, ... i. Label each edge with the absolute value of the difference between its endpoints. A labelling is called equitable if, for any two numbers a and b from 0 to i, the number of vertices with label a differs by at most one from the number with label b and a similar property holds for the number of edges with each label. It is conjectured that every tree has an equitable labelling for every i. We prove this conjecture for i=2.

    Software:


    I occasionally write Mathematica notebooks to aid in studying combinatorial problems. I ll use this section of my website to post notebooks that I think are of general interest and well enough commented to be useful to others.

  • SerParCore This note book lists the series-parallel matroids of given rank and number of elements. The number of such matroids is Sloane sequence A115594.

    I have some notebooks which compute random groves and enumerate groves in various settings. I'm not posting them right now because they are poorly commented and I think all of their functionality is subsumed by Gabriel Carroll's GrovePak software. If you think that you need something that GrovePak can't do, though, send me a note and I'll see if mine can.