ams fall sectional · pdf filebuilding polyhedra by self-assembly by govind menon (brown...

September 9–10, 2017 Central Meeting; (Saturday–Sunday)University of North Texas, Denton, TX

Semigroups of Branched Mapping Classes: Dynamics and Geometry by Kevin M. Pilgrim (Indiana University, Bloomington) page 824

Building Polyhedra by Self-Assemblyby Govind Menon (Brown University)page 822

The Protean Chromatic Polynomialby Bruce Sagan (Michigan State University)page 828

Strings, Trees, and RNA Foldingby Christine Heitsch (Georgia Institute of Technology)page 817

Realizing the Spectrum of Tensor Categoriesby Jonathan R. Kujawa (University of Oklahoma)page 820

AMS FALL SECTIONAL SAMPLER

For permission to reprint this article, please contact:[email protected]: http://dx.doi.org/10.1090/noti1595

Make the time to visit any of the AMS Fall Sectional Meetings.

In this sampler, the speakers above have kindly provided introductions to their Invited Addresses for the upcoming AMS Fall Sectional Meetings.

September 16–17, 2017 Eastern Meeting; (Saturday–Sunday)State University of New York–Buffalo, Buffalo, NY

September 23–24, 2017 Southeastern; (Saturday–Sunday)University of Central Florida–Orlando, Orlando, FL

*A sampler from the Fall Western Sectional Meeting willappear in the November issue of Notices.

From left to right: Christine Heitsch, Jonathan R. Kujawa, Govind Menon, Kevin Pilgrim, and Bruce Sagan.

This meeting's location was erroneously listed as University of Texas in the print edition. Notices apologizes for any confusion.

AMS FALL SECTION SAMPLER

Christine HeitschStrings, Trees, and RNA FoldingWe highlight some challenges and opportunities at theinterface of discrete mathematics and molecular biology,illustrating that this interaction motivates new com-binatorial theorems as well as advancing biomedicalapplications.

High school teaches us that RNA’s role is to mediatethe production of proteins from DNA. Closer inspection,however, reveals a vast complexity of structure anddiversity of function. As illustrated in Figures 1 and 2,RNAmolecules are essential to cellular processes rangingfrom bacterial communication to viral capsid assembly.It goes almost without saying that advancing knowledgeof how RNA functions in these diverse roles has thepotential for tremendous scientific impact.

Figure 1. How do bacteria communicate? Four RNAmolecules (diagrammed here through marine bio-luminescence) are essential to the quorum sensingprocess by which bacteria regulate collective behav-ior, ranging from this benign light display to choleratoxicity. Despite high sequence similarity and knownfunctional redundancy [1], the branching of the fourpossible RNA structures can vary significantly. Newresults in combinatorics and its applications provideinsight into this important, yet difficult-to-determine,molecular characteristic.

A basic biological principle is that “function followsform”; that is, to understand what a molecule does, one

Christine Heitsch is professor of mathematics at the GeorgiaInstitute of Technology. Her e-mail address is [email protected].


Figure 2. How do viral capsids assemble? The RNAgenome (gold) is partially visible inside the quasi-icosahedral protein capsid (purple). The resolutionof this detailed crystal structure [2] is such that thenumber and approximate length of many runs ofstacked base pairs are known, but the compositionand connectivity of these helices are not. New knowl-edge of RNA branching configurations is needed tounderstand how this viral sequence folds into itsdodecahedral cage.

must first know how it is structured. Generically, an RNAmolecule is a single nucleotide sequencewhich folds into a3D conformation via a set of noncrossing, intrasequencebase pairings known as a secondary structure. Giventhat the goal of 3D experimental determination remainsinaccessible for most RNA structures, computationalpredictions of possible base pairing configurations, suchas the fouroutlined in Figure1, are essential for generatingfunctional hypotheses.

Yet accurate prediction of the branching of thesestructures remains a fundamental open question. Giventhe combinatorial nature of the problem, this is anopportunity for discrete models, methods, and analysesto provide insights into a complex biomolecular process.

To begin to appreciate the challenge, first consider themodel sequence g6a4c6 = gggggg aaaa cccccc. Underthe Watson-Crick association of g and c, it folds into astructure with one helix (run of 6 stacked base pairs) andone loop (single-stranded region) known as a “hairpin.”Thus, the lower left structure in Figure 1 has 3 hairpins,1 internal loop, and the central loop from which branch3 helical arms.

To analyze this coding of 2D branching information ina linear biochemical chain, we model an RNA structure

September 2017 Notices of the AMS 817


Figure 3. How do branching configurations relate? Five arrangements of loops and helices for the RNA se-quence A4 (G6 A4 C6 A4)3, determined by the pairing of G (labeled 1, 3, 5) and C (labeled 2, 4, 6) segments.Arrows indicate movement between structures under allowable pairing exchanges. Two configurations witha graph geodesic of length 2 comprise a meander; the corresponding noncrossing perfect matchings form asingle closed loop when drawn on the same endpoints, one above and the other below, as illustrated in the in-set figure. Although these closed meanders arise in various mathematical settings, their exact enumerationproblem remains open.

as a plane tree—that is, a rooted tree whose subtreesare linearly ordered, by mapping loops to vertices andhelices to edges. In Figures 1 or 3 this would correspondto collapsing each circular region to a point and mergingtwo parallel lines into one. This abstraction preservesthe basic structural arrangement, including informationabout the sequence ordering and the energetic types ofdifferent loop structures.

Mathematics is avital source of newstructural insights.

We then generalizeour toy example fromabove to consider sat-urated (fully paired,hence the lowest freeenergy) structures for𝑅 = a4 (g6a4c6a4)𝑛and their correspond-

ing plane trees. This highlights one of the critical issues:there can be exponentially many different possible low-energy branching configurations for an arbitrary RNAsequence.

Hence, even in a situation, as in Figure 2, wheresome experimental information about the 3D structure isknown, mathematics is a vital source of new structuralinsights. As we will discuss, by using strings and trees asa combinatorial model of RNA folding, we can analyze dif-ferent possible branching configurations at viral genomelength scales. We prove theorems, using methods fromenumerative, probabilistic, and geometric combinatorics,which address questions such as: What are the trade-offs

among different types of loop structures? Is there a typicaldegree of loop branching? What is the dependence on thethermodynamic optimization parameters? Since the accu-racy of computational base pairing predictions decaysrapidly with sequence length, these theoretical resultshelp separate structural signals from thermodynamicnoise, thereby supporting alternative hypotheses in viralcapsid assembly.1

Conversely, under a suitable abstraction, the spaceof branching configurations reveals significant combi-natorial structure. The challenge of understanding thedifferent possible low-energy secondary structures foran RNA sequence motivates a new local move on planetrees/noncrossing perfect matchings. This yields graphs,as in Figure 3, isomorphic to the Hasse diagram for thelattice of noncrossing partitions. By recapitulating thiswell-known combinatorial structure, we gain insights thatnow allow us to count and characterize the orbits underthe Kreweras complementation operator. This result thennaturally leads to considering some new approaches tothe challenging open problem of meander enumeration.In this way, we illustrate that the interaction of the twodisciplines is fruitful formathematics as well as beneficialfor biology.

1For more on the mathematics of molecular machines, see “Build-ing Polyhedra by Self-Assembly” by Govind Menon in this issue(page 822).

818 Notices of the AMS Volume 64, Number 8


References[1] D. H. Lenz, K. C. Mok, B. N. Lilley, R. V. Kulkarni, N. S.

Wingreen, and B. L. Bassler, The small RNA chaperone Hfqand multiple small RNAs control quorum sensing in Vibrioharveyi and Vibrio cholerae, Cell 117 (2004), no. 1, 69–82.

[2] L. Tang, K. N. Johnson, L. A. Ball, T. Lin, M. Yeager, andJ. E. Johnson, The structure of Pariacoto virus reveals a do-decahedral cage of duplex RNA, Nat. Struct. Biol. 8 (2001),77–83.

Image CreditsFigure 1 courtesy of Henke and Bassler, Princeton Univer-

sity.Figure 2 courtesy of Nature Structural and Molecular Biol-

ogy, Nature Publishing Group, reprinted with permis-sion [2].

Figure 3 courtesy of Christine Heitsch, Georgia Instituteof Technology. Secondary structures predicted anddrawn by mfold software, available viaunafold.rna.albany.edu.

Photo of Christine Heitsch courtesy of James Heitsch,University of Illinois at Chicago.


http://unafold.rna.albany.edu


Jonathan R. Kujawa

Realizing the Spectrum of Tensor CategoriesThe usefulness of attaching geometry to algebraic objectsgoes back at least to Descartes. Using geometry we canobtain qualitative information about our original algebraicobject. Given a polynomial with real coefficients, 𝑝(𝑥), weteach schoolchildren to look at the graph of 𝑦 = 𝑝(𝑥) inℝ2. By examining the x-intercepts, y-intercepts, and endbehavior they can say things about the degree, leadingcoefficient, the constant term, and so on.

A more modern example is the prime ideal spectrum,Spec(𝑅), of a commutative ring 𝑅. More generally, given afinitely generated 𝑅-module𝑀, we can define the supportof 𝑀, supp(𝑀), to be the subset of Spec(𝑅) consistingof all prime ideals 𝑃 such that 𝑀 localized at 𝑃 doesnot vanish. The geometry of the spectrum and supportagain captures algebraic information. For example, fortwo modules the support of a direct sum is the union ofthe supports, and the support of the tensor product isthe intersection of the supports.

Now suppose𝒞 is a category which admits both a directsum and a tensor product. We also assume the tensorproduct is “commutative” in that there are canonicalisomorphisms 𝑋⊗𝑌 ≅ 𝑌⊗𝑋 for all pairs of objects 𝑋and𝑌. For example,𝒞 couldbe𝑘-vec, the category of finite-dimensional 𝑘-vector spaces over a fixed ground field 𝑘with the usual direct sum and tensor product operations.In this case the canonical isomorphism 𝑉⊗𝑊 → 𝑊⊗𝑉 isgiven by the “flip” map 𝑣⊗𝑤 ↦ 𝑤⊗𝑣. The ground fieldacts as the identity for the tensor product in that thereare canonical isomorphisms 𝑘 ⊗ 𝑉 ≅ 𝑉 ≅ 𝑉 ⊗ 𝑘. Suchtensor categories are common throughout mathematics.Another elementary example is the category of closed,orientable surfaces with direct sum given by disjointunion and tensor product given by connected sum.

Such a category can be thought of as a categoricalanalogue of a commutative ring, with the direct sum as the“addition” and the tensor product as the “multiplication.”With this in mind it is natural to ask for the notion ofan ideal. A tensor ideal of 𝒞 is a full subcategory 𝐼 whichhas the property that (1) if 𝑀 is an object of 𝐼 and 𝑁 an

Jonathan R. Kujawa is professor of mathematics at the Universityof Oklahoma. His e-mail address is [email protected].

The research of the author was partially supported by NSF grantDMS-1160763 and NSA grant H98230-16-0055.


object of 𝒞, then 𝑀⊗𝑁 is an object of 𝐼, and (2) 𝐴⊕ 𝐵is an object of 𝐼 if and only if both 𝐴 and 𝐵 are objectsof 𝐼. The second condition is the requirement that 𝐼 bea “thick” subcategory. Thinking of the kernels of ringhomomorphisms, the thick condition becomes plausibleonce we notice that for any functor of tensor categories,𝐹 ∶ 𝒞 → 𝒟, we have 𝐹(𝐴⊕ 𝐵) ≅ 0 if and only if 𝐹(𝐴) ≅ 0and 𝐹(𝐵) ≅ 0. Finite-dimensional 𝑘-vector spaces is acategorical version of a field in that it has no properideals. Namely, if 𝑉 is a nonzero vector space in 𝐼, then bywriting it as a direct sum of one-dimensional subspacesand using the thick condition, it follows that 𝑘 lies in 𝐼.Therefore every vector space 𝑊 ≅ 𝑊⊗𝑘 lies in the tensorideal 𝐼.

Things become more interesting when the category isnot semisimple. In this case the structure of the categorycan be quite complicated. Just classifying the objects isalready a hopeless task in all but the easiest examples.A more reasonable goal is to describe the tensor ideals.Doing so gives us an idea of the coarse structure of thecategory and, in particular, gives information about whenone object can be obtained from another by direct sums,direct summands, and tensor products.

In easy cases, the tensor ideals can be described byhand. For example, let 𝑘 be a fixed ground field whichis algebraically closed and of characteristic 𝑝 > 0. Let𝐶𝑝 be the cyclic group of order 𝑝 and let 𝐶𝑝-mod bethe category of finite-dimensional 𝐶𝑝-modules, that is,finite-dimensional 𝑘-vector spaces with a linear action bythe elements of 𝐶𝑝. Then 𝐶𝑝-mod again admits a tensorproduct. Namely, given 𝐶𝑝-modules 𝑀 and 𝑁, define𝑀⊗𝑁 to be the tensor product as vector spaces with 𝐶𝑝action given by the formula 𝑔.(𝑚 ⊗ 𝑛) = (𝑔.𝑚) ⊗ (𝑔.𝑛)for 𝑔 ∈ 𝐶𝑝. There are 𝑝 nonisomorphic 𝐶𝑝-modules,𝑄1,… ,𝑄𝑝, which cannot be written as a direct sum ofsmaller modules. The dimension of𝑄𝑑 as a 𝑘-vector spaceis 𝑑. In particular, 𝑄1 is the unique simple 𝐶𝑝-module,and 𝑄𝑝 is the unique projective indecomposable module.In this case, direct calculations show that there are twotensor ideals: the entire category and the full subcategoryconsisting of projective modules. A more general andmuch more difficult problem is to classify the tensorideals of 𝐺-mod when 𝐺 is any finite group in which 𝑝divides the order of 𝐺.

Now let𝒦 be a tensor triangulated category consistingof compact objects. That is, 𝒦 is a triangulated categorywith a compatible tensor product and subject to suitablefiniteness assumptions. Approximately ten years ago PaulBalmer introduced geometry by defining the spectrum of𝒦 in the spirit of commutative ring theory. In this settinga tensor ideal is taken to be a full triangulated subcategorywith properties (1) and (2) as above. A tensor ideal 𝐼 iscalled prime if it is a proper ideal and if whenever 𝐴⊗𝐵 isan object of 𝐼, either𝐴 or 𝐵 is an object of 𝐼. The spectrumof 𝒦, Spc(𝒦), is then the collection of all prime tensorideals with the Zariski topology. Balmer also defined thesupport of any object 𝑀 in 𝒦 as the set of all primetensor ideals which do not contain 𝑀.



Tensor triangulargeometry is abeautiful and

powerful theory.

Balmer proved thatthe spectrum andsupport for 𝒦 areuniversal in a precisesense among supporttheories for𝒦. He alsoproved that it providesa classification of thethick tensor ideals of𝒦 and, hence, that it provides a geometric answer to ourearlier question. Balmer, collaborators, and others havegone on to show that tensor triangular geometry is a richtheory which brings valuable new tools and insights toa variety of settings. Those who are able to attend theNovember Western Sectional Meeting at the University ofCalifornia, Riverside, will have the opportunity to hearthis story in person at Balmer’s Invited Address “Aninvitation to tensor-triangular geometry.”

Tensor triangular geometry is a beautiful and powerfultheory. However, for tensor triangular categories of inter-est it is desirable to have a concrete description of thespectrum. Such a realization is both useful for applica-tions and to connect it to existing theories. For example,when he introduced tensor triangular geometry, Balmerproved that if 𝒦 is the homotopy category of boundedcomplexes of finitely generated projective 𝑅-modules,then the spectrum and support recover Spec(𝑅) and itssupport. He also showed that if 𝒦 is the stable modulecategory for 𝐺-mod when 𝐺 is a finite group, then thespectrum and support match the long-studied spectrumof the cohomology ring of 𝐺 and cohomological supportvarieties for 𝐺-modules. In particular, this recovers theclassification of thick tensor ideals in this setting firstobtained by Benson-Carlson-Rickard twenty years ago.This shows that tensor triangular geometry encompassesknown theories. Additional examples have since beencomputed. Nevertheless, it remains a challenging prob-lem to give an explicit realization of the spectrum andsupport for tensor triangulated categories of interest.

In joint work with Brian Boe and Daniel Nakano, weprovide a description of the spectrum for several tensortriangulated categories which appear in nature. We giveexplicit, down-to-earth descriptions of the spectrum forthe stable category of finite-dimensional modules forthe complex Lie superalgebra 𝔤𝔩(𝑚|𝑛) and for the stablecategory of finite-dimensional modules for quantizedenveloping algebras at a root of unity. The goal of thetalk will be to describe these results through a gentleintroduction involving plenty of examples.

Photo CreditPhoto of Jonathan Kujawa by Anne Dunn, courtesy

of Jonathan Kujawa.



Govind Menon

Building Polyhedra by Self-AssemblyGromov begins an interesting—and speculative—recentarticle [2] with the question, “Is there mathematics inbiology?” The answer, I think, is yes, but this is notimmediately apparent, since the real underlying questionis whether modern biology can inspire new forms ofmathematics in a way that compares to the deep tiesthat bind mathematics and physics. If we believe thatan essential aspect of mathematics lies in the discoveryof abstract principles from empirical knowledge, thereis little doubt that biology today presents us with anabundance of the “raw stuff.” What seems much harderis to process this raw stuff into beautiful mathematics,especially if one begins with the genetic code and thetheory of evolution.

The topic of my talk is not true biology, but aninstance of “synthetic biology.” All biological organismsbuild themselves or “self-assemble.” This is, of course,familiar to us from our everyday experience, but mytalk will be about much smaller organisms. For thepast twenty years, nanotechnologists have been tryingto manufacture devices by mimicking biological self-assembly and the exquisite design of molecular machines.The goal ofmy talk is to advertise one aspect of this rapidlygrowing field and to explain how an important biologicalexample—the self-assembly of viruses with icosahedralsymmetry—can inspire and guide the development ofself-assembly in technology.

Viruses are biological organisms that lack the cellu-lar machinery necessary for independent existence. Thesimplest viruses consist of genomes contained within aprotein shield (the capsid). The capsid disassembles whenthe virus attacks a host cell; the virus genome then hijacksthe host cell and uses it to make many more copies ofvirus genome andproteins, which then rapidly reassembleinto new copies of the virus. The natural design of viruseshas two elegant features that should appeal to all math-ematicians: genetic economy and structural symmetry.The genetic sequences of primitive viruses are very short.For example, the genome of MS2, a well-studied virus, hasonly 3,569 nucleotides that code for four proteins (lysis,replicase, maturation, and coat protein), each of whichhas a very specific function. The lysis enzyme degradesthe cell wall of the host, and the replicase catalyzes thereproduction of the virus. The other two proteins are usedto build the MS2 capsid: it consists of 180 copies of thecoat protein, pinned at one end by the maturation protein,in a beautiful arrangement of dimers with icosahedralsymmetry (Figure 1). While the genome of MS2 has been

Govind Menon is professor of applied mathematics at Brown Uni-versity. His e-mail address is [email protected].


Figure 1. A ribbon-diagram showing the structureof the bacteriophage MS2. The coat protein existsin three distinct conformations (A, B, and C), whichmerge in pairs into A/B dimers (blue/green) and C/Cdimers (maroon). A/B dimers cluster into pentamersaround the 5-fold axes of an icosahedron, three alter-nating A/B and C/C clusters form at the 3-fold axes,and the C/C dimers sit as axes of 2-fold symmetry.

known since the mid-1970s, it is only recently that theintricate combinatorial structure of the co-assembly ofthe capsid with RNA folding was deciphered by ReidunTwarock and her colleagues [1].1

The self-assembly of viruses has inspired many exam-ples of synthetic self-assembly. My work has mainly beenin collaboration with David Gracias, an experimentalistat Johns Hopkins University. Over the past fifteen years,David has used photolithography to design many devicesand containers that fold themselves into a final shapeonce they are released from a substrate. The devices builtin his lab are small (a hair’s width and smaller), but muchlarger than viruses such as MS2. This allows us to observethe pathways of self-folding, unlike the process of self-assembly of viruses, which must be inferred indirectly(Figure 2).

The unfolding of a polyhedron into a planar net is aclassical problem in discrete geometry, and our collabora-tion began when David askedme what the best net shouldbe for a self-folding dodecahedron. The issue here is acombinatorial explosion. The cube has only 11 nets, each

1For more on connections between combinatorics and molecu-lar biology, see “Strings, Trees, and RNA Folding” by ChristineHeitsch in this issue (page 817).



Figure 2. Optical microscope images of surface-tension-driven self-assembly of a dodecahedron froma net. The sides of each face of the dodecahedron are300 μm.

of which may be tested in the lab. However, the dodeca-hedron has 43,380 nets, and, to my surprise and delight,simple heuristics along with our computations revealedthe best nets in the lab [4]. Since then ourwork has evolvedinto a study of the pathways of self-assembly [3]. Thishas required some surprisingly sophisticated mathemat-ics. My current goal is to understand the conformationaldiffusion of polyhedral linkages. More formally, this in-volves a rigorous formulation for Brownian motion onalgebraic varieties defined by polyhedral linkages, alongwith effective algorithms for simulation.

References[1] E. C. Dykeman, P. G. Stockley, and R. Twarock, Solving a

Levinthal’s paradox for virus assembly identifies a uniqueantiviral strategy, Proceedings of the National Academy ofSciences (2014).

[2] M. Gromov, Crystals, proteins, stability and isoperimetry,Bull. Amer. Math. Soc. (N.S.) 48 (2011), 229–257.

[3] Ryan Kaplan, Joseph Klobušický, Shivendra Pandey,David H. Gracias, and Govind Menon, Building polyhe-

dra by self-assembly: Theory and experiment, Artificial Life(2014). MR 2774091

[4] S. Pandey, M. Ewing, A. Kunas, N. Nguyen, D. H. Gracias,and G. Menon, Algorithmic design of self-folding polyhedra,Proceedings of the National Academy of Sciences 108 (2011),19885–19890.

Image CreditsFigure 1 ©Dr Neil Ranson, University of Leeds, UK. Usedunder the Creative Commons Attribution-Share Alike 3.0Unported License.Figure 2 courtesy of Shivendra Pandey.Photo of Govind Menon courtesy of Govind Menon.



Kevin M. Pilgrim

Semigroups of Branched Mapping Classes:Dynamics and GeometryA rational function of a single complex variable definesa continuous map of the Riemann sphere to itself. In theearly 1980s, W. Thurston gave a topological characteriza-tion of certain rational functions among the much largerset of self-branched coverings of the sphere. Implicit inhis development are generalizations of mapping classgroups. These generalizations will be the focus of mytalk.

Mapping Class GroupsThe simplest mapping class group is that of the torus.The mapping class group of the torus 𝑇2 is the groupMod(𝑇2) of orientation-preserving self-homeomorphismsof the torus, where two such maps are identified if theyare isotopic, i.e. connected by a continuous path ofhomeomorphisms.

Matrices provide lots of examples. Let 𝑇2 = ℝ2/ℤ2 bethe usual presentation of the torus as a quotient of theplane. Suppose 𝐴 = [ 𝑎 𝑏

𝑐 𝑑 ]. The linear map ℝ2 → ℝ2 givenby [ 𝑥

𝑦] ↦ 𝐴[ 𝑥𝑦 ]descends toahomeomorphism𝑓 ∶ 𝑇2 → 𝑇2

that preserves orientation if and only if it sends ℤ2 ontoitself and det(𝐴) > 0. Equivalently, 𝑎,𝑏, 𝑐, 𝑑 ∈ ℤ anddet(𝐴) = 1, i.e. 𝐴 ∈ SL2(ℤ). For example:

• 𝐴 = [ 0 −11 1 ]. A calculation shows 𝐴 has order 6.

The map 𝑓 is periodic.• 𝐴 = [ 1 1

0 1 ]. Since 𝐴𝑛 = [ 1 𝑛0 1 ], 𝑓 has infinite order.

The 𝑥-axis is an eigenspace of 𝐴. Its image underprojection to𝑇2 gives a simple closed curve whichis preserved by 𝑓. Infinite order maps for whichsome iterate preserves a curve are called aperiodicreducible.

• 𝐴 = [ 2 11 1 ]. Again 𝐴 has infinite order. The

eigenspaces have irrational slopes and projectto dense subsets of 𝑇2. If we use suitable localcoordinates (𝑢, 𝑣) given by the eigenspaces, 𝑓looks like (𝑢, 𝑣) ↦ (𝜆𝑢,𝜆−1𝑣) where 𝜆 = 3+√5

2 isthe larger eigenvalue. No iterate of 𝑓 fixes a curve.Such 𝑓 are called irreducible.

It turns out that every element of Mod(𝑇2) arises inthis way: the mapping class group Mod(𝑇2) is naturallyisomorphic to SL2(ℤ). Put another way, mapping classesof the torus are determined by their induced action onthe fundamental group ℤ2 of 𝑇2.

Kevin Pilgrim is professor of mathematics at Indiana University.His research is partially supported by the Simons Foundation. Hise-mail address is [email protected].


1

τ τ−1 + τ

!

0 −1

1 1

"

Figure 1. Shown are two fundamental domains forthe torus ℂ/⟨1,𝜏⟩ where 𝜏 = 𝑒2𝜋𝑖/6. The ℝ-linear mapinduced by 1 ↦ 𝜏 and 𝜏 ↦ −1 + 𝜏 is a rotation oforder 6 and descends to an isometry on the torus.

ConjugacyTwo elements 𝑓, 𝑔 ∈ Mod(𝑇2) are conjugate if 𝑔 = ℎ−1𝑓ℎfor some ℎ ∈ Mod(𝑇2). When does this happen? Thinkingdynamically, two maps 𝑓, 𝑔 are conjugate if they coincideafter a change of coordinates via an element ℎ of Mod(𝑇2).Properties like being periodic, reducible, or irreducible arethus invariant under conjugacy. The conjugacy problemasks, given 𝑓, 𝑔, can you tell if 𝑓 and 𝑔 are conjugate? Andif the answer is “yes,” can you produce such an elementℎ? For Mod(𝑇2) the answer is “yes”; the classification ofconjugacy classes goes back to Gauss.

GeometrizationThere is another way to think about the classificationof conjugacy classes, via geometrization. Roughly, ge-ometrization is theproblemoffindinga “nice”or “optimal”geometric structure for some topological object. Thinkabout the (surface of the) bagel you ate this morning.It inherits a metric from usual Euclidean 3-space. Thatmetric has variable curvature. In contrast, the Euclideanmetric on the planeℝ2 descends to ametric on𝑇2 which isflat: the curvature is zero everywhere. The map 𝑓 inducedby 𝐴 = [ 0 −1

1 1 ] seems simple enough, but it distorts thisEuclidean metric! There are, however, lots of flat metricson 𝑇2. They can be fruitfully organized by points in theupper-half-plane ℍ ⊂ ℂ. Given 𝜏 ∈ ℍ, we can identify ℝ2

as a real vector space with ℂ via (1, 0) ↦ 1 and (0, 1) ↦ 𝜏,and transport the Euclidean metric on ℂ to one on ℝ2

via this identification. If we take the Euclidean metricon ℂ and take 𝜏 = exp(2𝜋𝑖/6), 𝑓 becomes an isometricrotation of order 6; see Figure 1.

In general, if 𝑓 ∈ Mod(𝑇2) and if we equip 𝑇2 witha Euclidean metric corresponding to a point 𝜏 ∈ ℍ inthis way, then we can transport this metric via 𝑓 toget another metric on 𝑇2. This leads to an action ofMod(𝑇2) on the space of metrics ℍ. The action is givenby [𝑎 𝑏

𝑐 𝑑].𝜏=↦ 𝑎𝜏+𝑏𝑐𝜏+𝑑 .

Given 𝜏1, 𝜏2 ∈ ℍ, there is a natural notion of distance𝑑(𝜏1, 𝜏2) between the corresponding flat metrics on 𝑇2

that turns out to be the hyperbolic metric on ℍ; its



infinitesimal form is given by 𝑑𝑠 = 2|𝑑𝜏|ℑ(𝜏) . This yields

a natural invariant of 𝑓 ∈ Mod(𝑇2) via its minimumdisplacement inf{𝑑(𝜏,𝐴.𝜏) ∶ 𝜏 ∈ ℍ}. This can be (1) zeroand realized by some fixed-point 𝜏, (2) zero and notrealized by any point, or (3) positive and realized bypoints 𝜏 lying along some geodesic that is preservedby the action of 𝑓. These correspond to the three casesabove. The first and third cases are called geometrizable:an optimal metric exists. For reducible cases, we can getclose to optimal, but we cannot achieve it.General Mapping Class GroupsNow suppose 𝑆 is an arbitrary closed orientable surfaceand 𝑃 ⊂ 𝑆 is finite. The mapping class group Mod(𝑆,𝑃) isthe group of orientation-preserving homeomorphisms 𝑓 ∶𝑆 → 𝑆 for which 𝑓(𝑃) = 𝑃, where again two are identifiedif they are isotopic. Roughly summarizing thousands ofpages of hard work by many mathematicians: the flavorof the results for the torus outlined above extend toarbitrarymapping class groups. Elements ofMod(𝑆,𝑃) aredetermined by their action on the fundamental group. Ageometrizable element is either periodic and representedby an isometry or irreducible and represented by a mapwhich in (mildly singular local) Euclidean coordinateslooks again like (𝑥, 𝑦) ↦ (𝜆𝑥, 𝜆−1𝑦). A class that is notgeometrizable reduces canonically into geometrizablepieces by cutting along some finite invariant collectionof pairwise disjoint curves. The conjugacy problem issignificantly harder but solvable.Branched MappingsWe now drop the assumption that 𝑓 is a homeomorphismand require only that it is a finite branched coveringunramified away from 𝑃. Equivalently, 𝑓 ∶ 𝑆 − 𝑓−1(𝑃) →𝑆−𝑃 is a covering map of some degree 𝑑 ≥ 1; we requirestill that 𝑓(𝑃) ⊂ 𝑃. The Riemann-Hurwitz formula impliesthat if 𝑑 ≥ 2, then the surface 𝑆 is either the torus 𝑇2

or the sphere 𝑆2 and that if 𝑆 is the torus, the map 𝑓 isunramified. Composition descends to a well-defined mapon isotopy classes, and we obtain a countable semigroupBrMod(𝑆,𝑃).

Again the case of the torus is instructive. The conjugacyproblem in this semigroup boils down to the question:Given a pair of 2 by 2 integral matrices 𝐴,𝐵 of commondeterminant larger than one, when is 𝐵 = 𝑃−1𝐴𝑃 forsome 𝑃 ∈ SL2(ℤ)? This question is equivalent to theclassification of real quadratic fields and is still unsolved.

So I’ll focus on the case 𝑆 = 𝑆2. It turns out the case#𝑃 = 4 is already interesting.Complex DynamicsComplexdynamicsprovides anatural sourceof geometriz-able elements of BrMod(𝑆2, 𝑃). Let’s identify the Riemannsphere ℂ̂ with 𝑆2 in the usual way via stereographicprojection. For a parameter 𝑐 ∈ ℂ let 𝑝𝑐 ∶ ℂ̂ → ℂ̂ be givenby 𝑝𝑐(𝑧) = 𝑧2 + 𝑐. Dynamics is concerned with iteration.We study sequences like {𝑧, 𝑝𝑐(𝑧), 𝑝𝑐(𝑝𝑐(𝑧)),…}, calledthe orbit of 𝑧. On the one hand, an easy exercise shows

that if |𝑧| > max{2, |𝑐|}, then this sequence convergesto infinity. On the other hand, the quadratic formulashows that there are fixed points. So there is a nonemptycompact subset 𝐾𝑐 consisting of points 𝑧 for which theorbit of 𝑧 is bounded. The boundary 𝐽𝑐 of this locus isthe typically fractal Julia set and is the locus of chaoticbehavior. We have 𝑧 ∈ 𝐽𝑐 if and only if the orbit of 𝑧 isbounded, but there are arbitrarily small perturbations of𝑧 for which the orbit becomes unbounded.

Looking at the polynomials 𝑝∘𝑛𝑐 (0) − 𝑝∘𝑚

𝑐 (0), 𝑛,𝑚 =0, 1, 2,…, we find there is a countably infinite set ofparameter values 𝑐 for which the forward orbit of thebranch point at the origin is finite. These critically finiteparameters play a crucial role in complex dynamicsand provide a rich source of examples of semigroupsBrMod(𝑆2, 𝑃).An ExampleLet 𝑔(𝑧) = 𝑧2+𝑖. There are branch points at the origin andthe point at infinity where 𝑔 is locally 2-to-1. The point atinfinity ∞ is fixed by 𝑔, and 0 ↦ 𝑖 ↦ 𝑖 − 1 ↦ −𝑖 ↦ 𝑖 − 1.Let’s take 𝑃 ∶= {𝑖, 𝑖 − 1,−𝑖,∞}; note that 𝑔(𝑃) ⊂ 𝑃. Wehave just constructed an element of BrMod(𝑆2, 𝑃).

How might we get other elements? We can do so bytwisting via pre- and post-composition with elements ofMod(𝑆2, 𝑃). Let’s look at all such elements:ℋ𝑔 ∶= {[ℎ0∘𝑔∘ℎ1] ∶ ℎ0, ℎ1 ∈ Mod(𝑆2, 𝑃)} ⊂ BrMod(𝑆2, 𝑃).This set decomposes into conjugacy classes. How manyare there? As shown in a groundbreaking paper by L.Bartholdi and V. Nekrashevych, the set of conjugacyclasses in ℋ𝑔 consists of two geometrizable elementsclassified by polynomials 𝑧2±𝑖 and a ℤ’s worth of classesrepresented by reducible elements that fix a curve in away that provides an obstruction to geometrization.

What if we use a different polynomial? Now let 𝑓(𝑧) =𝑧2+𝑐where 𝑐 is the unique parameter for which ℑ(𝑐) > 0and the origin is periodic of least period 3; now put𝑃 = {0, 𝑐, 𝑐2 + 𝑐,∞}. The polynomial 𝑓 is called theDouady Rabbit polynomial, since its Julia set (shown inthe left-hand side of Figure 4) looks a bit like a rabbit.Now the corresponding set of conjugacy classes consistsof just three geometrizable elements, classified by thethree values of 𝑐 for which the origin has period 3 underiteration of 𝑧2 + 𝑐.

G. Kelsey and R. Lodge have just announced anextension of these results to all quadratics with #𝑃 = 4.HighlightsThere are several results about the semigroupBrMod(𝑆2, 𝑃) that mirror results for the mappingclass group Mod(𝑆,𝑃).

A first point—one exploited by L. Bartholdi and V.Nekrashevych—is that branched mapping classes arefaithfully encoded by algebraic data known as wreathrecursions; like induced maps on fundamental groups,they are well defined up to conjugacy. As an illustration,equation (1) gives wreath recursion data for the Rabbit



Figure 2. A computer program reads the group-theoretic data faithfully encoding Douady’s Rabbit.It produces a numerical approximation for the poly-nomial map. It then draws an approximation to itsfractal Julia set on the sphere.

polynomial 𝑓(𝑧); here 𝑎, 𝑏, 𝑐 are free generators for thefundamental group of 𝑆2 −𝑃. The upshot is that we canrepresent and compute with elements of BrMod(𝑆2, 𝑃). Aprogram by L. Bartholdi takes this algebraic data as inputand returns a numerical approximation for the rationalgeometrization (if it exists) and an approximation of itsassociated Julia set; see Figure 2:

(1)𝑎 = ⟨𝑎−1𝑏−1, 𝑐𝑏𝑎⟩(12),𝑏 = ⟨𝑎, 1⟩,𝑐 = ⟨𝑏, 1⟩.

A second point is that our understanding of thecorresponding geometrization questions is advancingrapidly. In our setting, typical geometrizable elements ofdegree larger than one are represented bymaps which areexpanding in a suitable sense. Critically finite polynomialsand rational maps in this sense are expanding. However,there do exist nonrational expanding maps. L. BartholdiandD. Dudko give a characterization of expanding classes.Recently N. Selinger andM. Yampolsky and independentlyL. Bartholdi and D. Dudko have shown that a general mapdecomposes algorithmically into geometrizable pieces.

A third point is that the semigroup BrMod(𝑆2, 𝑃) actsnaturally on the space of conformal structures on (𝑆2, 𝑃).When #𝑃 = 4 this is again the hyperbolic planeℍ. Figure 3shows the limit set of the action of the subsemigroup of

Figure 3. The limit set of the subsemigroup ofBrMod(𝑆2, 𝑃) generated by Mod(𝑆2, 𝑃) and theDouady Rabbit polynomial 𝑓(𝑧) = 𝑧2 + 𝑐, drawn inthe disk model.

BrMod(𝑆2, 𝑃) generated by the Rabbit polynomial 𝑓 andthe group Mod(𝑆2, 𝑃).

Finally, the subject is immensely rich. In joint workwithW.Floyd,G.Kelsey, S.Koch,R. Lodge,W. Parry, andE. Saenz,we systematically study elements of BrMod(𝑆2, 𝑃), #𝑃 = 4,for which the branch points are simple. Such maps turnout to be closely related to affine torus maps, andthis observation leads to them being computationallytractable. A database of tens of thousands of examplesis now online at www.math.vt.edu/netmaps/index.php.Figure 4 illustrates the circle of ideas. The first two figures(top left) show respectively the fractal Julia set for the

(0, 0)

(2, 1)

(0, -1)

-2 -1 -.5 0 .5-1.5

-1.5 -1.0 -0.5-0.5

-1.0

-1.5

0.5

1.5

1.0

0.5 1.0 1.5

-2

-2 -1 −1 0−12

α1α1

α2

α2α3 α3σf

0 –

Figure 4. Follow the Rabbit from its fractal Julia setto its induced action on the upper-half-plane.



Rabbit polynomial 𝑓 and a diagram encoding the relationbetween 𝑓 and an affine torus map. The (right top) figuredeals with the geometry of the induced map 𝜎𝑓 ∶ ℍ → ℍ:there is a unique fixed point in the white region. The finalpair (bottom) illustrates that 𝜎𝑓 maps an ideal triangle toa Schwarz triangle and is extended by reflection.Image CreditsFigure 1 courtesy of Kevin M. Pilgrim.Figure 2 courtesy of L. Bartholdi.Figure 3 courtesy of S. Koch.Figure 4 courtesy of W. Floyd et al.Photo of Kevin M. Pilgrim courtesy of Indiana University

College of Arts and Sciences.



Bruce SaganThe Protean Chromatic PolynomialI am very excited to have the opportunity to share someof the ideas surrounding one of my favorite objectsin combinatorics, the chromatic polynomial, during myInvited Address. Let me start by defining each of theterms in my title.

The Merriam-Webster Dictionary defines “protean” as“of or resembling Proteus in having a varied nature orability to assume different forms.” In Greek mythology,Proteus was one of the gods of the sea and thus wasassociatedwith its constantly changing nature. In a similarmanner, the chromatic polynomial gives one informationabout many things which, a priori, have nothing to dowith its original purpose, as described below.

𝐺 =

𝑥 𝑤

𝑢 𝑣

𝐺 =

𝑥 𝑤

𝑢 𝑣

𝐺 =

𝑥 𝑤

𝑢 𝑣

Figure 1. A graph and two colorings

“Chromatic” refers to color, and our general topic isthe coloring of the vertices of a graph. A (combinatorial)graph, 𝐺, consists of a set of vertices 𝑉 and a set of edges𝐸 which connect pairs of vertices. For example, the graphin the upper left in Figure 1 has vertex set 𝑉 = {𝑢,𝑣,𝑤, 𝑥}and edge set 𝐸 = {𝑢𝑣,𝑢𝑥, 𝑣𝑥, 𝑣𝑤}. A coloring of 𝐺 is afunction 𝑐 ∶ 𝑉 → 𝑆 where 𝑆 is called the color set. Thecoloring is proper if the endpoints of every edge havedifferent colors. The coloring in the upper right of Figure 1is proper, while the one on the bottom is not because theedge 𝑒 = 𝑣𝑤 has the same color on both endpoints. Thechromatic number, 𝜒(𝐺), is the smallest number of colorsneeded to properly color 𝐺. The graph in Figure 1 has𝜒(𝐺) = 3 since the upper right image exhibits a propercoloring with three colors, and the triangle 𝑢𝑣𝑥 cannotbe colored with fewer colors. Maybe the most famoustheorem in graph theory is the Four Color Theorem,which states that if a graph is planar (can be drawn inthe plane without edge crossings), then 𝜒(𝐺) ≤ 4. Thisstatement was a conjecture for over a hundred years until

Bruce Sagan is professor of mathematics at Michigan State Uni-versity. His e-mail address is [email protected].


it was finally proved by Wolfgang Haken and KennethAppel in 1976. Their proof caused quite a stir in themathematical community, because it was the first to usea substantial amount of computing time, and the largenumber of cases could not all be checked by hand.

The chromatic polynomial was introduced in 1912 byGeorge Birkhoff as a possible tool for proving the thenFour Color Conjecture. Although it did not turn out to beuseful for the eventual proof, it has more than justifiedits existence through its many other applications. Let 𝑡 bea nonnegative integer. The chromatic polynomial, 𝑃(𝐺; 𝑡),is the number of proper colorings 𝑐 ∶ 𝑉 → {1, 2… , 𝑡}. Itis not apparent at first blush why this cardinality shouldbe called a polynomial. However, this will become clearerif we compute 𝑃(𝐺; 𝑡) for the graph in Figure 1. Supposewe color the vertices in the order 𝑢,𝑣,𝑤, 𝑥. Then thereare 𝑡 choices for the color of 𝑢 since it is the first vertexto be colored. After that, there will be 𝑡 − 1 choices forthe color of 𝑣, since it cannot be the same color as 𝑢.Similar reasoning shows that there are 𝑡 − 1 choices for𝑤. Finally, 𝑥 is adjacent to both 𝑢 and 𝑣, and these twovertices have different colors, so we can color 𝑥 in 𝑡 − 2ways. The net result is that

𝑃(𝐺; 𝑡) = 𝑡(𝑡 − 1)2(𝑡 − 2) = 𝑡4 − 4𝑡3 +5𝑡2 − 2𝑡,which is a polynomial in 𝑡, the number of colors!

One can show that 𝑃(𝐺; 𝑡) is always a polynomial in 𝑡and give nice characterizations of its degree, coefficients,andotherproperties. Furthermore, it has connectionswithmanyother objects of study, including acyclic orientationsof graphs, hyperplane arrangements, and even Chernclasses in algebraic geometry. I will explain these duringmy lecture, as well as present some recent work withJoshua Hallam and Jeremy Martin relating 𝑃(𝐺; 𝑡) to yetanother graphical concept, increasing spanning forests. Ifyou are at the Buffalo AMS Sectional Meeting in September,I hope to see you at my talk.

Image CreditsFigure 1 courtesy of Bruce Sagan.Photo of Bruce Sagan by Robert Chandler, courtesy

of Bruce Sagan.


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