The of Matroids

Federico Ardila

Introduction Gian-Carlo Rota, who helped lay the foundations of Matroid theory is a combinatorial theory of independence the field and was one of its most energetic ambassadors, which has its origins in linear and rejected the “ineffably cacophonous” name of matroids. He and turns out to have deep connections with many other proposed calling them combinatorial instead.1 fields. There are natural notions of independence in This alternative name never really caught on, but the , graph theory, theory, the theory geometric roots of the field have since grown much of field extensions, and the theory of routings, among deeper, bearing many new fruits. others. Matroids capture the combinatorial essence that The geometric approach to matroid theory has recently those notions share. led to the solution of long-standing questions and to the development of fascinating at the intersec- Federico Ardila is professor of mathematics at San Francisco State tion of , algebra, and geometry. This article University and profesor adjunto at Universidad de los Andes in is a selection of some recent successes, stemming from Colombia. His email address is [email protected]. three geometric models of matroids. For permission to reprint this article, please contact: [email protected]. DOI: http://dx.doi.org/10.1090/noti1714 1It was tempting to call this note “The geometry of geometries.”

902 Notices of the AMS Volume 65, Number 8 Definitions Finally, the orthogonal matroid of 푀, denoted 푀⟂, is Matroids were defined independently in the 1930s by the matroid on 퐸 with bases Nakasawa and Whitney. A matroid 푀 = (퐸, ℐ) consists of ℬ⟂ = {퐸 − 퐵 ∶ 퐵 ∈ ℬ}. a finite set 퐸 and a collection ℐ of of 퐸, called the independent sets, such that Remarkably, this simple notion simultaneously general- izes orthogonal complements and dual graphs. If 푀 is (I-1) ∅ ∈ ℐ. the matroid for the columns of a whose rowspan (I-2) If 퐽 ∈ ℐ and 퐼 ⊆ 퐽, then 퐼 ∈ ℐ. is 푈 ⊆ 푉, then 푀⟂ is the matroid for the columns of (I-3) If 퐼, 퐽 ∈ ℐ and |퐼| < |퐽|, then there exists 푗 ∈ 퐽 − 퐼 any matrix whose rowspan is 푈⟂. If 푀 is the matroid such that 퐼 ∪ 푗 ∈ ℐ. for a 퐺, drawn on the plane without edge We will assume that every singleton {푒} is independent. intersections, then 푀⟂ is the matroid for the Thanks to (I-2), it is enough to list the collection ℬ of 퐺⟂, whose vertices and edges correspond to the faces and maximal independent sets; these are called the bases of edges of 퐺, respectively, as shown in Figure 2. 푀. By (I-3), they have the same size 푟 = 푟(푀), which we call the rank of the matroid. Our running example will be the matroid with (1) 퐸 = 푎푏푐푑푒, ℬ = {푎푏푐, 푎푏푑, 푎푏푒, 푎푐푑, 푎푐푒}, omitting brackets for easier readability. See Figure 1. Let us now discuss the two most important motivating examples of matroids; there are many others. Figure 2. The planar graph of Figure 1 and its dual ⟂ Vector Configurations graph, whose set of bases is ℬ = {푏푑, 푏푒, 푐푑, 푐푒, 푑푒}. Let 픽 be a field, let 퐸 be a set of vectors in a over 픽, and let ℐ be the collection of linearly independent Enumerative Invariants subsets of 퐸. Then (퐸, ℐ) is a linear matroid (over 픽). Two matroids 푀1 = (퐸1,ℐ1) and 푀2 = (퐸2,ℐ2) are isomor- phic if there is a relabeling bijection 휙 ∶ 퐸1 → 퐸2 that Graphs maps ℐ1 to ℐ2.A matroid invariant is a function 푓 on ma- Let 퐸 be the set of edges of a graph 퐺 and let ℐ be troids such that 푓(푀1) = 푓(푀2) whenever 푀1 and 푀2 are the collection of forests of 퐺, that is, the subsets of 퐸 isomorphic. Let us introduce a few important examples. containing no . Then (퐸, ℐ) is a graphical matroid. The 푓-vector and the ℎ-vector The independent sets of 푀 form a simplicial complex ℐ by (I-2); its 푓-vector counts the number 푓푘(푀) of independent sets of 푀 of size 푘 + 1 for each 푘. The ℎ-vector of 푀, defined by Figure 1. A linear and a graphical representation of 푟 푟 푓 (푞 − 1)푟−푘 = ℎ 푞푟−푘, the matroid of (1) with ℬ = {푎푏푐, 푎푏푑, 푎푏푒, 푎푐푑, 푎푐푒}. ∑ 푘−1 ∑ 푘 푘=0 푘=0 There are several natural operations on matroids. For stores this information more compactly. For example, the 푆 ⊆ 퐸, the restriction 푀|푆 and the contraction 푀/푆 are matroid of (1) has matroids on the ground sets 푆 and 퐸 − 푆, respectively, 푓(푀) = (1, 5, 9, 5), ℎ(푀) = (1, 2, 2, 0). with independent sets ℐ|푆 = {퐼 ⊆ 푆 ∶ 퐼 ∈ ℐ}, The Characteristic Polynomial We define the rank function 푟 ∶ 2퐸 → ℤ of a matroid 푀 by ℐ/푆 = {퐼 ⊆ 퐸 − 푆 ∶ 퐼 ∪ 퐼푆 ∈ ℐ} 푟(퐴) = largest size of an independent of 퐴, for any maximal independent subset 퐼푆 of 푆. When 푀 is a linear matroid in a vector space 푉, 푀|푆 and 푀/푆 are the for 퐴 ⊆ 퐸. Let 푟 = 푟(푀) = 푟(퐸) be the rank of 푀. linear matroids on 푆 and 퐸 − 푆 that 푀 determines on the When 푀 is a linear matroid, 푟(퐴) = dim span(퐴). The vector spaces span(푆) and 푉/span(푆), respectively. characteristic polynomial of 푀 is The direct sum 푀 ⊕ 푀 of two matroids 푀 = (퐸 ,ℐ ) 1 2 1 1 1 휒 (푞) = (−1)|퐴|푞푟(푀)−푟(퐴). and 푀 = (퐸 ,ℐ ) on disjoint ground sets is the matroid 푀 ∑ 2 2 2 퐴⊆퐸 on 퐸1 ∪ 퐸2 with independent sets The sequence 푤(푀) of Whitney numbers of the first kind 푟 푟−1 푟 0 ℐ1 ⊕ ℐ2 = {퐼1 ∪ 퐼2 ∶ 퐼1 ∈ ℐ1, 퐼2 ∈ ℐ2}. is defined by 휒푀(푞) = 푤0푞 − 푤1푞 + ⋯ + (−1) 푤푟푞 . For example, the matroid of (1) has Every matroid decomposes uniquely as a direct sum of its connected components. 푤(푀) = (1, 4, 5, 2).

September 2018 Notices of the AMS 903 The characteristic polynomial of a matroid is one of which the cost 푐(퐵) is minimized. To do this, one often 푑 its most fundamental invariants. For graphical and linear looks for a polytope 푃ℬ ⊂ ℝ modeling the family ℬ and matroids, it has the following interpretations. a linear function 푓 on ℝ푑 such that

• 푃ℬ has a vertex 푣퐵 for each object 퐵 ∈ ℬ, and Graphs • 푐(퐵) = 푓(푣퐵) for each 퐵 ∈ ℬ. If 푀 is the matroid of a connected graph 퐺, then 푞 휒푀(푞) If one can do this, then the optimal object(s) 퐵 corresponds is the of 퐺; it counts the colorings to the vertices of the face of the polytope 푃ℬ where the of the vertices of 퐺 with 푞 given colors such that no two linear function 푓 is minimized. This simple, beautiful idea neighbors have the same color. is the foundation of . There are many techniques to optimize 푓, whose efficiency depends on Hyperplane Arrangements the complexity of the polytope 푃ℬ. Suppose 푀 is the matroid of nonzero vectors 푣1, … , 푣푛 ∈ Edmonds observed that, given a matroid 푀 and a 픽푑, and consider the arrangement 풜 of hyperplanes cost function 푐 ∶ 퐸 → ℝ on its ground set, the bases 퐵 = {푏1, … , 푏푟} of 푀 of minimum cost 푐(퐵) ∶= 푐(푏1)+⋯+푐(푏푟) 퐻푖 ∶ 푣푖 ⋅ 푥 = 0, 1 ≤ 푖 ≤ 푛, can be found via linear programming on the matroid and its complement 푉(풜) = 픽푑 − (퐻 ∪ ⋯ ∪ 퐻 ). De- 1 푛 polytope 푃푀. pending on the underlying field, 휒푀(푞) stores different As a sample application, Edmonds used these ideas information about 푉(풜): to solve the problem for matroids 푀 (a) (픽 = 픽푞) 푉(풜) consists of 휒푀(푞) points. and 푁 on the same ground set. This problem asks us to (b) (픽 = ℝ) 푉(풜) consists of |휒푀(−1)| regions. find the size of the largest set which is independent in (c) (픽 = ℂ) The Poincaré polynomial of 푉(풜) both 푀 and 푁. 푘 푘 푑 ∑ rank 퐻 (푉(풜), ℤ)푞 = (−1) 휒푀(−1/푞). 푘≥0 Algebraic Geometry Instead of studying the 푟-dimensional subspaces of ℂ푛 Geometric Model 1. Matroid Polytopes one at a time, it is often useful to study them all at once. A crucial insight into the geometry of matroids came from They can be conveniently organized into the space of two seemingly unrelated places: combinatorial optimiza- 푟-subspaces of ℂ푛 called the Grassmannian Gr(푟, 푛); each tion and algebraic geometry. From both points of view, it point of Gr(푟, 푛) represents an 푟-subspace of ℂ푛. is natural to model a matroid in terms of the following A choice of a coordinate system on ℂ푛 gives rise to the polytope. Plücker embedding of

푝 푛 Definition 1 (Edmonds, 1970). Let 푀 be a matroid on the Gr(푟, 푛) ↪ ℙℂ(푟)−1 ground set 퐸. The as follows. For an 푟-subspace 푉 ⊂ ℂ푛, choose an 푟 × 푛 푃푀 = conv{푒퐵 ∶ 퐵 is a of 푀}, 푛 matrix 퐴 with 푉 = rowspan(퐴). Then for each of the (푟) 퐸 where {푒푖 ∶ 푖 ∈ 퐸} is the standard basis of ℝ , and we 푟-subsets 퐵 of [푛] let write 푒퐵 = 푒푏1 + ⋯ + 푒푏푟 for 퐵 = {푏1, … , 푏푟}. 푝퐵(푉) ∶= det(퐴퐵) Figure 3 shows the matroid polytope for example (1). be the determinant of the 푟 × 푟 submatrix 퐴퐵 of 퐴 whose columns are given by the subset 퐵. Although there are many different choices for the matrix 퐴, they can be obtained from one another by elementary row operations, which only change the Plücker vector 푝(푉) by multiplication by a global constant. Therefore 푝(푉) is well defined as an element of projective space. The map 푝 provides a realization of the Grassmannian as a smooth projective variety. The torus 핋 = (ℂ−{0})푛 acts on ℂ푛 by stretching the 푛 coordinate axes, thus inducing an action of 핋 on Gr(푟, 푛). This action gives rise to a moment map 휇 ∶ Gr(푟, 푛) → ℝ푛 given by Figure 3. The matroid polytope for our sample ∑ | det(퐴 )|2 휇(푉) = 퐵∋푖 퐵 for 1 ≤ 푖 ≤ 푛. matroid (1). The vertices exhibit which triplets form 푖 2 ∑ | det(퐴퐵)| bases. 퐵 Now consider the trajectory 핋 ⋅ 푉 of the 푟-subspace Combinatorial Optimization 푉 ∈ Gr(푟, 푛) as the torus 핋 acts on it, and take its . The central question of combinatorial optimization is the Where does the resulting toric variety 핋 ⋅ 푉 ⊂ Gr(푟, 푛) following: Given a family ℬ of combinatorial objects and go under the moment map? Precisely to the matroid a cost function 푐 ∶ ℬ → ℝ, find the object(s) 퐵 in ℬ for polytope!

904 Notices of the AMS Volume 65, Number 8 Define the matroid 푀(푉) of the subspace 푉 ⊂ ℂ푛 to be the matroid of the columns of 퐴; its bases 퐵 correspond to the nonzero Plücker coordinates 푝퐵(푉). Gelfand, Goresky, MacPherson, and Serganova showed that

휇(핋 ⋅ 푉) = 푃푀(푉).

Thus matroid polytopes arise naturally in this algebro- geometric setting as well. 푛 As a sample application, the degree of 핋 ⋅ 푉 ⊂ ℙℂ(푟)−1 is then given by the volume of the matroid polytope 푃푀(푉). Ardila, Benedetti, and Doker used this to find a purely combinatorial formula for deg(핋 ⋅ 푉) in terms of the matroid 푀(푉).2

A Geometric Characterization of Matroids Figure 4. The root system 퐴3 = {푒푖 − 푒푗 ∶ 1 ≤ 푖, 푗 ≤ 4}, In most contexts where polytopes arise, it is advantageous where 푒푖 − 푒푗 is denoted 푖푗. Root systems play an if they happen to have a nice structure. For example, in essential role in matroid theory, as demonstrated by optimization, the edges of the polytope are crucial to Theorem 2. various algorithms for linear programming. In geometry, Hopf Algebra they control the GKM presentation of the equivariant cohomology of the Grassmannian. Joni and Rota showed that many combinatorial families Matroid polytopes have the following beautiful com- have natural merging and breaking operations that give binatorial characterization, which was discovered in the them the structure of a Hopf algebra, with many useful consequences. In particular, in the 1970s and 1980s, Joni– context of toric geometry. Rota and Schmitt defined the Hopf algebra of matroids 필 Theorem 2 (Gelfand–Goresky–MacPherson–Serganova, as the span of the set of matroids modulo isomorphism, 1987). A collection ℬ of subsets of [푛] is the set of bases with the product ⋅ ∶ 필 ⊗ 필 → 필 and coproduct Δ ∶ 필 → of a matroid if and only if every edge of the polytope 필 ⊗ 필 given by

푛 푀 ⋅ 푁 ∶= 푀 ⊕ 푁 for matroids 푀 and 푁, 푃ℬ ∶= conv {푒퐵 ∶ 퐵 ∈ ℬ} ⊂ ℝ Δ(푀) ∶= ∑ (푀|푆) ⊗ (푀/푆) for a matroid 푀 on 퐸. 푆⊆퐸 is a translate of 푒푖 − 푒푗 for some 푖, 푗. For 필 to be a Hopf algebra, we require an antipode map This makes matroid polytopes a very useful model 푆, which is the Hopf-theoretic analogue of an inverse. for matroids. In fact, one could define a matroid to be General results of Schmitt and Takeuchi show that this 푛 a subpolytope of the cube [0, 1] that uses only these map exists. vectors as edges. Notice that from this polytopal point The antipode 푆 is a fundamental ingredient of a of view, even if one cares only about linear matroids, all Hopf algebra, so it is important to find an efficient matroids are equally natural. Matroid theory provides the formula for it. For the Hopf algebra of matroids 필, this correct level of generality. was only resolved recently, thanks to the new insight The theorem above shows that in matroid theory, a that the matroid polytope plays an essential role. An central role is played by one of the most important vector important preliminary observation, which readily follows configurations in mathematics, the root system for the from Theorem 2, is that every face of a matroid polytope special linear group SL푛: is itself a matroid polytope. Theorem 3 (Aguiar–Ardila, 2017). The antipode of the 퐴 = {푒 − 푒 ∶ 1 ≤ 푖, 푗 ≤ 푛}, 푛−1 푖 푗 Hopf algebra of matroids 필 is given by as shown in Figure 4 for 푛 = 4. From this point of view, it 푆(푀) = ∑ (−1)푐(푁)푁 is natural to extend this construction to other Lie groups. 푃푁 face of 푃푀 The resulting theory of Coxeter matroids, introduced by for any matroid 푀, where 푐(푁) denotes the number of Gelfand and Serganova, is ripe for further combinatorial connected components of 푁. exploration. This formula is the best possible: it involves no cancel- lation. It has the unexpected consequence that matroid 2This was the subject of Carolina Benedetti and Jeff Doker’s fi- polytopes are also algebraic in nature. In the Hopf al- nal project for the first course offered by the SFSU-Colombia gebraic structure of matroids, matroid polytopes are Combinatorics Initiative in 2007, as described in [2]. fundamental.

September 2018 Notices of the AMS 905 Geometric Model 2. Bergman Fans (3) The Bergman fan Σ푀 is a cone over a wedge of 푤푟 We now introduce a second geometric model of matroids, spheres of 푟 − 2, where 푤푟 is the last Whitney coming from tropical geometry. The flats of 푀 are an number of the first kind. important ingredient; these are the subsets 퐹 ⊆ 퐸 such b that 푟(퐹 ∪ 푒) > 푟(퐹) for all 푒 ∉ 퐹. We say 퐹 is proper if it does not have rank 0 or 푟. The lattice of flats of 푀, denoted 퐿푀, is the set of flats, partially ordered by inclusion, as shown in Figure 5. bcde ab

de

ade a c

Figure 5. The lattice of flats of our sample matroid (1).

When 푀 is the matroid of a vector configuration 퐸 in a vector space 푉, the flats of 푀 are the (subsets of 퐸 ac contained in the) subspaces spanned by 퐸. Figure 6. The Bergman fan of our sample matroid (1) is modeled after the lattice of flats of Figure 5. It has Tropical Geometry 8 rays and 9 facets. It is a cone over a wedge of Tropicalization is a powerful technique that turns an 푤3 = 2 circles. algebraic variety 푉 into a simpler, piecewise linear space A Geometric Characterization of Bergman Fans Trop 푉 that still contains geometric information about Tropical varieties have a natural notion of degree, analo- 푉. Tropical geometry answers questions in algebraic gous to the notion of the degree of an algebraic variety. geometry by translating them into polyhedral questions We have the following remarkable characterization. that can be approached combinatorially [4]. An important early success of the theory was Theorem 5 (Fink, 2013). A tropical variety has degree 1 Mikhalkin’s 2005 tropical computation of the Gromov- if and only if it is the Bergman fan of a matroid. 2 Witten invariants of ℂℙ , which count the plane curves We conclude that Bergman fans are also excellent of degree 푑 and genus 푔 passing through 3푑 + 1 − 푔 models for matroids. In fact, one could define a matroid general points. Since then, many new results in classical to be a tropical variety of degree 1; this is the tropical algebraic geometry have been obtained through tropical analogue of a linear space. Notice that, although Σ푀 only techniques. arises via tropicalization when 푀 is a linear matroid, one Tropical varieties are simpler than algebraic varieties, should really consider the Bergman fans of all matroids; but they are still very intricate. An important example they are equally natural from the tropical point of view. to understand is that of linear spaces. What is the Again, matroid theory really provides the correct level of tropicalization of a linear subspace 푉 of ℂ푛? Sturmfels generality. realized that the answer depends only on the matroid of The theorems above explain the important role that 푉. It can be described as follows. matroids play in tropical geometry. On the one hand, they provide a useful testing ground, providing hints for the Definition/Theorem 4 (Ardila–Klivans, 2006). kinds of general results that may be possible and the (1) The Bergman fan Σ푀 of a matroid 푀 on 퐸 is the sorts of difficulties that one should expect. On the other 퐸 polyhedral complex in ℝ / ⟨푒퐸⟩ consisting of the cones hand, they are fundamental building blocks; for instance, in analogy with the classical definition of a manifold, a 휎 = cone{푒 ∶ 퐹 ∈ ℱ} ℱ 퐹 tropical manifold is a tropical variety that locally looks for each flag ℱ = {퐹1 ⊊ ⋯ ⊊ 퐹푙} of proper flats of 푀. Here like a (Bergman fan of a) matroid, as in Figure 7.

푒퐹 ∶= 푒푓1 + ⋯ + 푒푓푘 for 퐹 = {푓1, … , 푓푘}. (2) The tropicalization of a linear subspace 푉 of ℂ푛 is the Bergman fan of its matroid:

Trop 푉 = Σ푀(푉).

906 Notices of the AMS Volume 65, Number 8 The Hodge–Riemann relations imply that for any 퐿1, … , 퐿푟−3, 푎, 푏 ∈ 퐾(푀), if we write 퐿 = 퐿1 ⋯ 퐿푟−3, we have (2) deg(퐿푎2) deg(퐿푏2) ≤ deg(퐿푎푏)2.

Unimodality and Log-Concavity

We say a sequence 푎0, 푎1, … , 푎푟 of nonnegative is unimodal if there is an index 0 ≤ 푚 ≤ 푟 such that

푎0 ≤ 푎1 ≤ ⋯ ≤ 푎푚−1 ≤ 푎푚 ≥ 푎푚+1 ≥ ⋯ ≥ 푎푟, and, more strongly, it is log-concave if for all 1 ≤ 푖 ≤ 푟−1, 2 푎푖−1푎푖+1 ≤ 푎푖 . It is flawless if we have

푎푖 ≤ 푎푠−푖 푠 for all 1 ≤ 푖 ≤ 2 , where 푠 is the largest index with 푎푠 ≠ 0. Many sequences in mathematics have these properties, Figure 7. A tropical manifold is a tropical variety that but proving it is often very difficult. Aside from their locally looks like a (Bergman fan of a) matroid. intrinsic interest, these kinds of questions have been a source of fresh mathematics, because their solutions The Chow Ring and Hodge Theory have often required a fundamentally new construction or connection and have given rise to unforeseen structural The Chow ring of the Bergman fan Σ푀 is defined to be results about the objects of interest. ∗ 퐴 (Σ푀) ∶= ℝ[푥퐹 ∶ 퐹 proper flat of 푀]/(퐼푀 + 퐽푀), For matroids, this Hodge theory provides such a where connection. Consider the elements of the Chow ring ∗ 퐴 (Σ푀), 퐼푀 = ⟨푥퐹1 푥퐹2 ∶ 퐹1 ⊊ 퐹2 and 퐹1 ⊋ 퐹2⟩ , 훼 = 훼푖 = ∑ 푥퐹, 훽 = 훽푖 = ∑ 푥퐹, 퐹∋푖 퐹 /∋푖 퐽푀 = ⟨∑ 푥퐹 − ∑ 푥퐹 ∶ 푖, 푗 ∈ 퐸⟩ . 퐹∋푖 퐹∋푗 which are independent of 푖 and lie in the cone 퐾(푀).A ∗ This ring has a natural geometric interpretation when 푀 is clever combinatorial computation in 퐴 (Σ푀) shows that ∗ 푘 푟−1−푘 푘 linear over ℂ: Feichtner and Yuzvinsky proved that 퐴 (Σ푀) deg(훼 훽 ) = | coeff. of 푞 in 휒푀(푞)/(푞 − 1)|. is the Chow ring of De Concini and Procesi’s wonderful As 푘 varies, this sequence of degrees is log-concave by compactification of the complement of a hyperplane (2). In turn, by elementary arguments, this implies the arrangement. following theorems, which were conjectured by Rota, Surprisingly, 퐴∗(Σ ) behaves as nicely as the coho- 푀 Heron, Mason, and Welsh in the 1970s and 1980s. mology ring of a smooth projective variety. This is one of the most celebrated recent results in matroid theory, Theorem 7 (Adiprasito–Huh–Katz, 2015). For any ma- since it provided the tools to prove several long-standing troid 푀 of rank 푟, the following sequences, defined in conjectures, as we now briefly explain. “Enumerative Invariants,” are unimodal and log-concave: Theorem 6 (Adiprasito–Huh–Katz, 2015). The Chow ring • the Whitney numbers of the first kind 푤(푀), and ∗ • the 푓-vector 푓(푀). 퐴 (Σ푀) of the Bergman fan of a matroid 푀 satisfies Poincaré , the hard Lefschetz theorem, and the Hodge–Riemann relations. Geometric Model 3. Conormal Fans We now introduce another polyhedral model of 푀 that The inspiration for this theorem is geometric, coming leads to stronger inequalities for matroid invariants. We from the Grothendieck standard conjectures on algebraic say that a flag ℱ = {퐹1 ⊆ ⋯ ⊆ 퐹푙} of nonempty flats of 푀 ⟂ cycles. The statement and proof are combinatorial. For and a flag 풢 = {퐺1 ⊇ ⋯ ⊇ 퐺푙} of nonempty flats of 푀 of further details and a precise statement, see [1], [3]. the same length are compatible if Let us focus on a comparatively small but very powerful 푙 푙 ∗ consequence. The Chow ring 퐴 (Σ푀) is graded of degree (퐹 ∪ 퐺 ) = 퐸, (퐹 ∩ 퐺 ) ≠ 퐸. 푟−1 ⋂ 푖 푖 ⋃ 푖 푖 푟 − 1, and there is an isomorphism deg ∶ 퐴 → ℝ 푖=1 푖=1 characterized by the property that deg(퐹1 ⋯ 퐹푟−1) = 1 All maximal compatible pairs have length 푛 − 2. for any full flag 퐹1 ⊊ ⋯ ⊊ 퐹푟−1 of proper flats. Say a 퐸 function 푐 ∶ 2 → ℝ is submodular if 푐∅ = 푐퐸 = 0 and Definition 8 (Ardila–Denham–Huh, 2017). The conormal fan Σ ⟂ of a matroid 푀 is the polyhedral complex in 푐퐴 + 푐퐵 ≥ 푐퐴∪퐵 + 푐퐴∩퐵 for any 퐴, 퐵 ⊆ 퐸, and let 푀,푀 퐸 퐸 ℝ / ⟨푒퐸⟩ × ℝ / ⟨푒퐸⟩ consisting of the cones 1 퐾(푀) = { ∑ 푐퐹푥퐹 ∶ 푐 submodular} ⊂ 퐴 (Σ푀). 퐹 flat 휎ℱ,풢 = cone{푒퐹푖 + 푓퐺푖 ∶ 1 ≤ 푖 ≤ 푙}

September 2018 Notices of the AMS 907 for each compatible pair of flags (ℱ, 풢). Here {푒푖 ∶ 푖 ∈ 퐸} Theorem 10 (Ardila–Denham–Huh, 2017). For any ma- and {푓푖 ∶ 푖 ∈ 퐸} are the standard bases for two copies of troid 푀 of rank 푟, the following sequences, defined in “Enu- ℝ퐸. merative Invariants,” are unimodal and log-concave: • the ℎ-vector of the broken circuit complex, and It would be interesting to find an intrinsic character- • the ℎ-vector ℎ(푀). ization of conormal fans of matroids, in analogy with Theorems 2 and 5. Theorem 10 is significantly stronger than Theorem 7. By work of Juhnke-Kubitzke and Le, it also implies a 2003 The Chow Ring and Hodge Theory conjecture of Swartz: These ℎ-vectors are flawless.

Consider the polynomial ring with variables 푥퐹,퐺 where 퐹 and 퐺 are nonempty flats of 푀 and 푀⟂ respectively, ACKNOWLEDGMENTS. I am very grateful to Gian- not both 퐸, such that 퐹 ∪ 퐺 = 퐸. When it is defined, we Carlo Rota, Richard Stanley, and Bernd Sturmfels for write 푥 = 푥 ⋯ 푥 for flags ℱ = {퐹 ⊊ ⋯ ⊊ 퐹 } and the beautiful mathematics and to my coauthors and ℱ,풢 퐹1,퐺1 퐹푙,퐺푙 1 푙 students for the great fun we’ve had together trying 풢 = {퐺 ⊋ ⋯ ⊋ 퐺 }. We also need the special elements 1 푙 to understand matroids a bit better every day. ′ This work was supported in part by NSF Award 푎푖 = ∑ 푥퐹,퐺, 푎푖 = ∑ 푥퐹,퐺, 푑푖 = ∑ 푥퐹,퐺. 퐸≠퐹∋푖 퐸≠퐺∋푖 퐹∩퐺∋푖 DMS-1600609, Award DMS-1440140 to MSRI, and the Simons Foundation. We define the Chow ring of the conormal fan of 푀 to be

∗ A version of this survey including a full list of references is 퐴 (Σ ⟂ ) ∶= ℝ[푥 ]/(퐼 ⟂ + 퐽 ⟂ ), 푀,푀 퐹,퐺 푀,푀 푀,푀 at: math.sfsu.edu/federico/matroidsnotices.pdf. where References 퐼푀,푀⟂ = ⟨푥ℱ,풢 ∶ ℱ and 풢 are not compatible⟩ , [1] K. Adiprasito, J. Huh, and E. Katz, Hodge theory of ′ ′ matroids, Notices Amer. Math. Soc. 64(1), 26–30. MR3586249 퐽 ⟂ = ⟨푎 − 푎 , 푎 − 푎 ∶ 푖, 푗 ∈ 퐸⟩ . 푀,푀 푖 푗 푖 푗 [2] F. Ardila-Mantilla, Todos cuentan: Cultivating diversity in combinatorics, Notices Amer. Math. Soc. 63(10), 1164–1170. The Chow ring of the conormal fan behaves as nicely [3] M. Baker, Hodge theory in combinatorics, Bull. Amer. Math. as the Chow ring of the Bergman fan, though proving it Soc. (N.S.) 55(1), 57–80. MR3737210 requires significant additional work. [4] D. Maclagan and B. Sturmfels, Introduction to Tropical Ge- ometry, Graduate Studies in Mathematics, vol. 161, American Theorem 9 (Ardila–Denham–Huh, 2017). The Chow ring Mathematical Society, Providence, RI, 2015. MR3287221 ∗ 퐴 (Σ푀,푀⟂ ) of the conormal fan of a matroid satisfies Poincaré duality, the hard Lefschetz theorem, and the Image Credits Hodge–Riemann relations. Figures 1–6 courtesy of Federico Ardila. ∗ Figure 7 by Johannes Rau. This Chow ring 퐴 (Σ푀,푀⟂ ) has degree 푛 − 2, and there Author photo courtesy of May-Li Khoe. is an isomorphism deg ∶ 퐴푛−2 → ℝ characterized by the property that deg(푥ℱ,풢) = 1 for any maximal pair of compatible flags ℱ and 풢. The inequality (2) is still ⟂ satisfied for elements of a suitable cone 퐾(푀, 푀 ). ABOUT THE AUTHOR

Unimodality, Log-Concavity, and Flawlessness Federico Ardila works in combina- torics and its connections to other We now apply (2) to the elements 푎 = 푎푖 and 푑 = 푑푖 of ∗ areas of mathematics and appli- the Chow ring 퐴 (Σ푀,푀⟂ ), which are independent of 푖 and ⟂ cations. He also strives to build lie in the relevant cone 퐾(푀, 푀 ). A subtle combinatorial joyful, empowering, and equitable argument shows that mathematical spaces. He has ad- 푘 푛−2−푘 푘+1 vised over forty thesis students in deg(푎 푑 ) = | coeff. of 푞 in 휒푀(푞 + 1)|. Federico Ardila the US and Colombia; more than As 푘 varies, this sequence of coefficients is the ℎ-vector of half of his US students are mem- the broken circuit complex 퐵퐶<(푀). This is the collection bers of underrepresented groups of subsets of 퐸 − min 퐸 that do not contain a broken and more than half are women. circuit; that is, a set of the form 퐶 − min 퐶 for a minimal Outside of work, he is often play- dependent set 퐶. The broken circuit complex depends ing fútbol or DJing with his wife on a choice of a linear order < on 퐸, but its ℎ-vector is May-Li, and Colectivo La Pelanga. independent of <. The inequalities (2) for the Chow ring 퐴푀,푀⟂ then imply the following theorems, which were conjectured by Brylawski and Dawson in the 1980s.

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