www.msri.org

EMISSARYM a t h e m a t i c a l S c i e n c e s R e s e a r c h I n s t i t u t e

Spring 2020

Higher Categories and CategorificationDavid Gepner and Claudia Scheimbauer

Categories arise everywhere in mathematics: things we study usu-ally come with a natural notion of maps between them, such assets and functions between them, vector spaces and linear trans-formation, groups and homomorphisms, . . . In practice, there areoften many natural choices of maps: for instance, in geometry, onecan take for morphisms all functions, continuous functions, differ-entiable functions, smooth functions, polynomial functions, andothers.

The first concepts in category theory were formulated by Eilenbergand Mac Lane in the early 1940s in an attempt to make precise thenotion of naturality in mathematics. They were driven by examplescoming from algebraic topology: in particular, studying how points,paths, triangles, and so on, relate when we continuously deform thespaces in which they live. (See the figure on page 4.)

Formally, a category C consists of classes of objects {A,B,C, . . .}and morphisms {f,g,h, . . .} between objects, together with a rulefor composing (uniquely) morphisms with compatible source andtarget: if f :A!B and g :B!C are morphisms then g�f :A!C

Cutting manifolds into simple pieces is one way higher cate-gories arise naturally.

is again a morphism. This composition rule is associative — that is,(h�g)� f= h� (g� f) — and has identities.

Categories arise from geometric objects themselves. From a sur-face, or more generally a topological space X, we can produce a

(continued on page 4)

Mathical Inspires Children NationwideThe Mathical Book Prize, now in its sixthyear, is growing in scope and reach throughpartnerships and recognition across thecountry. Mathical books can be poetry, biog-raphy, fiction, nonfiction — anything both“lyrical and logical.”

They are math-inspiring literary books ofall sorts and types that appeal to children’swide-ranging interests: from magic to base-ball to cooking to nature to mystery to ad-

venture, and much more. They are engagingand energizing — and we hope you investi-gate them if you haven’t already!

You can find a list of this year’s Mathi-cal winnners (pictured below) along withnews about Mathical’s reach to low incomeschools and students as well as its growingnetwork of partners on page 9.

In response to the COVID-19pandemic, MSRI programs andworkshops for Spring 2020 havemoved online. For news aboutchanges to the institute’s Summer2020 activities, visit msri.org. Weare very grateful to all the organiz-ers and speakers for their support,patience, and invaluable contribu-tions to this transition.

The View from MSRIDavid Eisenbud, Director

Over the last year and a half, I’ve been reporting in this column onthe state of our National Science Foundation recompetition proposal.I’m happy now to announce what I think is the penultimate chapter:MSRI is being recommended by the Division of Mathematical Sci-ences for a new grant of $5 million per year for the next five years,the maximum allowed in the call for proposals!

Needless to say, I’m both delighted and relieved at the securitythis gives us, temporary though it is. The NSF grant is now aboutone-half of our budget — and about two-thirds of our direct sciencebudget — so its loss would have been catastrophic. I believe that theonly long-term path to security is developing a larger endowment.We’ve made a start — the endowment we’ve built up over the lastdozen years now provides about 10% of our total support — but wehave a long way to go until it can guarantee the long-term ability ofMSRI to continue serving the mathematical sciences community.

Scientific ActivitiesOur programs this spring are “Higher Categories and Categorifica-tion” and “Quantum Symmetries,” the latter having much to do withmonoidal categories. As a student of Saunders Mac Lane, I had afair grounding in category theory. I didn’t look beyond 2-categoriesat the time, old-fashioned in this day of infinity-categories. I won-der whether Mac Lane would have been surprised that the push forsuch an extension would come from homotopy theory. Some thingsdon’t change so much, though: I enjoy seeing lecturers draw thepentagonal coherence axiom for monoidal categories that Mac Lanediscovered and taught me 50+ years ago.

A scientific activity of a very different kind centers around theannual prize for mathematical contributions to finance and eco-nomics that we give jointly with the CME Group (formerly calledthe Chicago Mercantile Exchange). The winner this year is a starof the economics and machine learning communities, Susan Athey(see page 3). The award ceremony and panel, originally plannedfor April 7, will be rescheduled, with more information expected inSummer 2020.

Underrepresented Groups in MathematicsMSRI has substantially expanded its effort to help women and othercurrently underrepresented groups in mathematics stay engaged inresearch. One such initiative is simply called Summer Researchin Mathematics (SRIM). Small groups consisting predominantlyof women mathematicians that have an established research col-laboration apply to come for a couple of weeks to continue theirwork. We provide comprehensive support for childcare, which hasturned out to be a tremendous draw: 80 groups, comprised of nearly

300 researchers, applied this year, almost double the numbers fromlast year. Thanks to generous gifts from the Lyda Hill and McGov-ern Foundations, we have been able to accept about 25% of thoseapplications.

Another such initiative is the African Diaspora Joint Mathemat-ics Workshop (ADJOINT). It is a two-week summer program forresearchers with Ph.D.s in the mathematical sciences interestedin conducting research in a collegial environment. We ran a pi-lot program with NSF support in the summer of 2019, and thissummer, depending on COVID-19, we will host or e-host over 20African-American mathematicians through generous support bythe Sloan Foundation and the NSA. The participants will work insmall groups on five different projects, each with a well-establishedresearch leader. See msri.org/adjoint for more information.

SRIM and ADJOINT, together with four two-week summer gradu-ate schools and the six-week REU program MSRI-UP, now keepMSRI buzzing during the summer months more than ever before!

Popularization and Public Understanding

Readers of this column have heard before that we were workingon the film Secrets of the Surface: The Mathematical Vision ofMaryam Mirzakhani. It was directed by George Csicsery (who alsodirected N is a Number about Paul Erdös, Counting from Infinityabout Yitang Zhang, and Navajo Math Circles). The film is done! Ithad its world premiere at the Joint Mathematics Meetings this yearin Denver before a standing-room-only crowd, and it will soon havescreenings all over the world. See the article on the back page formore information.

Another of MSRI’s ventures in popularization, the Mathical BookPrize, has achieved a new level of success this year, through a con-tinuing grant from the Firedoll Foundation and a new grant fromthe McGovern Foundation. For example, the centennial meetingof the National Council of Teachers of Mathematics, April 1–4 inChicago, was poised to feature a large group of prizewinning Math-ical authors reading and talking about their work. (We hope thatMathical will continue to be among the rescheduled components ofthe meeting.) MSRI is now able to enlarge the distribution of thebooks, for example with grants to school libraries in underservedsettings. There’s an in-depth look at Mathical’s partners and itsreach to underserved communities, along with a list of this year’swinners, on page 9.

There’s lots more going on, both at and through MSRI. You cankeep track of us on the web or, better, by coming to participate in aworkshop or program. I hope to see you here before too long!

The View from MSRI 2CME Group–MSRI Prize 3Congressional Briefings 3Named Positions 3

Clay Scholars 3Pacific Journal of Math. 3HCC Program (cont’d) 4David Gepner 7Call for Proposals 7

Call for Membership 7Spring Postdocs 8Mathical Prize 9QS Program 10Dmitri Nikshych 12

Puzzles Column 13Upcoming Workshops 132019 Annual Report 14Math Circles Library 15Secrets of the Surface 16

Contents

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CME Group–MSRI PrizeThe 14th annual CME Group–MSRI Prize in Innovative Quanti-tative Applications was awarded to Susan Athey (pictured). Theaward ceremony and panel, originally scheduled for April 7, will berescheduled, with more information expected in Summer 2020.

Dr. Athey is the Economics of Technology Professor; Professor ofEconomics (by courtesy), School of Humanities and Sciences; and

Senior Fellow, Stanford Institute forEconomic Policy Research, StanfordBusiness School. Her current researchfocuses on the economics of digitization,marketplace design, and the intersectionof econometrics and machine learning.She has worked on several applicationareas, including timber auctions, inter-net search, online advertising, the newsmedia, and the application of digitaltechnology to social impact applications.

As one of the first “tech economists,” she served as consulting chiefeconomist for Microsoft Corporation for six years, and now serveson the boards of Expedia, Lending Club, Rover, Turo, and Rip-ple, as well as nonprofit Innovations for Poverty Action. She isthe founding director of the Golub Capital Social Impact Lab atStanford GSB, and associate director of the Stanford Institute forHuman-Centered Artificial Intelligence.

The annual CME Group–MSRI Prize is awarded to an individ-ual or a group to recognize originality and innovation in theuse of mathematical, statistical, or computational methods forthe study of the behavior of markets, and more broadly of eco-nomics. You can read more about the CME Group–MSRI Prize atat tinyurl.com/cme-msri.

Congressional BriefingsMSRI and the American Mathematical Society (AMS) host twocongressional briefings on mathematical topics each year in Wash-

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ington, DC, to inform members ofCongress and Congressional staffabout new developments made possiblethrough federal support of basic scienceresearch. On December 5, 2019, JillPipher (pictured), Vice President forResearch and Elisha Benjamin AndrewsProfessor of Mathematics at Brown Uni-versity, spoke to congressional staff andthe public about her research in her talk“No Longer Secure: Cryptography in the

Quantum Era.” Pipher traced the impact of modern cryptographyon network security, cybersecurity, financial transactions, privatecommunications, and digital currencies, outlining the complexenvironment the U.S. faces at the beginning of the quantum era.Learn more and view upcoming events: msri.org/congress.

Named Positions — Spring 2020MSRI is grateful for the generous support that comes from en-dowments and annual gifts that support faculty and postdocmembers of its programs each semester.

Chern, Eisenbud, and Simons ProfessorsQuantum SymmetriesTerry Gannon, University of AlbertaVaughan Jones, Vanderbilt UniversityVictor Ostrik, University of OregonSorin Popa, CNRS / University of California, Los AngelesKevin Walker, Microsoft Research Station QSarah Witherspoon, Texas A&M University

Higher Categories and CategorificationClark Barwick, University of EdinburghDavid Ben-Zvi, University of Texas, AustinDavid Gepner, University of MelbourneTeena Gerhardt, Michigan State UniversityEmily Riehl, Johns Hopkins UniversityUlrike Tillmann, Oxford University

Named Postdoctoral FellowsQuantum SymmetriesDella Pietra: Colleen Delaney, Indiana UniversityHuneke: Cris Negron, University of North CarolinaStrauch: David Reutter, Max-Planck-Institut für Mathematik

Higher Categories and CategorificationBerlekamp: Nicolle Sandoval González, UCLAStrauch: David Reutter, Max-Planck-Institut für MathematikViterbi: Alexander Campbell, Macquarie University

Clay Senior Scholars for 2020–21The Clay Mathematics Institute (www.claymath.org) SeniorScholar awards provide support for established mathemati-cians to play a leading role in a topical program at an insti-tute or university away from their home institution. Here arethe Clay Senior Scholars who will work at MSRI in 2020–21.

Random and Arithmetic Structures in Topology(Fall 2020)Uri Bader, Weizmann Institute

Decidability, Definability and Computability inNumber Theory (Fall 2020)François Loeser, Université Pierre et Marie Curie

Mathematical Problems in Fluid Dynamics(Spring 2021)Jean-Marc Delort, Université Paris 13

Pacific Journal of MathematicsFounded in 1951, The Pacific Journal of Mathematics has published mathematics research for more than 60 years.PJM is run by mathematicians from the Pacific Rim and aims to publish high-quality articles in all branches ofmathematics, at low cost to libraries and individuals. PJM publishes 12 issues per year. Please consider submittingarticles to PJM. The process is easy and responses are timely. See msp.org/publications/journals/#pjm.

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Higher Categories andCategorification(continued from page 1)

category whose objects are the points of X, and whose morphisms(between a pair of points) are the paths in X (formally, continuousmaps [0,1] ! X) between these points. Composition is by con-catenation of paths. Observe that there is a subtlety here: namely,the composition of the path ↵ : [0,1] ! X followed by the path� : [0,1] ⇠= [1,2]! X is a map � : [0,2]! X, which we may viewas a path after choosing a reparametrization [0,2] ⇠= [0,1]. Thereader may like to check for themselves that this composition is notassociative.

Surfaces produce a “category” whose objects are the pointsof the surface and morphisms are the paths on the surface.

From Categories to Higher Categories

Historically, the first attempt to fix this was to redefine the mor-phisms as equivalence classes of paths (path homotopies), leadingto the fundamental groupoid ⇧1(X). However, it is a fundamentalprinciple in higher category theory, the idea of which goes backat least to Grothendieck, to retain this information, leading to thefundamental 1-groupoid ⇧1(X) (which retains the homotopy typeof X). In particular, we now view a path homotopy as a morphismbetween morphisms, which we call a 2-morphism. More gener-ally, an n-morphism is a morphism between (n- 1)-morphisms.A higher category has objects, and n-morphisms for all n, whichcompose appropriately.

•••

• ••

(a) (b)

V

(c)

(a) Two objects, a morphism, and a 2-morphism in the fun-damental 1-groupoid of the torus; (b) a 2-morphism in ahigher category; (c) a composition of a 3-morphism and a2-morphism in a higher category.

Higher categories also arise when keeping track of the symme-tries of an object. For example, let us look at commutativity of abinary operation, say, addition or multiplication of any two realnumbers a and b. If a and b are natural numbers, we can lookat two sets A and B which have cardinality a and b, respectively.The identity a ·b= b ·a tells us that the cardinalities of A⇥B and

B⇥A are equal. But we can say more. If A = {a1,a2, . . .} andB= {b1,b2, . . .}, then

A⇥B= {(a1,b1),(a2,b1), . . .}

B⇥A= {(b1,a1),(b2,a1), . . .} ,

so we have a bijection 'A,B : A⇥B ! B⇥A, which is not theidentity. These bijections are natural in A and B, which we nowexplain.

Let Fin denote the category of finite sets and bijections. The assign-ments (A,B) 7! A⇥B and (A,B) 7! B⇥A are also compatiblewith morphisms, which is summarized by saying they are functorsFin ⇥Fin ! Fin . The assignments (A,B) 7!'A,B form a naturaltransformation of functors, which means that any maps f :A!A 0

and g : B! B 0 induce maps between both products, and all of thesemaps are compatible; that is, the square

A⇥B B⇥A

A 0⇥B 0 B 0⇥A 0

'A,B

f⇥g g⇥f

'A 0,B 0

commutes (going around either way yields the same result). Thisillustrates that categories themselves form a higher category: ob-jects are categories, morphisms are functors, and 2-morphisms arenatural transformations.

Geometry and Field Theory

Cobordism categories. Any 1-dimensional smooth manifold withboundary is a disjoint union of copies of the circle and the closedinterval. We can view them as morphisms in a category in the fol-lowing way: choose a decomposition of its boundary into two parts,which we will call incoming and outgoing boundaries, respectively,to obtain a cobordism. The objects in the category are finite sets ofpoints (viewed as a 0-dimensional manifold); in particular, the in-and outgoing boundaries are objects. We compose by gluing along acommon boundary. (Technically, we should identify diffeomorphiccobordisms for this to be well-defined.)

For example, consider the following two diagrams:

; ;;

The left figure depicts a morphism from two points to the empty set,where the arrow indicates the direction. The right figure depicts acomposition of two elbows along two points, giving a circle. Theviewpoint of thinking of this as cutting a circle into easier pieceswill turn out to be useful below.

This category is called the category of cobordisms. It comes inmany different flavors: we can increase the dimension n of the man-ifolds and add orientations, framings, spin structures, or even non-topological, that is, geometric, decorations. We will use the notationBordn in the following to denote generically one of these cobordismcategories or its higher cousins, which we will encounter below.

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Furthermore, taking disjoint unions gives a symmetric monoidalstructure.

Duality. The symmetric monoidal (higher) category Bordn enjoysa universal property, relating the geometry to algebraic properties.To see this, let us begin with the prototypical example.

Consider the category of (real) vector spaces Vect . Let’s look at theobject Rn, and observe that we have linear maps

A : Rn2⇠= Rn⌦Rn⇠= Hom(Rn,R)⌦Rn!R, (',v) 7!'(v),

B : R!Hom(Rn,Rn) ⇠= Rn⌦Hom(Rn,R), 1 7! id 2 Hom(Rn,Rn).

Let us pick the standard basis {ei} of Rn and denote vectors in thedual basis by ei, so that ei(ej) is 1 if i= j and 0 otherwise. Theyform a basis of Hom(Rn,R) ⇠= Rn. The maps A and B enjoy anice property (which we will label †):

Rn B⌦id�! Rn⌦Hom(Rn,R)⌦Rn id⌦A�! Rn,

ej 7�! (P

i ei⌦ei)⌦ej 7�!P

i ei⌦ (ei(ej)) = ej,

so this composition is the identity! Similarly, we could havestarted with Hom(Rn,R) and first used id⌦B and then A⌦ id,and we would also have gotten the identity. This property exhibitsHom(Rn,R) as a dual of Rn.

We can now ask the question: when does a vector space V havea dual in the sense that there is another vector space W and mapscoev : R ! V ⌦W and ev : W⌦V ! R, satisfying the triangleidentities (namely, the two compositions alluded to above are theidentity). The answer is: exactly for the finite-dimensional vectorspaces!

Our definition of having a dual used that the category of vectorspaces has a tensor product ⌦, so we can repeat the definition inany monoidal category C, a monoidal 1-category, or even in amonoidal (1,n)-category. Moreover, it is an important insight thatthis notion is an instance of having adjoints in a higher category(which is analogous to a functor having an adjoint).

This is the lowest-dimensional example of a beautiful and deepconnection between higher category theory and smooth manifolds,called the cobordism hypothesis.

Topological field theories. A mathematical framework for fieldtheories following Atiyah and Segal is to define it to be a symmetricmonoidal functor

Z : Bordn �! Vect ,

which, roughly speaking, assigns a vector space of states to an object(space) and a linear evolution operator to a cobordism (spacetime).

The cobordism hypothesis now says that a one-dimensional orientedtopological field theory corresponds exactly to a vector space Vwhich has a dual. Concretely, it assigns

(evV :V_⌦V ! C)

(coevV : C!V⌦V_)and ⇠= ⇠=

unravel to exactly our property (†) from above.

There are two ways in which higher categories appear naturallywhen studying bordism categories. First, when increasing the di-mension, we might like to cut our manifolds into simple pieces bycutting in several directions, as shown in the figure of the torus onthe cover. This naturally has the structure of an n-category, forwhich we have objects, morphisms, and morphisms between mor-phisms (called 2-morphisms), and so on. Then each of the pieces inthe figure are 2-morphisms in this higher category of cobordisms.The boundary of each of these pieces is one-dimensional, albeit inan interesting way. One can read off an incoming and outgoing parton the left and right, and each of these parts themselves look likethe pictures we had before.

Second, we might want to consider a moduli space of n-dimensionalmanifolds rather than a set thereof, by including information aboutdiffeomorphisms in the form of a classifying space. The mo-tivation from physics for this is that the theory should dependsmoothly on the space or spacetime. This is implemented by using(1,n)-categories as the categorical framework. We denote thesymmetric monoidal (1,n)-category of cobordisms equipped withn-framings by Bord fr

n . A fully extended n-dimensional topologi-cal field theory valued in C is a symmetric monoidal functor fromBord fr

n to C . With this at hand, the Baez–Dolan–Lurie cobordismhypothesis reads as follows:

Cobordism hypothesis. Evaluation at a point with a standardn-framing gives an equivalence of 1-groupoids betweenfully extended n-dimensional topological field theories andn-dualizable objects in C .

The condition of being n-dualizable is a generalization of havinga dual, and should be thought of as a categorical, hence algebraiccondition. In Bordn the 2-dualizability of a point can be visual-ized by the following diffeomorphism between (compositions of)cobordisms:

⇠=

Thus, a higher categorical property can be understood via geometry,and vice versa.

Several research members of this semester’s program work onprojects related to the cobordism hypothesis; on the proof andvariations, on verifying dualizability in various settings and appli-cations, in particular to topological field theories, knot theory, andrepresentation theory. The latter also has interesting connections toresearch interests of our partner program on Quantum Symmetries.

Algebra: Groups and SpheresIf X is a space with a chosen basepoint x0 2 X, elements of thekth homotopy group ⇡k(X,x0) are represented by continuous mapsf : Dk ! X, which send the boundary to x0. (Any two such mapsrepresent the same elements if they are homotopic, that is, if onecan be continuously deformed into the other.) In fact, the elements

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of ⇡k(X,x0) form a group, which is even commutative if k > 1.The conceptual reason for this is fundamentally higher categoricalin nature, as we illustrate below.

The group structure is given as follows: consider a pair of el-ements of ⇡k(X,x0), represented by a pair of continuous mapsf,g : Dk ! X that send everything on the boundary of these disksto x0. Now embed these two disks disjointly into a bigger disk. Theproduct of these two elements is represented by the continuous mapDk ! X obtained by sending everything outside the little disks tothe basepoint x0 of X. However, this depends on the embedding ofthe little disks into the bigger disk. By deforming the embedding,we can assume that one of the disks sits at the origin. The otherdisk is free to move around the complement, which is homotopic toSk-1.

If k = 1, this is a discrete space, so that we cannot move the freedisk past the fixed one, and indeed the so-called fundamental group⇡1(X,x0) need not be commutative! For k > 1, this is a connectedspace, so we can always move the free disk past the fixed one, butin an Sk-1-parameter space worth of ways. This space is invisi-ble in ⇡k(X,x0) because we identified homotopic maps, makingit commutative. (This is the classical Eckmann–Hilton argument.)However, if we retain this information, which is natural from theperspective of higher category theory, we obtain something that ismore than associative, but not quite commutative.

• •

Moving disks in the discrete space S0 (left) and in the con-nected space Sk-1 (right).

As k!1, the space Sk-1 becomes increasingly connective, andin the limit, S1 is contractible, so that (in the appropriate highercategorical sense) there is a unique witness for the commutativityrelation [↵] · [�] = [�] · [↵]. This behavior can be formalized in termsof the little k-disks operad, a fundamental object of study by variousmembers of this semester’s program. It records the embeddings ofk-dimensional disks, allowing for successive embeddings, as in thefigure at the top of the next column.

Higher Algebraic StructuresAlgebraic structures, such as groups and rings, also exist highercategorically. The higher categorical notion of a group is a (based)loop space. That is, the space

⌦X := Map⇤(S1,X)

of loops in X, based at the basepoint x0 2 X, admits a group struc-ture in which the multiplication is given by concatenation of loops.This is only a group in the higher categorical sense, as this opera-tion requires a choice of reparametrization of the circle. Note theresemblance to the fundamental group: indeed, ⇡1X ⇠= ⇡0⌦X. Thegroup structure follows from what is arguably the most importantresult in basic category theory, the Yoneda lemma.

A composition of operations in the little 2-disks operad (left),and a depiction of three little 3-disks embedding into a larger3-disk (right).

If G is a group (or even an 1-group), one can build a contractiblespace, called EG, on which G acts freely. The quotient spaceBG = EG/G then fits into a fibration sequence G ! EG ! BG.Since ⇡nEG ⇠= 0 for all n, we have ⇡n+1BG ⇠= ⇡nG, so thatG'⌦BG.

From this point of view, n-fold loop spaces X0 '⌦X1 '⌦2X2 '· · · '⌦nXn correspond to spaces with n operations which com-mute “up to Sn-1 ⇢ Rn”. As n!1, the space Sn-1 becomescontractible, so that an infinite loop space is the higher categoricalincarnation of a commutative group. The precise relation betweenthese commutative 1-groups and spectra is as follows: any com-mutative 1-group G naturally defines a spectrum {G ' ⌦BG '⌦2B2G · · · }, and a spectrum X = {X0 ' ⌦X1 ' ⌦2X2 · · · } is inthe image of this functor if and only if ⇡mXn = 0 for m<n.

Classically, a commutative group is the same thing as a Z-module.Higher categorically, Z is no longer the unit of the tensor producton commutative 1-groups (or spectra), as there is a more initialring, denoted S, and called the sphere spectrum. (This is not aclassical ring; rather, it has nontrivial higher homotopy groups, thecalculation of which is a major open problem in algebraic topology.)The reason for this, ignoring inverses for the moment, is that thegroupoid Fin of finite sets and bijections is the free commutativemonoid on one generator, and not N. While the group completionof N is Z, the group completion of Fin is S. The reason for thisseemingly mysterious terminology is that the underlying space of Sis the union

S' colimn!1 Map⇤(Sn,Sn)

of the spaces of (pointed) self-maps of the n-sphere Sn.

As envisioned by Waldhausen, one can do brave new algebra overthe sphere, or, more generally, a commutative ring spectrum, just asclassically one can do ordinary algebra over the integers or someother base commutative ring. This has led to a paradigm shift in sev-eral areas of mathematics. For instance, Waldhausen showed that ifM is a pointed manifold, then ⌦M is an 1-group, and the K-theoryof the group-ring spectrum K(S[⌦M]) (also obtained by a groupcompletion process) is related to the diffeomorphism groups of M.In fact, the cohomology theory represented by the sphere spectrumS is framed cobordism, another mysterious connection betweenalgebra and geometry. Another derived algebraic invariant of a ringR, called its topological Hochschild homology THH(R), appears innumber theory and arithmetic algebraic geometry in several surpris-ing ways. This is an active research area, and features prominentlyin this semester’s program and its affiliated workshops.

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Focus on the Scientist: David GepnerDavid Gepner is an Eisenbud Professor in this semester’s pro-gram on Higher Categories and Categorification. He has madedynamic contributions in and around higher categories, withwork ranging from higher algebraic K-theory and derived al-gebraic geometry, through stable, equivariant, and motivic

David Gepner

homotopy theory, to the foundationsof 1-category theory. An enthu-siastic collaborator with seventeencoauthors and counting, Gepner hasemerged as one of the strongest andmost versatile homotopy theoristsof his generation, with a lengthy listof highly-cited papers on a broadrange of topics.

David grew up in the western sub-urbs of Chicago where he liveduntil attending Reed College. Heearned his Ph.D. in 2006 from theUniversity of Illinois at Urbana-Champaign. He has held faculty positions at the Universityof Sheffield, the University of Illinois at Chicago, UniversitätRegensburg, Purdue University, and the University of Melbourne,where he is currently based.

David’s most highly-cited work includes “A universal charac-terization of higher algebraic K-theory” with Andrew Blumberg

and Gonçalo Tabuada; a pair of famous papers with Matt Ando,Andrew Blumberg, Mike Hopkins, and Charles Rezk on Thomspectra; and a foundational paper, “Enriched 1-categories vianon-symmetric 1-operads,” joint with Rune Haugseng, who isalso in residence at MSRI this semester.

While it’s not possible to convey the depth and originality ofDavid’s work in this short space, a recent paper, “1-Operadsas analytic monads,” with Rune Haugseng and Joachim Kock,highlights one of his myriad contributions to the foundations ofhigher category theory. The paper addresses classical operadslike the “little k-discs operad” and the monad endofunctors thatare built from an operad’s composition maps. Gepner, Haugseng,and Kock show that when operads and monads are redevelopedin the less rigid setting of infinite-dimensional category theory,an 1-operad can in fact be recovered from its correspondingendofunctor and those endofunctors that arise from 1-operadsare exactly those that define analytic 1-monads (Cartesian mon-ads whose underlying endofunctors preserve sifted colimits andweakly contractible limits).

David brings the same spirit of exploration and collaboration tohis extracurricular interests, as a pianist who enjoys dabbling inall genres of music. He is grateful to spend yet another semesterat MSRI, and his colleagues are all grateful for the opportunityto work with and learn from him!

— Emily Riehl

Call for ProposalsAll proposals can be submitted to the Director or Deputy Directoror any member of the Scientific Advisory Committee with a copyto proposals@msri.org. For detailed information, please see thewebsite msri.org/proposals.

Thematic ProgramsThe Scientific Advisory Committee (SAC) of the Institute meetsin January, May, and November each year to consider letters ofintent, pre-proposals, and proposals for programs. The deadlines tosubmit proposals of any kind for review by the SAC are March 1,October 1, and December 1. Successful proposals are usually de-veloped from the pre-proposal in a collaborative process betweenthe proposers, the Directorate, and the SAC, and may be consideredat more than one meeting of the SAC before selection. For completedetails, see tinyurl.com/msri-progprop.

Hot Topics WorkshopsEach year MSRI runs a week-long workshop on some area of in-tense mathematical activity chosen the previous fall. Proposalsshould be received by March 1, October 1, and December 1 forreview at the upcoming SAC meeting. See tinyurl.com/msri-htw.

Summer Graduate SchoolsEvery summer MSRI organizes several two-week long summergraduate workshops, most of which are held at MSRI. Proposalsmust be submitted by March 1, October 1, and December 1 forreview at the upcoming SAC meeting. See tinyurl.com/msri-sgs.

Call for MembershipMSRI invites membership applications for the 2021–22 academicyear in these positions:

Research Professors by October 1, 2020Research Members by December 1, 2020Postdoctoral Fellows by December 1, 2020

In the academic year 2021–22, the research programs are:

Universality and Integrability in Random Matrix Theory andInteracting Particle SystemsAug 16–Dec 17, 2021Organized by Ivan Corwin, Percy Deift, Ioana Dumitriu, AliceGuionnet, Alexander Its, Herbert Spohn, Horng-Tzer Yau

The Analysis and Geometry of Random SpacesJan 18–May 27, 2022Organized by Mario Bonk, Joan Lind, Steffen Rohde, Eero Saks-man, Fredrik Viklund, Jang-Mei Wu

Complex Dynamics: From Special Families to Natural Gener-alizations in One and Several VariablesJan 18–May 27, 2022Organized by Sarah Koch, Jasmin Raissy, Dierk Schleicher, Mit-suhiro Shishikura, Dylan Thurston

MSRI uses MathJobs to process applications for its positions. Inter-ested candidates must apply online at mathjobs.org after August 1,2020. For more information about any of the programs, please seemsri.org/programs.

7

Named Postdocs —Spring 2020BerlekampNicolle González is the Berlekamp Postoc-toral Fellow in the Higher Categories andCategorification program at MSRI. Nicollegrew up on the border between Venezuelaand Colombia. Shecompleted her un-dergraduate studiesat the University ofOregon and did herPh.D. at the Uni-versity of South-ern California un-der the supervisionof Aaron Laudaand Sami Assaf. One important result ofher thesis is a categorification of the famedBoson–Fermion correspondence. She is nowa University of California Presidential Post-doctoral Fellow at UCLA, under the mentor-ship of Raphaël Rouquier. Nicolle’s princi-pal research interests lie in the categorifica-tion of structures arising in representationtheory, topology, and combinatorics, in par-ticular those related to actions of Heisenbergalgebras and the theory of Macdonald poly-nomials. The Berlekamp Postdoctoral Fel-lowship was established in 2014 by a groupof Elwyn Berlekamp’s friends, colleagues,and former students whose lives he touchedin many ways. He was well known for hisalgorithms in coding theory, important con-tributions to game theory, and his love ofmathematical puzzles.

Stephen Della PietraColleen Delaney is the Stephen Della PietraPostdoctoral Fellow in the Quantum Sym-metries program. Colleen was an undergrad-uate physics major in Ricketts Hovse at Cal-tech before pursuing a Ph.D. in mathematicswith Zhenghan Wang at UC Santa Barbaraas an NSF Graduate Research Fellow andMicrosoft Station Q Graduate Research Fel-low. After the Quantum Symmetries pro-gram they will resume their postdoctoralposition at Indiana University with JuliaPlavnik and Noah Snyder as an NSF Mathe-matical Sciences Postdoctoral Research Fel-low. Colleen’s research interests lie in al-gebraic and topological aspects of quantumfield theory, most recently in applying fusioncategories to better understand the physics ofquasiparticles and defects in topologicallyordered quantum phases of matter in twospatial dimensions. They also work with

computational approaches to quantum in-variants of knots and links and their cate-gorifications to homology theories. Colleenrequested not to have a photo run with thisprofile. The Stephen Della Pietra fellow-ship was established in 2017 by the DellaPietra Family Foundation. Stephen is a part-ner at Renaissance Technologies, a boardmember of the Simons Center for Geometryand Physics, and treasurer of the NationalMuseum of Mathematics in New York.

HunekeCris Negron is the Huneke Postdoctoral Fel-low in this semester’s Quantum Symmetriesprogram. Cris received his Ph.D. from theUniversity of Washington in 2015, under thesupervision ofJames Zhang. Hewas an NSF post-doc at LouisianaState University,and a CLE Mooreinstructor at theMassachusetts In-stitute of Technol-ogy. He recentlyjoined the faculty at the University of NorthCarolina, Chapel Hill, as an assistant profes-sor. In his earlier works, Cris studied derivedinvariants for near-geometric objects suchas orbifolds and Azumaya algebras. Morerecently, he has studied exotic quantumgroups and their related logarithmic confor-mal field theories. He has also worked onemerging geometric approaches to studiesof finite tensor categories. The Huneke Post-doctoral Fellowship is funded by a generousendowment from Professor Craig Huneke,who is internationally recognized for hiswork in commutative algebra and algebraicgeometry.

StrauchDavid Reutter is this semester’s StrauchPostdoctoral Fellow in both of this Spring’sprograms at MSRI,Higher Categoriesand Categorifica-tion and QuantumSymmetries. Davidearned his bach-elor’s and mas-ter’s degrees inphysics from theETH Zürich andcompleted Part 3 of the Mathematics Tri-pos at Cambridge before completing hisOxford D.Phil. in Computer Science underthe direction of Jamie Vicary in 2019. Heis currently a postdoctoral fellow at the

Max Planck Institute for Mathematics underthe mentorship of Peter Teichner, who isalso in residence at MSRI this semester. Inaddition to his D.Phil. dissertation, Davidis the author of nine papers in quantuminformation theory and mathematics on theboundary of higher category theory andtopology, most recently settling in the neg-ative the quarter-century old question ofwhether Crane–Yetter theory, constructedfrom what is now called a non-modularbraided fusion category, could detect exoticsmooth structures. David is very involved inpublic mathematical outreach, including thedevelopment of a public workshop in whichparticipants explore quantum concepts us-ing hand-held q-bit simulators. The StrauchFellowship is funded by a generous annualgift from Roger Strauch, Chairman of TheRoda Group. He is a member of the Engi-neering Dean’s College Advisory Boards ofUC Berkeley and Cornell University, and isalso currently the chair of MSRI’s Board ofTrustees.

ViterbiAlexander Campbell is this semester’sViterbi Postdoctoral Fellow and a memberof the Higher Categories and Categorifica-tion program. Alexander is a category the-orist from Sydney,Australia. From2009–12, he wasan undergraduate atthe University ofSydney, and from2013–19, he was amember of the Cen-tre of AustralianCategory Theory atMacquarie University, first as a Ph.D. stu-dent under the supervision of Ross Street,and subsequently as a postdoctoral researchfellow with Richard Garner. Alexander’sresearch interests are in category theory, es-pecially higher and enriched category the-ory, and categorical aspects of homotopytheory. In 2019, he solved two longstand-ing open problems in the theory of quasi-categories (aka, infinity-categories) posedby André Joyal. While at MSRI, Alexanderis working on a number of projects concern-ing higher infinity-categories jointly withother members of the Higher Categories pro-gram. The Viterbi postdoctoral fellowshipis funded by a generous endowment fromDr. Andrew Viterbi, well known as the co-inventor of Code Division Multiple Accessbased digital cellular technology and theViterbi decoding algorithm, used in manydigital communication systems.

8

Mathical Continues to Grow in Stature and Reach

A new collection of Mathical titles at the ELG Rapid RehousingCenter, a family homeless shelter in the Bronx. Image courtesyof the Books for Kids Foundation.

Core and Founding PartnersMathical is hitting a growth spurt, with greater recognition amongpublishers, librarians, teachers, and parents. We are grateful toour core partners. One is the Children’s Book Council, whichhelps us get the word out to publishers; others are the NationalCouncil of Teachers of English (NCTE) and the National Coun-cil of Teachers of Mathematics (NCTM), for opening doors witheducators.

NCTM teamed up with MSRI, and together we were poised tohave eight Mathical authors at the NCTM Centennial conferencein Chicago, in April 2020. Many components of the event havebeen postponed to Fall 2020, and we hope that Mathical contentwill continue to be among them.

We are also grateful to the Simons Foundation for its foundingsupport of the prize, and for support along the way from manyincluding the Heising-Simons Foundation and individual donorsincluding former MSRI board chair Roger Strauch, who believedstrongly in the mission of the prize from the start.

For the newest batch of Mathical winners (see inset; their coversare shown on this issue’s cover), we took the announcement eventonline in March — with great success! — and you can find the videoat tinyurl.com/mathicalCIME2020.

Reaching Low-Income Schools and ChildrenMSRI has a new partnership with the School Library Journal, asavvy group who are helping us reach low-income kids at Title Ischools by administering a new joint program, the Mathical BookPrize Collection Development Awards. With their help, up to 25schools nationwide will receive $700 awards this spring to purchaselarge sections of the Mathical List for their school libraries.

MSRI is working newly with the Books for Kids Foundation todistribute Mathical titles nationally this spring to libraries serving

low-income preschool children. In addition, the Books for KidsFoundation is placing a range of Mathical titles in newly establishedfamily libraries in homeless shelters in New York City.

Both of these new partnerships are made possible through the gen-erous support of the Patrick J. McGovern Foundation.

MSRI also continues our longstanding partnership with First Book,which provides high-quality new books and educational resources tolow-income children in programs and schools, reaching an estimatedone in three children in need through its ever-growing national net-work. Our work with them focuses on kids in need in the Bay Area,and is generously supported by the Firedoll Foundation.

A Growing Network of PartnersMathical’s other partnerships are almost too many to mention, butwe will try to remember them all:

The Association of Children’s Museums is hosting a webinar formuseum educators this month to learn more about Mathical andearly math learning; Bring Me a Book, a Bay Area nonprofit, isworking with MSRI to exchange resources and information; Booke-licious is a new app for children’s book pairings that will launch inQ2 of 2020; Development and Research in Early Math Educa-tion (DREME, out of Stanford) is working together with MSRI tocreate more reading guides for Mathical titles to make it easier forparents and non-math teachers to enjoy teaching the books; we areworking with the Internet Archive to exploring getting Mathicaltitles online in a new way; Reach Out and Read shares Mathicalinformation with physicians in early child well visits; TeachingBooks features a Mathical collection that serves as a pointer foreducators looking to stock their classrooms.

2020 Mathical Book Prize WinnersMSRI’s Mathical Book Prize recog-

nizes outstanding fiction and literarynonfiction for youth aged 2–18. Theprize is selected annually by acommittee of pre-K-12 teachers,librarians, mathematicians, early

childhood experts, and others. Thisyear’s winners are:

Pre-K: One Fox: A Counting Book Thriller by Kate ReadGrades K–2: Pigeon Math by Asia CitroGrades 3–5: Solving for M by Jennifer SwenderGrades 9–12: Slay by Brittney Morris

Have a book you hope we’ll consider? Everyone can submityour favorite math-inspiring titles for possible inclusion as aMathical Honor Book or in the Mathical Hall of Fame.

The Mathical Book Prize is awarded by MSRI in partnershipwith the National Council of Teachers of English and theNational Council of Teachers of Mathematics, and in coordi-nation with the Children’s Book Council. The Mathical listis intended as a resource for educators, parents, librarians,children, and teens. Download the list at mathicalbooks.org.

9

Quantum Subgroups: Back to the FutureTerry Gannon and Cris Negron

In order to make sense of modern physics, mathematicians are beingchallenged to generalize their classical notions. Quantum symme-tries, the theme of this semester’s program, is what is evolvingfrom groups and Lie theory. It covers a wide range: quantum com-puters, string theory, knots, subfactors, topological quantum fieldtheory, Hopf algebras, tensor categories, . . . The following articlefocuses on one influential storyline running through the history ofthe subject, which has been shaped by several participants of thisprogram.

Modular Invariants and the Classification ofConformal Field TheoriesThe language of modern physics is quantum field theory, and un-derstanding it is a major challenge for mathematics. The mostaccessible class of quantum field theories are the conformal fieldtheories (CFTs), as they are especially simple (only one space andone time dimension), and especially symmetrical (the conformaltransformations are the maximal space-time symmetries allowed byrelativity). CFTs arise for instance in perturbative string theory andcondensed matter.

In 1986 Cappelli, Itzykson, and Zuber addressed the classificationof the possible (rational) CFTs associated to the Lie algebra sl(2).They reduced it to the following problem:

For each integer > 3, define matrices

Sa,b =

r2

sin(

⇡ab

) , Ta,b = �a,be

-⇡i/4e⇡ia2/2 ,

for 16 a,b < . Find all matrices M such that(i) SM=MS and TM=MT ;(ii) Ma,b 2 Z>0;(iii) M1,1 = 1.

Any such matrix M is called a modular invariant for sl(2). Equiva-lently, we can write M in the form

Z=-1X

a,b=1

Ma,b�a�⇤b

for formal variables �a,�⇤b. For example, the identity M = I

is a modular invariant, corresponding to Z =P-1

a=1�a�⇤a =P-1

a=1 |�a|2.

Intriguingly, the list of sl(2) modular invariants fits into an A-D-Epattern: two infinite sequences of graphs Am and Dn, as well asthree exceptionals called E6,E7,E8. Some of these are displayed inthe figure in the next column. A-D-E is a common answer to seem-ingly different questions in math and mathematical physics. Theyclassify the simply-laced simple Lie algebras, and also parametrizethe finite subgroups of the matrix group SL2(C). The key propertyof these graphs, which is responsible for their ubiquity, is that theiradjacency matrices have largest eigenvalue < 2.

Each solution M can be assigned an A-D-E graph which sharessome of its properties: is the so-called Coxeter number of the

graph, and the values of a with Ma,a 6= 0 are its so-called expo-nents. For example, M= I is assigned the A-1 graph, while D5

and E8 are assigned to

Z= |�1|2+ |�3|

2+ |�4|2+ |�5|

2+ |�7|2+�2�

⇤6+�6�

⇤2 ,

Z= |�1+�11+�19+�29|2+ |�7+�13+�17+�23|

2 ,

which work for = 8 and 30, respectively.

A CFT itself consists of two halves, called vertex operator algebras(VOAs), and a matching of the irreducible representations of thoseVOAs. A VOA is an infinite-dimensional space with infinitely manybilinear products — a complicated enough beast that its formal defi-nition is best avoided. parametrizes the different sl(2)-type VOAsV(sl(2),). The S and T matrices define a representation of themodular group SL2(Z) fundamental to the theory. The modularinvariant explains how to extend V(sl(2),) to get those two halvesand gives the matching.

For example, the A-1 modular invariants involve trivial VOAextensions and the identity matching. For E8, both extensions are tothe VOA V(G2,5), with trivial matching. The D5 modular invarianthas trivial extensions but nontrivial matching.

A4 D5 E6

E7 E8

The A-D-E diagrams.

Nim-Reps and Boundary CFT

To deepen the connection between graphs and modular invariants,we need to give meaning to the edges. One classical story, McKay’sassociation of graphs to finite subgroups of SL2(C), suggests how.Consider the symmetry group of the icosahedron for concrete-ness. This size-120 subgroup has nine irreducible representations⇢1,⇢2,⇢2 0 , . . . ,⇢6 (labelled by dimension), which form the vertices.Call ⇢2 the defining representation in SL2(C). Draw a directededge from vertex a to b if ⇢b occurs in the decomposition of thetensor product ⇢2⌦⇢a. For example, ⇢2⌦⇢2 ⇠= ⇢1�⇢3, so wedraw directed edges from vertex 2 to vertices 1 and 3. The arrowson an edge directed in both directions are erased. The resultinggraph is extended E8. The largest eigenvalue of this graph (namely,2) is the dimension of ⇢2.

Something similar happens in CFT. The VOA representations(�1, . . . ,�-1 in our case) act on the boundary states of the the-ory. This action is called a nim-rep, short for “nonnegative integermatrix representation.” The number of indecomposable boundarystates equals the trace of the modular invariant. If we draw the

10

McKay graph for this nim-rep, with vertices labelled by the bound-ary states and edges describing how �2 acts, we recover preciselythe A-D-E graph assigned to that modular invariant. The largesteigenvalue of the graph is the quantum-dimension 2cos(⇡/) ofrepresentation �2. The A-D-E classification of sl(2) CFT is a sortof quantum-deformation of the A-D-E classification of subgroupsof SL2(C), hence the term, quantum subgroups.

2 0 4 0 6 5 4 3 2 1

3 0

extended E8

McKay graph for binary icosahedral group.

Galois and Higher Rank

Because the sl(2) classification is so interesting, we should con-sider the modular invariants for other Lie algebras g. Again, foreach , we get another VOA V(g,), with its own S and T matrices.Unfortunately, the method of Cappelli–Itzykson–Zuber completelyfails even for sl(3). New ideas were needed to explore the modularinvariant classifications for higher rank g.

It was discovered within this context that the S and T matrices al-ways have a very special Galois symmetry. For sl(2), this impliesthe following condition: M1,a 6= 0 requires

cos✓⇡`

a-1

◆> cos

✓⇡`

a+1

◆

for all ` coprime to . Using this condition, the proof of the clas-sification for sl(2) can be shortened by a factor of over 15. Moreimportantly, this Galois symmetry led in 1994 to the classificationof sl(3) modular invariants. These don’t fall into an A-D-E pat-tern, nor seem connected to finite subgroups of SL3(C), but havemysterious relations with simple factors in the Jacobians of Fermatcurves. A nim-rep graph for sl(3) is given in the figure below.

Nim-rep graph for M= I and V(sl(3),6).

For higher rank Lie algebras, the Galois condition remains powerful,but is hard to use. So for a while, the modular invariant classifica-tions stopped at sl(3). The hard part was controlling the possibleVOA extensions of V(g,). New ideas again were needed.

Operator Algebra Takes OverAlready in 1990, Vaughan Jones in his ICM plenary talk mentionedthe Cappelli–Itzykson–Zuber result as hinting at deep connectionsbetween subfactors and CFT. The next decade or so fleshed out arich structure underlying modular invariants and their associatednim-reps. VOA representations �a and CFT boundary states weremodelled by morphisms between von Neumann algebras, so thattensor product simplifies to composition. But perhaps the mostimportant discovery was alpha-induction.

Classically, group representations restrict to subgroup representa-tions. This preserves dimension and respects tensor product. Ingroup theory, restriction has an adjoint, called induction, whichcanonically lifts a representation of the subgroup to that of thegroup. Implicit in modular invariants are VOA extensions; is therea VOA analogue of induction?

The answer the subfactor community discovered was yes, thoughwith a difference. The target of this new induction isn’t the represen-tations of the larger VOA, but a larger, looser class called twistedrepresentations. Here induction, and not restriction, preserves di-mensions and respects tensor product.

For concreteness consider the E8 modular invariant. The verticescorrespond to the twisted representations of the VOA V(G2,5). Themodular invariant only sees the true representations (“11” and “12”).Beside each node are the restrictions to V(sl(2),30). Representa-tion �2 of V(sl(2),30) induces to the twisted representation ⌧1, sowe find that, for example, ⌧ 0

1⌦⌧ 01⇠= (11)�⌧ 0

1.

The Categorical InterpretationWe now know, based on a number of fundamental works from theearly 2000s, that the representation theory of the VOA V(g,) formsa modular tensor category (MTC), and the twisted representationsform a fusion category. The nim-rep and modular invariant arecaptured by a module category for the MTC. (A module categoryover an MTC is the straightforward categorification of the notionof a module over a commutative ring.) The matching mentionedearlier is a braided tensor equivalence between the MTCs of thetwo extended VOAs. Alpha-induction is a tensor functor (adjoint torestriction) from the MTC of say V(sl(2),30) to the fusion categoryof twisted representations of V(G2,5). This reinterpretation is im-portant in that it makes the deep ideas of the subfactor communitymuch more widely accessible. We now know that the full structureof a CFT can be recovered from the initial VOA and the modulecategory.

It took 20 years, but we now know the question Cappelli–Itzykson–Zuber were asking: Find all module categories of the V(sl(2),)modular tensor category.

The Endgame: Induction Meets GaloisThe modular invariant classifications have essentially been stalledfor 25 years at sl(3), despite the fact that large swaths of nim-reps

11

Focus on the Scientist: Dmitri NikshychDmitri Nikshych is a world-renowned researcher in quantumsymmetries. He has made substantial contributions to weak Hopfalgebras, quantum groups, and tensor categories, most notably inhis joint paper with Pavel Etingof and Victor Ostrik, “On FusionCategories” (Annals, 2005).

Dmitri Nikshych

His recent book, Tensor Cate-gories (AMS, 2015), co-writtenwith Etingof, Ostrik, and ShlomoGelaki, is now widely consideredto be “The Book” for those (in-terested) in quantum symmetries.Dmitri’s work is exciting, deep,and innovative, and is greatly ap-preciated by algebraists, analysts,topologists, and physicists aroundthe globe.

Dmitri was raised in Kiev,Ukraine. His parents fosteredhis mathematical interests at a young age, and with this support,he was admitted into a competitive high school that specializes inmathematics. He then enrolled at the Kiev Polytechnic Institute(KPI) to study applied math, and received great training in func-tional analysis during this time. This drew him to the precisionand beauty of theoretical mathematics, especially to quantumgroups via the generous guidance of Leonid Vainerman.

After KPI, Dmitri began his graduate studies at UCLA in 1997with advisor Edward Effros. During his second year, Effrospointed out to Dmitri a course on quantum groups that Etingofwas giving at MIT, and Dmitri soon wrote Etingof to ask if hecould visit to attend the course. His wish was granted and heended up sticking around MIT for nearly two years! Both Etingofand Ostrik (who was also at MIT at the time) introduced Dmitri tomany new ideas, and this sparked a transformative collaborationand continuing friendship.

Dmitri defended his thesis at UCLA in 2001 and joined the Uni-versity of New Hampshire as an assistant professor that year.With having a child on the way, he was grateful for the terms andstability of his new position.

A year later, Dmitri’s ground-breaking preprint with Etingofand Ostrik on fusion categories appeared. Other notable worksappearing since then include “On braided fusion categories I”(Selecta, 2010) joint with Gelaki, Ostrik, and Vladimir Drinfeld,and “The Witt group of non-degenerate braided fusion categories”(Crelle, 2013) joint with Ostrik, Alexei Davydov, and MichaelMüger.

In New Hampshire, Dmitri lives with his wife, three kids, andseveral pets including a chicken named Nightfury. He enjoysspending time with his family, reading history books, and pickingmushrooms in his spare time.

— Chelsea Walton

11

1,11,19,29

⌧1

2,10,12,18,20,28

⌧01

3,9,11,13,17,19,21,27

⌧001

4,8,10,... ,22,26

⌧02

5,7,... ,25⌧2

6,8,.... ,24

12

7,13,17,23

⌧002 6,10,14,16,20,24

The E8 example revisited.

(indeed, module categories) have been worked out for a few otherLie algebras. However, recent breakthroughs, in part from twopostdocs in our program, promise a glut of new module categoryclassifications in the near future.

Recall that classifying the module categories requires two things:determining the possible VOA extensions of V(g,) and deter-mining all braided tensor equivalences between the MTC of thoseextensions.

Schopieray used alpha-induction to find an explicit bound on forg = sp(4) and G2, beyond which any VOA extension of V(g,)would have to be of an especially nice type. In principle this boundcould be computed for any Lie algebra, though it grows exponen-tially with the rank. However, it has since been shown that Galoissymmetry can be combined with alpha-induction to come up withvastly smaller bounds, growing only cubically with the rank. For

example, for V(G2,) Schopieray’s -bound is 18 million, whileGalois and alpha-induction reduce it to 129.

Using these bounds, the classification of all possible extensionsof V(g,), for all Lie algebras g of small rank (for example, 6 8)should be imminent. The result in small rank is that there are veryfew extensions which are not of the especially nice type.

Furthermore, Edie-Michell has recently worked out the autoequiv-alences of the MTC for V(g,). The final step in the quantumsubgroup (aka, module category) classifications for all Lie algebrasof small rank, will be the application of Edie-Michell’s methods tothe especially nice extensions.

Quantum subgroups for sl(2) fall into an A-D-E pattern. Thosefor sl(3) seem directly related to Jacobians of Fermat curves. It isexciting to speculate what these relate to in higher rank! We shouldsoon know!

12

Puzzles ColumnJoe P. Buhler and Tanya Khovanova

1. Find all possible positive integers a, b, c, and d such thatab= c+d and cd= a+b.

2. Let C be the surface of a cube in 3-space. What is the largest nsuch that there is a regular n-gon P, not lying entirely on one faceof C, such that all P’s vertices are on C?

3. Here is a different sort of hat problem. There are n hats, eachwith a different color. These hats are placed on the heads of n sages.All of the sages know all the colors: their own hat and everyoneelse’s hats. A referee then announces the “correct” hat color thatshould be on the head of each sage.

The sages are then allowed “swap” sessions: in one session disjointpairs of sages are allowed to interchange their hats. Can the sagesfully correct their hat colors in two swap sessions?

A CN

BM

4. Line segments AB and AC have equal length and have an angleof one degree at A. Let M be the midpoint of AB and N be themidpoint of AC. Erase AM and AN, so that the remaining seg-ments MB and NC are disjoint and at an angle of one degree. (Thisis illustrated, for a much larger angle, in the figure.) We think ofthese line segments as mirrors in that they reflect light perfectly inthe usual way.

A light ray in the plane containing ABC enters quadrilateral MBCNat the thin end, between M and N. Then it bounces back and forthuntil it exits at the thick end, between B and C. What is the maxi-mum possible number of reflections that it can make?

Comment: This appeared in a 2001 book of 50 mathematical puz-zles, edited by G. Cohen, and we first heard about it from StanWagon and Dan Velleman.

?

5. A race of aliens tours our galaxy looking for planets with ediblehumanoids. When they visit a planet they take an instantaneous cen-sus of all humanoids, and classify each as either edible or inedible.If the number of pairs of edible humanoids is strictly bigger thanone-half of the total number of pairs of humanoids, then all of itsedibles are promptly hunted and eaten.

Earth was visited by these aliens sometime in the last century. We(or, at least those of us that were judged to be edible) escaped bythe narrowest possible margin. What year did the aliens visit?

Comment: This problem is due to Steve Silverman.

6. N people are randomly placed on a line segment. Each personturns to face their closest neighbor (with probability 1, all the dis-tances are distinct, so this is sufficiently well-defined). What is theexpected number of “lonely people” who are looking at a neighborwho is not looking at them?

Forthcoming WorkshopsMay 4–8, 2020: Hot Topics: OptimalTransport and Applications to MachineLearning and Statistics

Jun 13–Jul 26, 2020: MSRI-UP 2020:Branched Covers of Curves

Aug 20–21, 2020: Connections Workshop:Decidability, Definability andComputability in Number Theory

Aug 24–28, 2020: Introductory Workshop:Decidability, Definability andComputability in Number Theory

Sep 3–4, 2020: Connections Workshop:Random and Arithmetic Structures inTopology

Sep 8–11, 2020: Introductory Workshop:Random and Arithmetic Structures inTopology

Nov 30–Dec 4, 2020: Structure andRandomness in Locally Symmetric Spaces

Dec 7–11, 2020: Topical Workshop:Decidability, Definability andComputability in Number Theory

Summer Graduate Schools

Jun 8–19, 2020: Combinatorial andDG-Algebra Techniques for FreeResolutions (Tianjin, China)

Jun 15–26, 2020: Geometric Flows(Athens, Greece)

Jun 15–26, 2020: Algebraic Theory ofDifferential and Difference Equations,Model Theory, and their Applications

Jun 29–Jul 10, 2020: New Directions inRepresentation Theory (AMSI, Brisbane,Australia)

Jun 29–Jul 10, 2020: Random Graphs

Jun 29–Jul 10, 2020: Séminaire deMathématiques Supérieures 2020: DiscreteProbability, Physics and Algorithms(Montréal, Canada)

Jun 29–Jul 10, 2020: Algebraic Curves(Hainan, China)

Jun 29–Jul 10, 2020: Foundations andFrontiers of Probabilistic Proofs (Zurich,Switzerland)

Jul 6–17, 2020: Metric Geometry andGeometric Analysis (Oxford, UnitedKingdom)

Jul 13–Jul 24, 2020: Sums of SquaresMethod in Geometry, Combinatorics, andOptimization

Jul 27–Aug 7, 2020: Introduction to WaterWaves

To find more information about any of theseworkshops, as well as a full list of all up-coming workshops and programs, please seemsri.org/workshops and msri.org/programs.

13

2019 Annual ReportWe gratefully acknowledge the supporters of MSRI whose generosity allows us to fulfill MSRI’s mission to advance andcommunicate the fundamental knowledge in mathematics and the mathematical sciences; to develop human capital for thegrowth and use of such knowledge; and to cultivate in the larger society awareness and appreciation of the beauty, power,and importance of mathematical ideas and ways of understanding the world.

This report acknowledges grants and gifts received from January 1–December 31, 2019. In preparation of this report, we havetried to avoid errors and omissions. If any are found, please accept our apologies, and report them to development@msri.org.If your name was not listed as you prefer, let us know so we can correct our records. If your gift was received after December31, 2019, your name will appear in the 2020 Annual Report. For more information on our giving program, please visitwww.msri.org.

Current Foundation &Corporate SponsorsAnonymous (3)Alfred P. Sloan FoundationAmazon SmileAmerican Mathematical SocietyArkay FoundationBlackRockBrazeCitadel Derivatives GroupDella Pietra Family FoundationDella Pietra FoundationEllen Fletcher Middle SchoolElwyn & Jennifer Berlekamp

FoundationFiredoll FoundationGoldman SachsThe John Smillie and Karen

Vogtmann FundThe Kavli FoundationKevin and Deborah Bartz

FoundationMagnitude Capital, LLCMake and TakeMath for AmericaMeyer Sound Laboratories, Inc.Microsoft CorporationMorgan StanleyThe Oshay Family FoundationPacific Journal of MathematicsPatrick J. McGovern FoundationS. S. Chern Foundation for

Mathematical ResearchSECOR Asset ManagementSimons FoundationSvrcek FoundationSylvester Meister Foundation

* Includes gifts made in 2019 tothe MSRI Endowment Fund.Gifts to the endowment arecarefully and prudently investedto generate a steady stream ofincome.

Hilbert Society$100,000 per year and/or$1 million lifetime givingWe recognize our mostgenerous and loyal supporterswhose leadership andcommitment ensure MSRIcontinues to thrive as one ofthe world’s leadingmathematics research centers.

AnonymousEdward Baker & Annie Yim*Elwyn & Jennifer BerlekampPaul & Susan Chern*May & Paul Chu*William & Margaret HearstMark & Stephanie Lynn

Kleiman*Marilyn & Jim SimonsRoger Strauch & Julie KulhanjianSandor & Faye StrausAndrew Viterbi

Museion Society$5,040 – $25,000+The Museion Society,named after Musaeum, theHall of the Muses in ancientAlexandria, recognizes ourleadership donors in annualand endowment giving.

EuclidAnonymous (2)Sean Fahey & Robin Luce FaheyNaren & Vinita GuptaIrwin & Joan Jacobs

HypatiaElizabeth Barton & Robert BealsAbib BocresionRay IwanowskiRon Kahn & Julia RoweSorin & Marie-Anne PopescuColin Rust & Jeannie Tseng

PlatoAnonymous (4)Ian Agol & Michelle McGuinnessFilippo AltissimoJishnu & Indrani BhattacharjeeRobert Bryant & Réymundo

GarcíaJennifer Chayes & Christian

BorgsHerman ChernoffThomas Davis & Ellyn BushDavid desJardins & Nancy

BlachmanDaniel Freed & Sonia PabanPaul GaliettoHatch GrahamAshwini & Anita GuptaJoe HarrisTom Kailath & Anu MaitraMaria Klawe & Nicholas

PippengerWilliam LangKristin LauterAlex Lesin*David & Cynthia LippeDiviya & Alexander MagaroR. Preston & Kristin McAfeeCalvin C. & Doris L. Moore*Howard & Eleanor MorganEmil & Irene MoshkovichThomas & Peggy RikeMyron & Jan ScholesAaron Siegel & Olya GurevichPeter & Angela Yianilos

Archimedes Society$55 – $5039The Archimedes Societyrecognizes MSRI’s annualsupporters.

RamanujanAnonymous (1)Ken BaronGeorgia BenkartDavid J. Benson

David & Monika EisenbudGisela Fränken & Walter

OhlemutzDon & Jill KnuthHoward Masur*Hugo & Julia Rossi*Ronald & Sharon SternJustin WalkerHung-Hsi Wu & Kuniko

Weltin-Wu

NoetherAnonymous (1)Hélène Barcelo &

Steven Kaliszewski*Norman Bookstein & Gillian

KuehnerIain & Jane BratchieMikhail BrodskyMicha Brym & Faye DuchinJoe & Danalee BuhlerRuth Charney & Stephen

CecchettiDr. Paul FongAlfred & Virginia HalesPat Hanrahan & Delle

MaxwellJonathan HaylockBengt HolmstromJohn HosackMonish Kumar & Sumita

BhattacharyaJoseph & Betty LangsamDusa McDuff & John MilnorAndrew OdlyzkoHenry Schenck*Michael Singer & Jean

BlackwellLance & Lynne Small*Jan SpethTimothy Sylvester & Barbara

MeisterRichard & Christine TaylorShanker TrivediJonathan Zhu

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FermatAnonymous (3)Bedros AfeyanValery AlexeevElliot Aronson & Shirley Hoye Annie Averitt*Sheldon Axler & Carrie Heeter Dave Bayer & Laurie Mitsanas Thomas & Louise Burns Charles & Elizabeth Curtis Anthony D’AristotileManoj DasDouglas DunhamRobert & Judith Edwards Stewart EthierE. Graham & Kaye Evans Mark Feighn & Edith Starr John FriedmanSylvester & Dianna Abney

GatesWilliam & Emily Goldman Daniel & Ryoko Goldston Curtis GreenePhillip & Judith Griffith Robert & Louise Grober Robin HartshorneKevin IgaCaroline KlivansRobert KorsanRobert Kusner & Denise Kim Suzanne Langlois & John

McCarthyWilliam J. LewisRafe MazzeoJon McCammondDonald McClureAndrew OggSandra Peterson*Winston A. RichardsMarc RieffelVincent RodgersFreydoon Shahidi*James SmithLouis & Elsbeth Solomon Hugo & Elizabeth Sonnenschein John StroikChuu-Lian TerngWolmer VasconcelosCharles WeibelPeter & Gloria YuJulius & Joan Zelmanowitz

FibonacciAnonymous (5)Mina AganagicAnuradha AiyerHyman BassEthan & Kyra Berkove Kenneth BrownBradley Carroll

Richard ChampionJean Bee ChanAndreas ChatziafratisRichard ChurchillKen CollinsDavid CoxSteven & Margaret CrockettMartin & Virginia DavisEmily Dryden*Patrick Eberlein & Jennifer

Pedlow EberleinPeter EggenbergerLawrence EinBahman EnghetaHans EnglerDavid FaulknerRaluca FeleaMichael H. FreedmanTeena GerhardtAgastya GoelDavid HarbaterBrian HayesNancy HingstonStanley IsaacsKiran KedlayaEllen KirkmanAleksandr KoldobskiyMichel LapidusSee Pheng LeeGail LetzterClinton G. McCroryRobert & Kathy MegginsonJanet Mertz & Jonathan KaneCarlos & Ofelia MorenoWalter NeumannYong-Geun OhRichard & Susan OlshenSusan PisaniEric Todd QuintoMalempati RaoBruce ReznickJonathan RosenbergWilliam Rowan & Charlene

QuanDavid RowlandMary & James SalterTimo Seppalainen*William SitBruce SolomonJames T. SotirosLee & Maryse StanleyHarold M. StarkJames StasheffTroy StoryJussi SuontaustaEarl & Hessy TaftTed TheodosopoulosPham TiepTatiana Toro & Daniel Pollack*

Burt Totaro & Diana GilloolyMariel Vazquez & Javier

Arsuaga*Bill VedermanStanley Wagon & Joan

HutchinsonJonathan WahlBrian WhiteSusan WildstromErling WoldJinze WuLev Zelenin

Book, Journal, andMedia GiftsPrivate DonorsEdward D. BakerHélène BarceloHeidrun BrieskornMay ChuDavid EisenbudWilliam FultonTakayuki HibiSteven G. KrantzTsit-Yuen LamKenneth Ribet

There are many ways to support MSRI with a donation or aplanned gift. A planned gift in the form of a bequest may enableyou to make a substantial gift to MSRI, ensuring that the instituteremains strong in the future. Planned gifts may help reducecapital gains and estate taxes. Friends who include MSRI in theirwill or have made other stipulations for a gift from their estatebecome members of the Gauss Society and receive invitations toour Museion dinners and to a reception during the annual JointMathematics Meeting. To learn more about the Gauss Societyand other ways to support MSRI, please contact Annie Averitt,Director of Advancement and External Relations, at 510-643-6056or aaveritt@msri.org.

Now in the Math Circles Library!The latest volume in the Mathemat-ical Circles Library series has justbeen released. MSRI and the AMSpublish the series as a service toyoung people, their parents and teach-ers, and the mathematics profession.Inspiring Mathematics: Lessons fromthe Navajo Nation Math Circles,edited by Dave Auckly, Bob Klein,Amanda Serenevy, and Tatiana Shu-bin, shares interactive mathematical explorations developedby educators and mathematicians in the NNMC programs.The problem sets included are good for puzzling over withstudents interested in exploring the art, joy, and beauty inmathematics. Explore the entire collection of MSRI Math-ematical Circles Library titles at bookstore.ams.org/MCL.

PublishersAmerican Mathematical SocietyBasic BooksBelgian Mathematical SocietyCambridge University PressCentre de Recherches

MathématiquesCRC Press/Taylor & FrancisElsevierEuropean Mathematical SocietyFields InstituteHiroshima UniversityInstitut de Mathématiques

de JussieuInstitute for Advanced StudyInternational PressJapan AcademyKyoto UniversityKyushu UniversityMathematical Society of JapanPrinceton University PressSpringerTohoku UniversityUniversity of TokyoWorld Scientific

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MSRI’s Newest DocumentaryShares the Life and Work ofMaryam MirzakhaniFilmed in Canada, Iran, and the United States, Secrets of the Surface: TheMathematical Vision of Maryam Mirzakhani (2020) examines the life andmathematical work of Maryam Mirzakhani, an Iranian immigrant to theUnited States who became a superstar in her field. In 2014, she was boththe first woman and the first Iranian to be honored with one of mathe-matics’ highest prizes, the Fields Medal.

Maryam’s contributions are explained in the film by leading mathe-maticians and illustrated by animated sequences. Her mathematicalcolleagues from around the world, as well as former teachers, classmates,and students in Iran today, convey the deep impact of her achievements.The path of her education, success on Iran’s Math Olympiad team, andher brilliant work, make Mirzakhani an ideal role model for girls lookingtoward careers in science and mathematics.

Secrets of the Surface is now available at www.zalafilms.com/secrets.If you would like to arrange a screening of the film at your institution,contact the film’s director, George Csicsery, at geocsi@zalafilms.com.

Make a Donation to MSRI

Individuals can make a significant impact on the life of the Institute, on mathematics, and on scientific research by contributing to MSRI.