AMS SPRING SECTIONAL SAMPLER Photo Courtesy of College Charleston

AMS SPRING SECTIONAL SAMPLER Photo courtesy of College Charleston. Image courtesy of Indiana University Communications. Spring Southeastern Sectional Meeting Spring Central Sectional Meeting March 10–12, 2017 (Friday–Sunday) April 1–2, 2017 (Saturday–Sunday) College of Charleston, Charleston, SC Indiana University, Bloomington, IN Image courtesy of Hunter College, CUNY. Image courtesy of WSU Photo Services. Spring Western Sectional Meeting *Spring Eastern Sectional Meeting April 22–23, 2017 (Saturday–Sunday) May 6–7, 2017 (Saturday–Sunday) Washington State University, Pullman, WA Hunter College, City University of New York, New York, NY * A sampler from this meeting will appear in the May issue of Notices. Make the time to visit any of the AMS Spring Sectional Meetings listed above. In this sampler, the speakers below have kindly provided introductions to their Invited Addresses for the upcoming AMS Spring Central Sectional (Indiana University, April 1–2) and the AMS Spring Western Sectional (Washington State University, April 22–23) Meetings.

Spring Central Sectional Meeting Spring Western Sectional Meeting

Sarah C. Koch Michael Hitrick Postcritical Sets in Complex Dynamics Spectra for Non-self-adjoint Operators and page 339 Integrable Dynamics page 334

Andrew Raich – 2 n Closed Range of ∂ in L on Domains in ℂ page 335

For permission to reprint this article, please contact: Daniel Rogalski [email protected]. Noncommutative Projective Surfaces DOI: http://dx.doi.org/10.1090/noti1530 page 336 AMSSPRINGSECTIONALSAMPLER

Michael Hitrik constant (quantum mechanics), but also the temperature (Fokker–Planck equation), or the reciprocal of the square root of an eigenvalue (spectral geometry), of frequency Spectra for Non-self-adjoint Operators and (scattering theory), of Reynolds number (fluid mechanics), Integrable Dynamics or of the power of a line bundle (complex geometry). Non-self-adjoint operators Roughly speaking, ℎ measures the slow variation of the arise in many settings, medium compared to the length of the waves, hence This creates a from linearizations of the alternative name short wave approximation. In PDE challenge, but equations of mathematical theory, this is the basis of microlocal analysis, where physics to operators defin- ℎ ∼ 1/ |휕푥|. Adopting the philosophy of the quantum- also provides an ing poles of Eisenstein classical correspondence principle of Niels Bohr, whereby series. Roughly speaking, the behavior of quantum mechanical systems should opportunity. any physical system where reduce to classical physics in the semiclassical limit the energy is not con- ℎ → 0, we are then led to the following basic problem: can served, typically due to we describe the distribution of complex eigenvalues of a some form of damping or a possibility of escape to non-self-adjoint operator, in the semiclassical régime, in infinity, will be governed by a non-self-adjoint operator. terms of the underlying classical dynamics? The latter is From the point of view of mathematics, an essential given by a Hamilton flow on the classical phase space, feature making the study of non-self-adjoint operators and to implement the quantum-classical correspondence notoriously difficult is the potential instability of their in a non-self-adjoint environment, one frequently needs spectra. When 퐴 is a self-adjoint matrix, we know from the to consider complex such flows on the complexification spectral theorem that the operator norm of the resolvent of the real phase space. (퐴 − 푧퐼)−1 is equal to the reciprocal of the distance from In joint work with Johannes Sjöstrand, we have carried 푧 to the spectrum of 퐴. In contrast, let us consider a out a detailed spectral analysis for non-self-adjoint pertur- (non-self-adjoint) Jordan block matrix bations of self-adjoint operators in dimension two under suitable assumptions of analytic and dynamical nature. 0 1 0 … 0 Specifically, let us consider a non-self-adjoint operator of ⎛0 0 1 … 0⎞ ⎜ ⎟ the form 퐽 = ⎜ ...…. ⎟ 푛 ⎜ ⎟ 푃 = 푃 + 푖휀푄, ⎜0 0 0 … 1⎟ 휀 0 ⎝0 0 0 … 0⎠ acting on a compact two-dimensional real-analytic Rie- mannian manifold 푀. Here, to fix the ideas, we may take of size 푛, with 푛 large. The spectrum of 퐽푛 is equal to 푃 = −ℎ2Δ to be the semiclassical Laplacian, and 푄 is a −1 0 {0}, while the operator norm of the resolvent (퐽푛 − 푧퐼) multiplication by an analytic real-valued function 푞 on 푀, 푛 grows exponentially with 푛, roughly like 1/ |푧| , for 푧 which we can think of as a damping coefficient or a poten- in the open unit disc in the complex plane. As one tial. The small parameter 휀 > 0 measures the strength of can imagine, things are even more intricate for, say, the non-self-adjoint perturbation, which should not be too differential operators. This creates a challenge, but also weak. The eigenvalues of 푃휀 (see Figure 1) are confined to provides an opportunity, accounting for some of the a thin band near the real domain, and thanks to a classical complex and fascinating traits in the spectral behavior of result originating in work of Torsten Carleman, we know non-self-adjoint operators. that the distribution of the real parts of the eigenvalues For most operators, we cannot compute the spectra of 푃휀 near the energy level 퐸 = 1, say, is governed by the exactly, and, as is natural in many physical situations, one same Weyl law as that for the unperturbed self-adjoint studies asymptotic properties for problems depending on operator 푃0. The distribution of the imaginary parts of a parameter. To make things uniform, we consider a small the eigenvalues of 푃휀 is much more subtle, however, and parameter ℎ and refer to the study of the limit ℎ → 0 as the is expected to depend on the dynamics of the geodesic semiclassical approximation. It is important to remember, flow and its relation to the non-self-adjoint perturbation. however, that ℎ may mean different things in different In order to obtain a rigorous justification of the intuition settings: as the name suggests, it could be the Planck above, let us assume that the geodesic flow is completely integrable. The phase space 푇∗푀 is then foliated by Michael Hitrik is professor of mathematics at the University of Cali- Lagrangian tori, invariant under the geodesic flow, and the fornia Los Angeles. His e-mail address is [email protected]. flow on each such torus becomes linear when expressed I am most grateful to Johannes Sjöstrand, Joe Viola, and Maciej in suitable canonical coordinates. Following the ideas of Zworski for their very helpful comments on this note. the method of averaging (Birkhoff normal forms), we are For permission to reprint this article, please contact: led to consider the long-time averages of 푞 along the [email protected]. geodesic flow. The following two radically different cases DOI: http://dx.doi.org/10.1090/noti1495 occur, depending on the type of the dynamics: periodic

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Spectrum of p + i*epsilon*q,, epsilon=0.01, h=0.01, kappa=2, F=2 3.5 sometimes be used to obtain spectral results that are more precise than those for real Hamiltonians. The spectral structures described above are expected to 3 occur also for other genuinely two-dimensional non-self- adjoint analytic spectral problems and notably for the

2.5 distribution of scattering resonances for convex obstacles in R3, assuming that the boundary geodesic ﬂow is completely integrable or is a small perturbation of it. The 2 study of these fascinating issues, which is very much at the heart of the current research, will also be touched Im z/epsilon upon in the talk. 1.5

Credits 1 All images are courtesy of Michael Hitirik.

0.5 0.85 0.9 0.95 1 Re z Figure 1. Numerical computation of the real and nor- malized imaginary parts of the eigenvalues of the non-self-adjoint operator 푃휀 = 푃0 + 푖휀푄 on the two- dimensional torus, when 푄 is multiplication by a trigonometric polynomial 푞 of degree 2. The legs of the spectral centipede represent the inﬂuence of rational tori for the classical ﬂow.

(rational tori) or dense (irrational or, better, Diophantine tori). In joint work with Sjöstrand and San Vũ Ngọc, we have obtained a complete semiclassical description of the spectrum of 푃휀 near a complex energy level of the Andrew Raich form 1 + 푖휀퐹, assuming that 퐹 is given by the average ̄ 2 푛 of 푞 along a Diophantine torus and that no other torus Closed Range of 휕 in 퐿 on Domains in ℂ has this property, roughly speaking. It turns out that the At the AMS Western Sectional Meeting, I will talk about 2 spectrum in this region has the structure of a distorted my recent work on the 퐿 theory of the Cauchy–Riemann ̄ two-dimensional lattice, given by a quantization condition operator 휕 in several complex variables and the solv- of Bohr–Sommerfeld type. ability/regularity of solutions 푢 to the Cauchy–Riemann 푛 What about spectral contributions of rational tori? equation 휕푢̄ = 푓 in a domain Ω in ℂ . These have remained rather mysterious for quite some In one complex variable, we have the Cauchy kernel time, until recent work with Sjöstrand demonstrated as an incredibly powerful tool to solve 휕푢̄ = 푓. of the that rational tori can produce eigenvalues of 푃휀 close to Cauchy kernel rests largely on the facts that it is holo- the edges of the spectral band, isolated away from the morphic and can be used to build a variety of operators. bulk of the spectrum, and that a complete semiclassical Integrating against the Cauchy kernel on Ω produces description of these extremal, “rational” eigenvalues can a solution to the Cauchy–Riemann operator, while in- be achieved. Somewhat surprisingly perhaps, the extremal tegrating against the Cauchy kernel on the boundary eigenvalues turn out to have the structure of the legs in bΩ builds an operator that (re)produces holomorphic a spectral “centipede” as in Figure 1, with the body of the functions. Also, for dimensional reasons, the boundary centipede agreeing with the range of torus averages of 푞. of any domain Ω ⊂ ℂ lacks any complex structure The following vague philosophical remark may perhaps clarify the reason for a detailed study of dimension two: as one knows from basic quantum mechanics texts, Andrew Raich is associate professor in the Department of Mathe- for problems in dimension one, the Bohr–Sommerfeld matical Sciences at the University of Arkansas. His e-mail address quantization rules work very well to determine spectra is [email protected]. of self-adjoint observables, whose energy surfaces are For permission to reprint this article, please contact: real curves. For complex Hamiltonians in dimension [email protected]. two, energy surfaces are complex curves, and this can DOI: http://dx.doi.org/10.1090/noti1494

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(a one real dimensional object on forms, and componentwise it is nothing more than cannot carry any complex struc- the ordinary Laplacian. However, inverting □ never gains The Riemann ture), which, philosophically, two derivatives. Sometimes the solving operator gains Mapping allows for a result like the one derivative, sometimes it gains a fractional derivative, Riemann Mapping Theorem to and sometimes the inverse is just continuous on 퐿2. It Theorem fails hold. is in the domain of 휕∗̄ that the geometric information of In several variables, the the boundary is encoded, and it is the geometry of the in the most Cauchy kernel has no higher boundary that determines how nice solutions of 휕푢̄ = 푓 dimensional analog. All of the can be. spectacular replacements have serious defi- In the decades since the 1960s the exploration into the ciencies; for example, they are inhomogeneous Cauchy–Riemann equation 휕푢̄ = 푓 has way possible. noncanonical or not holomor- expanded to various function spaces, complex manifolds, phic. Moreover, domains in ℂ푛 the tangential Cauchy–Riemann equations on CR mani- have boundaries that are (2푛 − 1)-real dimensional manifolds, and, more recently, nonpseudoconvex domains and folds, so there is an (푛−1)-complex dimensional structure unbounded domains. This is where my colleague Phil Har- as part of the boundary. Not coincidentally, the Riemann rington and I enter the picture. To understand the more Mapping Theorem fails in the most spectacular way subtle geometry, we work on answering the question, possible—the unit ball and unit polydisk (product of What is the replacement condition for pseudoconvexity ̄ 2 real 2D disks) are not biholomorphic. We are left in a for optimal solvability of 휕푢 = 푓 in 퐿0,푞(Ω) for a fixed 푞, landscape where the solvability and regularity of the 1 ≤ 푞 ≤ 푛? In my talk, I will outline the progress that Cauchy–Riemann equations 휕푢̄ = 푓 depend on subtle we have made on this problem and put our results in geometric and potential-theoretic quantities associated context. Time permitting, I will also discuss our progress to the boundary bΩ. This makes the analysis much more on solving the inhomogeneous Cauchy–Riemann equation difficult than either the one complex dimensional case or on unbounded domains and the additional complications the analogous real problem 푑푢 = 푓 on Ω. that this entails. To discuss the 휕-problem̄ in any depth, we must frame the problem more carefully. Given a (0, 푞)-form 푓 on a Photo Credit domain Ω, the problem is to find a (0, 푞 − 1)-form 푢 so that 휕푢̄ = 푓. As with 푑, 휕2̄ = 0, so solving 휕푢̄ = 푓 has the Photo of Andrew Raich is courtesy of Shauna Morimoto. necessary condition 휕푓̄ = 0. The question then becomes to solve 휕푢̄ = 푓 when 휕푓̄ = 0. Before the 1960s, the inhomogeneous Cauchy–Riemann equation was studied in the 퐶∞-topology. In the 1960s, Hörmander and Kohn pioneered the use of 퐿2 techniques to solve the equation. From a functional analytic viewpoint, the switch from 퐶∞ to 퐿2 is a major improvement, because the space 퐶∞(Ω) is nonmetrizable, while 퐿2(Ω) is a Hilbert space, which allows for the use of tremendously powerful tools. ̄ 2 Hörmander showed that 휕푢 = 푓 can be solved in 퐿0,푞(Ω) at every form level (1 ≤ 푞 ≤ 푛) if and only if Ω satisfies a curvature condition called pseudoconvexity. Moreover, he also showed that there is no 퐿2 cohomology for 휕̄ on pseu- doconvex domains. Pseudoconvexity can be understood as the complex analysis version of convexity. For example, pseudoconvexity is invariant under biholomorphisms while convexity is not. Daniel Rogalski A common approach to solving the inhomogeneous Cauchy–Riemann equations is to establish a Hodge theory Noncommutative Projective Surfaces for 휕.̄ To do this, we need to introduce the 퐿2 adjoint of 휕,̄ 휕∗̄ . In this note we introduce the reader to the subject Computationally, 휕∗̄ is obtained via integration by parts, of noncommutative projective geometry, which will be and as every calculus student knows, integration by parts the focus of our talk at the AMS regional meeting in introduces a boundary term. For a form to be in the domain of 휕∗̄ this boundary term must vanish. This vanishing Daniel Rogalski is professor of mathematics at the University of condition is the source of trouble for proving regularity California, San Diego. His e-mail address is [email protected]. results. Although 휕̄ ⊕ 휕∗̄ provides an elliptic system, the For permission to reprint this article, please contact: inverse to 휕̄ (when it exists) never has an elliptic gain. [email protected]. The 휕-Neumann̄ Laplacian □ = 휕휕̄ ∗̄ + 휕∗̄ 휕̄ acts diagonally DOI: http://dx.doi.org/10.1090/noti1502

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Pullman, WA, in April 2017. In the talk we will begin with connection between varieties and rings becomes more an overview of noncommutative geometry and then give important, and one learns how to define a projective a survey of some results about noncommutative surfaces variety all at once using a graded ring. In this note, which have been the subject of our own research. In this a graded ℂ-algebra will be a ℂ-algebra 푅 with a ℂ- short space, however, we primarily describe the main vector space decomposition 푅 = 푅0 ⊕ 푅1 ⊕ …, where idea of the subject and then at the end briefly mention 푅푛푅푚 ⊆ 푅푛+푚 for all 푛, 푚, and where for simplicity we one of the interesting examples that will be explored always assume that 푅0 = ℂ and that 푅 is generated by in more detail in the talk. Given our informality here, 푅1 as an algebra over ℂ. For a commutative graded ℂ- we do not provide references. The reader can find more algebra, one defines Proj 푅 to be the set of homogeneous information in the author’s more extensive course notes prime ideals in 푅, that is, those prime ideals 푃 such on the subject [BRSSW]. that 푃 = ⨁푛≥0(푃 ∩ 푅푛), excluding the irrelevant ideal We assume the reader is familiar with the definition 푅≥1 spanned by all elements of positive degree. The of a ring. We always assume that rings have an identity maximal elements of this space Proj 푅 correspond to element 1, and in fact all of our examples will be algebras the points of a pro- over the complex numbers ℂ, which means that the ring jective variety, while contains a copy of the field ℂ and can be thought of as a the other points are Noncommutative ℂ-vector space. In a first course in abstract algebra, the called “generic” and discussion of rings invariably focuses almost exclusively are important to the projective on commutative rings, where 푎푏 = 푏푎 for all elements 푎 scheme-theoretic point geometry is an and 푏. Perhaps the matrix ring 푀푛(ℂ) might be mentioned of view. For example, as an example of a noncommutative ring. Of course, it the polynomial ring 푅 = attempt to makes sense to get comfortable first with commutative ℂ[푥0, 푥1, … , 푥푛] in 푛 + 1 rings, which are essential to many applications of ring variables over ℂ is graded, understand theory. Plenty of interesting rings arising in nature are where 푅푛 is simply the noncommutative, however, such as the various quantum span of monomials of de- noncommutative algebras that arise in physics to describe the interaction of gree 푛, and in this case operators (such as position and momentum) that do not Proj 푅 = ℙ푛. As a more graded rings in a necessarily commute. One simple but already nontrivial explicit example, the ho- geometric way. example of a noncommutative algebra is the quantum mogeneous primes of plane, 퐴푞 = ℂ⟨푥, 푦⟩/(푦푥 − 푞푥푦), for a nonzero scalar ℂ[푥, 푦] are (0), (푎푥 + 푏푦) 푞 ∈ ℂ. Here, ℂ⟨푥, 푦⟩ is the free associative algebra in two for 푎, 푏 ∈ ℂ not both 0, and (푥, 푦). Removing the irrelevant variables, that is, the ℂ-vector space with basis all words ideal (푥, 푦), the maximal elements are those ideals of the in the (noncommuting) variables 푥 and 푦, with product form (푎푥 + 푏푦) which correspond to points (푎 ∶ 푏) in ℙ1. given by concatenation. To obtain 퐴푞 we mod out by Noncommutative projective geometry is an attempt to the ideal generated by one relation, which says that the understand noncommutative graded rings in a geometric variables commute “up to scalar.” It is not hard to see way or to associate some kind of interesting geometric that the elements of 퐴푞 can be identified with the usual space to a noncommutative ring, similar to the way that 푖 푗 commutative polynomials ∑푖,푗≥0 푎푖푗푥 푦 in the variables, the Proj construction associates a projective variety to a but the multiplication is changed so that every time we commutative graded ring. The development of the subject move a 푦 to the right of an 푥, a factor of 푞 appears. We was surely delayed by the fact that the most obvious describe some properties of this algebra below. analog of the Proj construction for a noncommutative Commutative ring theorists have long enjoyed the ring does not lead to an interesting theory, even for such advantage of the tight connection of their subject with fundamental examples as the quantum plane. There is a algebraic geometry. This means both that ring-theoretic natural definition of prime ideal for a noncommutative results may have an application to geometry and that ring: 푃 is prime if whenever 퐼퐽 ⊆ 푃 for ideals 퐼 and geometric intuition can help to prove and interpret 퐽, 퐼 ⊆ 푃 or 퐽 ⊆ 푃 (this definition reduces to the usual results about rings. In a course on algebraic geometry one for commutative rings). However, in contrast to the over the complex numbers, a student learns first about commutative case, the quantum plane 퐴푞 usually has very the most fundamental varieties: affine 푛-space 픸푛 = few prime ideals. In particular, if 푞 is not a root of unity, 푛 {(푎1, 푎2, … , 푎푛)|푎푖 ∈ ℂ}, and projective 푛-space ℙ , which the only homogeneous prime ideals of 퐴푞 are (0), (푥), (푦), can be identified with the set of lines through the origin and (푥, 푦). Removing the irrelevant ideal (푥, 푦), one is not 푛+1 in 픸 , with the line through the point (푎0, 푎1, … , 푎푛) left with a very interesting space. On the other hand, 퐴푞 written as (푎0 ∶ 푎1 ∶ ⋯ ∶ 푎푛). Both affine and projective behaves a lot like a commutative polynomial ring in two spaces have a natural topology called the Zariski topology, variables in many other ways, so one might hope it would and affine varieties are defined to be closed subsets of have a geometry more similar to Proj ℂ[푥, 푦] = ℙ1. affine spaces, while projective varieties are defined to be As a general principle, often one has to find a particular closed subsets of projective spaces. In the more modern way of formulating a concept in the commutative case, approach to the subject via the theory of schemes, the not always the most elementary or obvious way, to

April 2017 Notices of the AMS 337 AMSSPRINGSECTIONALSAMPLER get something that generalizes to a useful notion for Sklyanin algebra noncommutative rings. In fact, there are two more subtle 2 ways of thinking about how ℙ1 is connected with the 푆 = ℂ⟨푥, 푦, 푧⟩/(푎푥 + 푏푦푧 + 푐푧푦, ring ℂ[푥, 푦] which do generalize well to the quantum 푎푦2 + 푏푧푥 + 푐푥푧, 푎푧2 + 푏푥푦 + 푐푦푥) plane. First, given any graded ℂ-algebra 퐴, a point module for general scalars 푎, 푏, 푐 ∈ ℂ. It turns out that noncom- over 퐴 is a graded left module 푀 = ⨁ 푀푛 (that 푛≥0 mutative ℙ2s can have fewer points than one might at first is, 퐴푖푀푗 ⊆ 푀푖+푗 for all 푖, 푗) such that 푀 = 퐴푀0 and expect. In particular, the point modules for 푆 turn out to dimℂ 푀푛 = 1 for all 푛 ≥ 0. The point modules for ℂ[푥, 푦] be parametrized by an elliptic curve 퐸, and the beautiful are the cyclic modules 푀(푎∶푏) = ℂ[푥, 푦]/(푎푥 + 푏푦), as theory of elliptic curves in algebraic geometry plays a big (푎 ∶ 푏) varies over points in ℙ1. One can even say that role in the analysis of the properties of the algebra 푆. the point modules are parametrized by ℙ1 in a way that The noncommutative projective scheme 푆-qgr is similar is made more precise in algebraic geometry. It turns out in some ways to the category of coherent sheaves on ℙ2, that the point modules for 퐴 are also parametrized by 푞 but is not the same—it is a genuinely noncommutative ℙ1, being of the form 푁 = 퐴 /퐴 (푎푥 + 푏푦) where (푎∶푏) 푞 푞 ℙ2, with only an elliptic curve’s worth of points! 퐴 (푎푥 + 푏푦) is the left ideal generated by 푎푥 + 푏푦. Thus, 푞 In the talk we plan to describe our long-term program to the space of point modules associated to 퐴 is always the 푞 develop a minimal model program for noncommutative projective line ℙ1. surfaces, which has driven a lot of our recent research. An even more subtle approach involves generalizing To close this note, we wish to mention a weird example the idea of sheaves. A sheaf on a projective variety that is particularly close to our heart. One of the joys of can be defined by taking an open cover by rings and noncommutative algebra is the wide abundance of exam- taking a module over each ring such that the modules ples and counterexamples which constantly point out the “glue together” appropriately. This is the most natural myriad ways in which noncommutative rings are more definition geometrically, but it turns out that there is a complicated (or we could say, with obvious bias, more purely algebraic way of getting at this notion. Given a interesting!) than commutative rings. Because of this zoo commutative graded ℂ-algebra 퐴, we define the category of examples, there are still many foundational questions of finitely generated ℤ-graded left 퐴-modules 퐴-gr and in noncommutative geometry about what assumptions define 퐴-qgr to be the quotient category of 퐴-gr by the on graded rings are necessary to get a good theory. It subcategory of modules 푀 which have 푀푛 = 0 for 푛 ≫ 0. is natural to restrict to the study of graded rings which The category 퐴-qgr turns out to be the same as the category are Noetherian (after Emmy Noether), which means that of coherent sheaves on Proj 퐴. The category of coherent they do not have infinite ascending chains of left or right sheaves on a variety is extremely fundamental to the study ideals. This is a basic structural assumption, just as in of the variety, similar to the way that the representations the commutative case, and one which is satisfied by most of a finite group are fundamental to the study of the reasonable examples. However, it is not enough to prevent group. Fortunately, the category 퐴-qgr as defined above certain kinds of pathologies in noncommutative geometry. makes perfect sense for any noncommutative graded In particular, there turn out to be many Noetherian graded algebra 퐴, and thus it intuitively represents some kind algebras 푅 whose space of point modules is so wild that of “category of coherent sheaves” even though there is no it is not parametrized by a projective variety. One simple noncommutative space that these objects are sheaves on! example of such an 푅 can be obtained as follows. First de- Thus the category becomes the fundamental object of fine 퐵 = ℂ⟨푥, 푦, 푧⟩/(푧푥 − 푝푥푧, 푥푦 − 푟푦푥, 푦푧 − 푞푧푦), where study in the noncommutative case, and in fact 퐴-qgr is 푝, 푞, 푟 ∈ ℂ are general scalars satisfying 푝푞푟 = 1. This is a called the noncommutative projective scheme associated very nice noncommutative ℙ2; in fact, 퐵 has point modules to the algebra 퐴. One may prove that the noncommutative parametrized by ℙ2, and 퐵-qgr is equivalent to coherent 2 projective scheme of the quantum plane, 퐴푞-qgr, is in fact sheaves on ℙ . The subalgebra 푅 = ℂ⟨푥 − 푦, 푦 − 푧⟩ ⊆ 퐵 equivalent to ℂ[푥, 푦]-qgr, the usual category of coherent generated by 푥 − 푦 and 푦 − 푧 inside 퐵 is Noetherian, yet 1 sheaves on ℙ . We say that 퐴푞 is a noncommutative has point modules corresponding to an infinite blowup of coordinate ring of ℙ1. ℙ2, where each of infinitely many points in the orbit of Most of the attention in noncommutative geometry has a certain automorphism of ℙ2 gets replaced by a whole focused on analogs of surfaces and higher-dimensional projective line. The noncommutative projective scheme varieties, where things become more complicated. Artin 푅-gqr also has unusual properties. Ring theoretically, the and Schelter defined a notion of regular algebra which strangeness of 푅 shows up in the failure of 푅 to be captures those noncommutative graded algebras which strongly Noetherian; that is, the base extension 푅 ⊗ℂ 퐶 behave homologically like commutative polynomial rings; is not Noetherian for some Noetherian commutative ℂ- these are the rings that correspond geometrically to algebra 퐶. There is now a whole theory of rings like this, noncommutative projective spaces. The noncommutative which are called naïve blowups. They form one of the main ℙ2s were classified by Artin, Schelter, Tate, and Van den classes of examples in the theory of what are known as Bergh. One of the most important such examples is the birationally commutative surfaces, as have been classified

338 Notices of the AMS Volume 64, Number 4 AMSSPRINGSECTIONALSAMPLER by the author and Staﬀord, and in a more general case by Sierra. References [BRSSW] G. Bellamy, D. Rogalski, T. Schedler, J. Toby Stafford, and M. Wemyss, Noncommutative Algebraic Geometry, Mathematical Sciences Research Institute Publications, vol. 64, Cambridge University Press, Cambridge, 2016. Figure 1. Filled Julia sets 퐾푐 of three quadratic polynomials 푝 are drawn in black. The critical point Photo Credit 푐 푧0 = 0 is in the center of each of these pictures, Photo of Daniel Rogalski is courtesy of Daniel Rogalski. and its orbit is drawn in white. Left: The set 퐾푐 is connected and has no interior. The critical point eventually maps to a repelling cycle of period 2. Center: The set 퐾푐 is a Cantor set. The critical point escapes to inﬁnity under iteration. Right: The set 퐾푐 is connected, and every interior point (including the critical point) is attracted to an attracting cycle of period 4.

퐾푐 is connected if and only if the orbit of the critical point 푧0 = 0 is bounded, that is, if and only if 0 ∈ 퐾푐. As another example, if 푝푐 possesses an attracting periodic cycle, then the critical point is necessarily attracted to it under iteration. The Mandelbrot Set Moving from the 푧-plane to the 푐-parameter plane, one considers the Mandelbrot set Sarah C. Koch 푀 ∶= {푐 ∈ ℂ | 퐾푐 is connected} = {푐 ∈ ℂ | 0 ∈ 퐾푐}. Postcritical Sets in Complex Dynamics The Mandelbrot set (see Figure 2) is a fundamental object I’ll begin by introducing some key objects and ideas in the field of complex dynamics. It has been thoroughly from the field of complex dynamics, specifically looking studied over the past few decades with much success, and at the dynamics of quadratic polynomials. I will then discuss the work I plan to present at the AMS meeting in April 2017. Iterating Quadratic Polynomials For each complex number 푐, consider the map 2 푝푐 ∶ ℂ → ℂ given by 푝푐 ∶ 푧 ↦ 푧 + 푐.

Although 푝푐 is just a quadratic polynomial, a wealth of complicated and deep behavior can emerge when it is iterated. Given 푧0 ∈ ℂ, the orbit of 푧0 is the sequence 푧푛 ∶= 푝푐(푧푛−1). There is a nonempty compact subset 퐾푐 ⊆ ℂ associated to iterating 푝푐 called the filled Julia set of 푝푐; see Figure 1. By definition 퐾푐 consists of all 푧0 ∈ ℂ such that the orbit of 푧0 is bounded. In complex dynamics, the orbits of the critical points (those points where the derivative of the map vanishes) play an important role. For example, the filled Julia set

Sarah C. Koch is associate professor of mathematics at the Univer- Figure 2. The picture in the center is the 푐-parameter sity of Michigan. Her research is supported in part by the NSF and 2 plane for the family 푝푐(푧) = 푧 + 푐; the Mandelbrot set the Sloan Foundation. Her e-mail address is [email protected]. is drawn in black. Each of the surrounding pictures For permission to reprint this article, please contact: contains a ﬁlled Julia set 퐾푐, and the arrows indicate [email protected]. the corresponding value of 푐. DOI: http://dx.doi.org/10.1090/noti1493

April 2017 Notices of the AMS 339 AMSSPRINGSECTIONALSAMPLER it continues to be a central topic of research. It is compact Postcritically Finite Rational Maps and connected. Among the connected components of its Let 푓 ∶ ℙ1 → ℙ1 be a rational map on the Riemann sphere interior are the hyperbolic components, which consist of of degree 푑 ≥ 2. By the Riemann–Hurwitz formula, 푓 has those parameters 푐 such that the polynomial 푝 possesses 푐 2푑 − 2 critical points, counted with multiplicity; let 퐶푓 be an attracting periodic cycle. the set of critical points of 푓. The map 푓 is postcritically Postcritically Finite Quadratic Polynomials finite if the postcritical set 푛 Of particular interest are the polynomials 푝푐 for which the 푃푓 ∶= ⋃ 푓 (퐶푓) 푛>0 orbit of the critical point 푧0 = 0 is finite. Such a polynomial is said to be postcritically finite. If 푝푐 is postcritically finite, is finite. then the critical point is either periodic or preperiodic; In My Talk that is, it eventually maps to a periodic cycle under 푝 . 푐 Based on joint work with L. DeMarco and C. McMullen, Therefore, if 푝 is postcritically finite, then 0 ∈ 퐾 , so 퐾 푐 푐 푐 we study the subsets 푋 ⊆ ℙ1 that arise as 푃 for some is connected. It follows that the parameters 푐 for which 푝 푓 푐 postcritically finite rational map 푓 and also investigate is postcritically finite are all contained in 푀. In fact, they the extent to which the combinatorics of 푓 ∶ 푃 → 푃 can can be found explicitly, as demonstrated in the following 푓 푓 be specified. We employ a variety of results to explore example. this problem, ranging from Belyi’s celebrated theorem [B] Example. Suppose we are interested in finding all parame- to analytic techniques used in the proof of Thurston’s ters 푐 ∈ 푀 for which the critical point 푧 = 0 is periodic of 0 topological characterization of rational maps [DH], one period 3. The equation defined by the critical orbit relation of the most central theorems in complex dynamics. (푝 )3(0) = 0 is 푐 References 푐(푐3 + 2푐2 + 푐 + 1) = 0. [B] G. V. Belyi, On Galois extensions of a maximal cyclotomic field, Math. USSR-Izv. 14 (1980), 247–256. MR0534593 We are interested in the three roots of 푐3 + 2푐2 + 푐 + 1 [DH] A. Douady and J. H. Hubbard, A proof of Thurston’s topo- (there is a degenerate solution at 푐 = 0 which we ignore). logical characterization of rational functions, Acta Math. 171 One root is real, and the other two are complex conju- (1993), 263–297. MR1251582 gates. Thus, there are three quadratic polynomials 푝푐 for which 0 is periodic of period 3. The filled Julia sets of Credits these polynomials are drawn in Figure 2: the two at the Figures 1 and 2 are produced with FractalStream, which is top, and the one at the lower right. written by M. Noonan and freely available online. More generally, each 푐 ∈ 푀 for which 0 is periodic un- Photo of Sarah Koch is courtesy of Sarah Koch. der 푝푐 is an algebraic integer, as it is determined by the 푛 critical orbit relation (푝푐) (0) = 0. Each of these parameters is contained in a hyperbolic component of 푀, and every hyperbolic component contains one and only one such parameter, called its center. This has been incredibly fruitful for understanding the structure of 푀; indeed, using the dynamics of 푝푐 as 푐 runs over all centers, one can essentially catalog and organize all hyperbolic components. The arrows in Figure 2 point to four centers of hyperbolic components in 푀. The parameters 푐 ∈ 푀 for which 0 is not periodic itself but strictly preperiodic are also algebraic integers. Indeed, each of these parameters is determined by a critical or- 푚 푛 bit relation (푝푐) (0) = (푝푐) (0). The strictly preperiodic parameters are dense in the boundary of 푀. If 푐 is algebraic, then the critical orbit 0 ⟼ 푐 ⟼ 푐2 + 푐 ⟼ (푐2 + 푐)2 + 푐 ⟼ ⋯ is also algebraic. Therefore, if 푝푐 is postcritically finite, then the critical orbit is a finite algebraic subset of ℂ. One question to explore is, Which finite algebraic subsets of ℂ arise as critical orbits of 푝푐 as 푐 ∈ 푀 runs over all postcritically finite parameters? There is no reason to restrict our attention to just postcritically finite quadratic polynomials; there is a completely analogous discussion for rational maps.

340 Notices of the AMS Volume 64, Number 4