Here is a complete annotated listing of my papers as of November 15, 2023.

We prove that the biclosed sets in affine weak order form a complete lattice, proving a conjecture of Dyer. We provide a strategy for reducing the question to computations in rank three subgroups, which could hypothetically work in wild type as well. Interestingly, our previous work classifying biclosed sets in affine type is only used mildly, and if we had been willing to use more brute force, we could have side stepped it all together.

Nathan Reading introduced shards as cones in finite Coxeter hyperplane arrangements which describe the quotients of the corresponding weak orders. Weak orders on Coxeter groups are closely related to the representation theory of preprojective algebras. In the finite simply laced types, we understand the connections between shards and preprojective algebras very well. In this paper, we do the corresponding work for (crystallographic) root systems which are not necessarily simply laced or finite. We also give counterexamples to several plausible conjectures.

A survey prepared for the Handbook of Combinatorial Algebraic Geometry. We describe Richardson and projected Richardson varieties in type A, including (1) results on coordinate rings, vanishing of cohomology, standard monomial theory and Frobenius splitting (2) Grobner theory, toric and Stanley-Reisner degenerations (3) parametrizations using Bott-Samelson varieties (4) decomposition into Deodhar strata and (5) total positivity. We also provide an overview of Postnikov's theory of plabic graphs and how it connects to the last three topics. There are detailed examples throughout. I am very grateful to everyone who has given me feedback on this, and am glad to get more.

This paper resolves issues that have been open in the field of cluster algebras for a long time: We construct cluster structures on open Richardson varieties, and more generally braid varieties, in type A, and prove that they are locally acyclic. Moreover, we give a combinatorial model for these cluster varieties based on a non-planar version of Postnikov's plabic graphs. (My co-authors rejected my suggestion that we call them bic graphs.) There is a sequel to this paper, by the other three authors, which does the other Lie types. This paper is strongly indebted to my student Gracie Ingermanson whose Ph. D. dissertation constructed a cluster structure on open Richardson varieties.

The first of what will be two papers with Grant Barkley on Dyer's conjectures on biclosed sets of roots. This one classifies the biclosed sets in affine type. This work started as an REU project!

My second paper with Thomas on cohomology of cluster varieties! In this case, we build an explicit complex that computes the cohomology for the acyclic case, and discover that the cohomology lives in a surprisingly narrow range of weights.

We give a combinatorial description of the highest degree part of a Grothendieck polynomial. This computes the Castelnuovo-Mumford regularity of a matrix Schubert variety, answering a question of Jenna Rajchgot, and we discover a lot of surprising combinatorics as a result.

Thomas Lam developed a theory of total positivity for electrical networks. We explain how it fits into the usual Lie theoretic story of total positivity, making a connection to the Lagrangian (Type C) Grassmannian.

Another paper on matroids and tropical geometry. My co-author figured out how to compute matroid subdivisions rapidly. This lets us work out a lot of examples, including breaking some plausible conjectures.

Matroids are really hard, but positroids are easy! This is yet another paper establishing this pattern -- everything that looks like it is a tropicalization of a point on the positive Grassmannian actually is one.

Nathan and my work on Cambrian lattices became stuck because we didn't have good ways to talk about lattice structure in infinite Coxeter groups. We now think that we might be able to get past this. In order to do so, we needed to learn how to think about semidistributive lattices. This paper explains how to make a general semidistributive lattice look like both (a) the downsets of a poset and (b) the torsion clases of an abelian category. We also connect to structural results for many other classes of lattices.

This is one of two papers proving conjectures of Stanley, inspired by his talk at Sergey Fomin's birthday conference. In this one, Stanley defines an array of numbers: The first few rows are (1), (1,1,1), (1,2,1,2,1), (1,3,2,3,1,3,2,3,1), with each row obtained by the previous one by inserting, between every two numbers, the sum of those numbers. Stanley shows that the sum of the r-th powers in each row obeys a linear recursion, but the actual linear recursions which are obeyed are much shorter than the ones he proof produces. I show that this follows by thinking systematically about the representation of SL

This is the other paper proving a conjecture of Stanley. Let W(l,n) be the vector space with basis the permutations of n elements of length l. Stanley defines matrices mapping W(l,n) to W(l+1,n) whose nonzero elements are in positions (u,v) where v covers u in weak orer, and conjectures that the composite map W(l,n) to W(n(n-1)/2-l,n) are invertible with an explicit determinant. We prove this conjecture by finding an underlying SL

Since the symmetric group S

This project started in 2009 when Noah Snyder and I were discussing certain tensor categories (blogpost 1, blogpost 2) on the Secret Blogging Seminar. Sami and I started working together at MIT a little later (when I asked these questions (MO 1, MO 2)), and our UROP student, John Schneider, computed a lot of the polynomials. In 2013, my student John Wiltshire Gordon started computing projective resolutions of the simple representations of the category of finite sets and noticed the same polynomials occuring. In 2015, Rose Orellana and Mike Zabrocki (paper 1, paper 2) started thinking about the same problem. So Sami and I have finally gotten our act together and written the paper. Thanks to everyone we talked with!

As an example of what this paper does, consider the adjacency matrix of the complete graph. Its eigenvalues are -1 and n-1; in particular, the depend algebraically on n. We formulate a precise statement which roughly says that constructions which are "uniform in n" lead to matrices with algebraic dependence on n.

We answer some commutative algebra questions which have bothered me for a long time, regarding why the cohomology class used by June Huh and Eric Katz is so similar to the K-theory class in my paper with Nick Proudfoot. I viewed this as a warm up to study some analogous questions relevant to my work with Alex Fink, but we did not make progress on the harder questions. This paper was developed in a working group at the Fields Institute Special Semester on Combinatoral Algebraic Geometry; thank you very much to the Fields Institute and to all the participants in the working group.

Suppose you have an algebraic variety equipped with a Frobenius splitting and you want to list all the compatibly split subvarieties. I show that, with little harm, you can normalize the ambient variety first. This is convenient, because we often want to talk about divisors and valuations in this context. This is a question Allen Knutson asked me ages ago and I finally got around to publishing it.

I prove a bunch of conjectures of Dinesh Thakur. As an example of the sort of result, if you sum up 1/(P+1), where P runs over all irreducible polynomial in F

My first extremal combinatorics paper! This is one of the papers pursuing the exciting new tools for bounding the size of sets in (Z/p)

Amer. Math. Monthly

Let G be a finite group and let n divide the order of G. Frobenius showed that the number of solutions to g

We fill in the last missing gap in Postnikov's positroid varieties paper -- discovering the analogue of the chamber ansatz and thus showing how to invert the boundary measurement map. This has a number of important consequences. We show that the image of the boundary measure map is open immersed torus, and that the domain where any set of Plucker cluster variables is nonzero is another torus — and they are not the same one! This involves lots of pretty combinatorics. We tried to write this to be a helpful reference for people who have ben confused by how key results are spread across the literature, and I have had a number of grad students tell me they found this paper a great place to start learning the positroid story. Give it a try!

We work to describe the mixed Hodge structure on cluster varieties. The main results (with some hypotheses omitted): The mixed Hodge structure is entirely of Tate type and split over the rationals, and we can prove the "curious Lefschetz property". There will be at least one major sequel to this, with explicit combinatorial rules in the acyclic setting.

I provide a quick proof of Kasteleyn's theorem that the adjacency matrix of a planar graph can be decorated with signs so that its determinant computes the number of perfect matchings of the graph. Then I show how this result and several related ones are key to Postnikov's parametrization of positroid varieties.

Until we develop something new, this is the last of Nathan and my papers on frameworks. We explain how to merge the theory of affine frameworks from our previous paper, and the theory of sortable elements for quivers with cycles that we developed several years prior. It was one of the real pleasures of this project to see our old definitions prove right. Sadly, I currently think we have reached the limits of these methods.

We explain how to take a degree

Nathan and I explain how our theory works in the affine acyclic case. It's very pretty!

My first physics paper! We rewrite some old computations in scattering theory and show to use the same method for some new ones. I much admit that there are parts of this paper I understand very well and other parts which I understand very little.

We show that, if we fix a weakly separated collection B, the set of all maximal weakly separated collections containing B is connected under mutation operations. This is a start to the task of understanding the topology of the simplicial complex of weakly separated collections, a task which has also been tackled by Hess and Hirsch. By using the technology from our earlier paper with Alex Postnikov, the proof becomes incredibly short; in my opinion, even in the case where B is empty, which was done by Postnikov, our argument is much simpler.

We show that cluster algebras of positroid varieties are locally acyclic, a condition introduced by Greg which makes them much easier to analyze. At the time we wrote the paper, it was not quite know that positroid varieties had cluster structure, but that has since been remedied by Leclerc.

As the title says, I construct an infinitely generated upper cluster algebra, answering a question left open in

Given a point z on the projective line, the osculating flag at z is spanned succesively by (1,z,z

We develop tools to describe higher dimensional Dressians (a combinatorial object a bit larger than the tropical Grassmannian), and in particular to describe their rays. In particular, we describe a construction for turning subdivisions of a product of two simplices into points of the Dressian. This bridges the gap between the setup in the "tropical convexity" and "tropical rank of a matrix" papers and in the "tropical Grassmannian" papers.

My involvement with this paper is a little odd; there was already a version written and on the arXiv with the other two authors, and I pointed out some improvements that lead to me being added as a co-author. As a result, I understand sections 2, 3 and 5 very well, and know very little about the computational details in section 6.

In

In my frameworks paper with Nathan Reading, we explain that the best combinatorial data with which to describe a cluster algebra is to assign an

We study Postnikov's stratification of the totally nonnegative Grassmannian from the persepcitve of algebraic geometry. We classify singularities, compute cohomology classes, give equations and describe Groebner degenerations. There are connections to the mathematics of juggling and to Stanley symmetric functions.

This material earlier appeared in our 2009 preprint, together with some lengthy combinatorial arguments and a number of results on projected Richardson varieties. The former are removed and replaced by better proofs; the latter now appear in a separate paper.

The next paper in Nathan and my series on describing combinatorics of cluster algebras in terms of Coxeter combinatorics. This is the big one — we finally prove that the combiantorial objects we constructed do match of with the cluster algebra. We also spend a lot of time building general technology to describe combinatorial models for cluster algebras; we hope this will streamline future papers both by us and by others.

The next paper will be on some special combinatorial constructions that occur in affine type.

We explain how, given a maximal weakly separated collection, to construct a plabic graph. (And thus to construct an alternating strand diagram, and any of the other objects from Postnikov's massive work.) This closes a gap in the description of the cluster theory of the Grassmannian which has been open since Josh Scott's thesis 1 2 ten years ago. In particular, we now know that every cluster consisting of Plucker coordinates comes from the plabic graph technology; and we have now proven Scott's purity and connectedness conjectures.

This paper has been a long time in production, and incorporates discussions I've had with Andre Henriques and Dylan Thurston. It also has heavy overlap with work of Danilov, Karzanov and Koshevoy; see our paper for a discussion of how our work is related to theirs.

This paper takes those results from "Positroid Varieties I" which are true in general type and gives uniform, cleaner proofs of them. There is a second paper which focuses on the combinatorial aspects which are unique to Grassmannians.

Expository pieces on approximation, written for high school or advanced middle school students.

In my previous paper "A Matroid Invariant

Some cute problems about affine reflection groups, motivated by work of Qëndrim on the Kottwitz-Rappaport conjecture.

We explain how to extend our Cambrian technology to the case of cycles, which is important for many applications to cluster algebras. While the definitions seem too simple to work, they do, due to some nontrivial results about Bruhat order.

We have split this paper into two papers 1 2, with better results and proofs in both. You probably don't want to read this one.

One of the most classical topics in combinatorial commutative algebra is the standard monomial basis for the flag variety. This is a basis for the Plucker algebra indexed by semi-standard Young tableaux, useful for computations in representation theory and algebraic geometry. Pavlo introduced objects he calls non-nesting tableaux which are a "non-nesting" version of semi-standard Young tableaux. In this paper, we explain the corresponding commutative algebra. We hope our work will be useful in the investigation of the cluster algebra structures on flag varieties and realted spaces, and of LeClerc and Zelevinsky's weakly seperated sets.

This is the first of a series of papers where Nathan and I take connections between Coxeter groups and cluster algebras that have been proven in finite type and generalize them to all types. This paper is purely on the Coxeter combinatorics side. We prove that Nathan's definitions of sortable elements, and the Cambrian lattice, work with almost no modification in any Coxeter group. Among our key techinical tools are (1) the use of a skew symmetric form on the root space to impose pattern avoidance conditions, giving us a type-free description of the "aligned" condition in Nathan's earlier work, and (2) an explicit description of normal vectors to any cone in the Cambrian fan, in terms of "forced and unforced skips".

Let

I want to emphasize that the main result of this paper was obtained earlier by Kleiner and Pelley. My argument is inspired by theirs, but it removes the use of quiver theory and simplifies the argument on several other points.

This is the first of what will be a series of two papers explaining how to use classical parameterizations of algebraic curves to write down curves with specified tropicalizations. In this paper, we deal with genus zero and one curves. The genus zero case, in particular, is very concrete. This material is drawn from the final chapter of my thesis.

This paper returns to themes I was thinking about in 2002, when I wrote the first octahedron and cube recurrence papers. In those paper, we studied three dimensional recurrences, whose initial conditions live on a two dimensional surface. Since then, Andre Henriques and Joel Kamnitzer have taken the octahedron recurrence and generalized it to a recurrence in any number of dimensions, whose initial conditions still live on a two dimensional lattice. This recurrence computes the associator and commutator in the category of gl

In this paper, Andre and I introduce a recurrence which relates to the cube recurrence as his and Joel's work relate to the octahedron recurrence. We show that this recurrence has the same combinatorial properties as the cube recurrence — well definedness, propogation of inequalities, and Laurentness. A special case gives a coordinatization of the isotropic grassmannian. I don't know what the underlying representation theory, or the underlying algebraic geometry, is. I also don't know how to extend Gabriel and my grove technology, Andre, Dylan Thurston and I are working on this.

We take Postnikov's positroid varieties and describe how to parameterize them by toric varieties. In particular, we can describe the totally nonnegative part of these varieties as a (toplogical) quotient of a polytope. The underlying combinatorics involves matching polytopes.

Let

In this paper, we show that the

Nathan and I are engaged in a long term research project to extend the results of this paper to infinite Coxeter groups. Our first paper on this subject is Sortable elements in infinite Coxeter groups.

Let

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!

Let

Erin and I announce the beginning of a collaboration. Possibly confusing point: Erin has since changed her name to Lark-Aeryn Speyer

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";

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.

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.

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. For further developments on this subject, consult David Helm and Eric Katz's paper Monodromy Filtrations and the Topology of Tropical Varieties.

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

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

I define tropical analogues of the notion of "linear space" and "Plucker coordinates" and basic constructions for working with them. The arXiv version of this paper is an exhaustive introduction that tells almost everything I have figured out. The most interesting aspect of the paper is the

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

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.

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

Presented at Formal Power Series and Algebraic Combinatorics (FPSAC) 2004.

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.

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 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.

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

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

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.

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.

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.

This is a note that I wrote up back when I was an undergrad on how to use transfer matrices to prove results like Propp's reciprocity principle for domino tilings. Since then, a few people have cited it as "D. Speyer, unpublished note on transfer matrices", so I figured I should at least host a copy on my website. Rereading it today, I see a number of typos but no mathematical errors. I am puzzled as to why I said that these results were a special case of Propp's; it seems to me that the reverse is true.

If anyone has the TeX original, I'd love it if they'd send me a copy so that I could fix the typos without retyping everything.

Let