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Events for January – May 2017

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Jan. 17, 2017 3pm (MATH 350)	Topology Angès Beaudry (CU Boulder) An Overview of Spectra
Jan. 24, 2017 1pm (MATH 220)	Peter Mayr (CU Boulder) X We discuss D. Zhuk's proof that for every finite algebra A the growth of generating sets of direct powers of A is either bounded above by a polynomial or bounded below by an exponential function. Size of generating sets of powers
Jan. 24, 2017 2pm (Math 220)	Keith Kearnes (CU) Equational classes whose finitely generated members are free.
Jan. 24, 2017 3pm (MATH 350)	Topology Sebastian Bozlee and Cherry Ng (CU Boulder) Cohomology Theories and Brown Representability
Jan. 24, 2017 3pm (Math 220)	Functional Analysis Judith Packer (CU Boulder) X This will be an organizational meeting for the seminar. We will be scheduling speakers, etc. Organizational Meeting
Jan. 31, 2017 11am (MATH 350)	Number Theory Tien Trinh (University of Colorado, Boulder) X I will talk about the growth of Fourier coefficients of automorphic eigenfunctions on SL(3,R) to verify part of a conjecture of Miatello and Wallach, who posit that the moderate growth condition is superfluous for real semi-simple Lie groups with reduced R-rank >=2. Estimates on Non-decaying Whittaker functions and the Miatello-Wallach’s conjecture
Jan. 31, 2017 12:10pm (MATH …	Kempner Jim Gates (University of Maryland) Career Talk Sponsored by the Meyer Fund
Jan. 31, 2017 3pm (MATH 220)	Topology Nikki Sanderson and Katharine Adamyk (CU Boulder) X * Note the Room Change! * Prespectra and Spectra
Feb. 01, 2017 5pm (Math 350)	MathClub Markus Pflaum (CU Boulder) X Despite topology being regarded as a very abstract mathematical theory it has surprising applications. In particular in physics and chemistry, topological effects appear abundantly. In chemistry, the topological phenomenon of chirality has been known for a long time. In this talk some fundamentals of topology are explained and it is shown how topology can be used to study and describe molecules. With the help of examples it will be explained that the topological study of molecules is not only of theoretical and scientific interest, but has applications in the life sciences and medicine. Chemical Topology
Feb. 07, 2017 2pm (MATH 220)	None. We'll skip this week.
Feb. 07, 2017 2pm (MATH 350)	Lie Theory Richard Green (CU) X Generalized nil Temperley--Lieb algebras, which are defined by generators and relations, are certain associative algebras arising from Coxeter systems. I will discuss the extent to which it is possible to classify the finite dimensional representations of such algebras. This will involve a review of some important topics in representation theory, including Morita equivalence and representation type. Representation theory of nil Temperley--Lieb algebras and related algebras
Feb. 07, 2017 3pm (MATH 350)	Topology Matthew Pierson (CU Boulder) The Stable Homotopy Category as a Triangulated Symmetric Monoidal Category
Feb. 07, 2017 3pm (Math 220)	Functional Analysis Alex Kumjian (University of Nevada, Reno) X Let $G$ be an amenable locally compact groupoid, and let $A$ be a closed subgroup of a locally compact abelian group $B$ . Given a $B/A$ -valued $1$ -cocycle $ϕ$ on $G$ , there is a central extension $Σ_{ϕ}$ of $G$ by $A$ that is trivial iff $ϕ$ lifts to a $B$ -valued cocycle. We prove that $C(Σφ)$ is isomorphic to the induced algebra of the natural action of $B/A$ on $C(G)$ . We also consider a simple class of examples arising from Cech $1$ -cocycles. This is joint work with Marius Ionescu. Obstructions to Lifting Cocycles on Groupoids and the Associated $C^{*}$ -Algebras (Part I)
Feb. 08, 2017 4pm (MATH 350)	Grad Student Seminar Cliff Blakestad X We will introduce the notion of formal group laws, which are power series somehow carrying the "essence" of a group. Originally developed to study Lie groups and Lie algebras, their use has extended into number theory, topology, and algebraic geometry, but they are also interesting in their own right. For us, they will provide a very concrete example of a category whose objects are not sets and how one can make progress in studying such a category. Only basic familiarity with groups and power series is assumed. Groups without Sets
Feb. 09, 2017 3pm (MATH 350)	Probability Soumik Pal (University of Washington) X A popular family of models of random partitions is called the Chinese Restaurant Process. We imagine n customers being seated randomly and sequentially at tables of a restaurant according to a fixed stochastic rule. Counting customers by the tables gives us a composition of n. Consider a Markov chain on such compositions where we remove a randomly chosen customer and reseat her. How can one describe the limit of such a Markov chain as n tends to infinity? We will construct such limits as diffusions on partitions of the unit interval. The stationary distributions of such diffusions are given by the complement of the zeros of the Brownian motion or the Brownian bridge. The processes of ranked interval lengths are members of a family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009). Our construction is a piece of a more elaborate diffusion on the space of real trees, stationary with respect to the law of the Brownian Continuum Random Tree, whose existence has been conjectured by David Aldous. Joint work with Noah Forman, Doug Rizzolo, and Matthias Winkel. Up-down Markov chains on partitions and their diffusion analogs
Feb. 14, 2017 1pm (MATH 220)	Robert Gray (University of East Anglia, UK) X Two fundamental finiteness properties in group theory are those of being finitely generated and of being finitely presented. These two finiteness properties were generalised to higher dimensions by C. T. C. Wall in 1965. A group G is said to be of type Fn if it has an Eilenberg–MacLane complex K(G,1) with finite n-skeleton. (Here K(G,1) is a certain nice topological space with fundamental group G.) It may be shown that a group is finitely generated if and only if it is of type F1 and is finitely presented if and only if it is of type F2. Related to this is a certain homological finiteness property called FPn. This property is defined for a monoid S in terms of the existence of certain resolutions of free left ZS-modules. The property FPn was introduced for groups by Bieri in (1976). In monoid and semigroup theory the property FPn arises naturally in the study of string rewriting systems. The connection between complete rewriting systems and homological finiteness properties is given by a result of Anick (1986) which shows that a monoid that admits such a presentation must be of type FPn for all n. For groups, Fn and FPn are equivalent for n = 0,1, while results of Bestvina and Brady (1997) show that FP2 is definitely weaker than F2. For higher n there are no further differences, and in general a group G is of type Fn (for n > 1) if and only if it is finitely presented and of type FPn. Because of the connection with rewriting systems, the the finiteness property FPn for monoids has received a great deal of attention in the literature. It is sometimes easier to establish the topological finiteness properties Fn for a group than the homological finiteness properties FPn, especially if there is a suitable geometry/topological space on which the group acts in a nice way. Currently no theory of Fn exists for monoids. In this talk I will describe some recent joint work with Benjamin Steinberg (City College of New York) which was motivated by the question of developing a useful notion of Fn for monoids. This led us to develop a theory of monoids acting on CW complexes. I will explain the ideas we have developed and some of their applications. M-CW complexes and topological finiteness properties of monoids Sponsored by the Meyer Fund
Feb. 14, 2017 2pm (MATH 350)	Lie Theory Richard Green (CU) X Generalized nil Temperley--Lieb algebras, which are defined by generators and relations, are certain associative algebras arising from Coxeter systems. I will discuss the extent to which it is possible to classify the finite dimensional representations of such algebras. This will involve a review of some important topics in representation theory, including Morita equivalence and representation type. Representation theory of nil Temperley--Lieb algebras and related algebras II
Feb. 14, 2017 3pm (Math 220)	Functional Analysis Alex Kumjian (University of Nevada, Reno) X Let $G$ be an amenable locally compact groupoid, and let $A$ be a closed subgroup of a locally compact abelian group $B$ . Given a $B/A$ -valued $1$ -cocycle $ϕ$ on $G$ , there is a central extension $Σ_{ϕ}$ of $G$ by $A$ that is trivial iff $ϕ$ lifts to a $B$ -valued cocycle. We prove that $C(Σφ)$ is isomorphic to the induced algebra of the natural action of $B/A$ on $C(G)$ . We also consider a simple class of examples arising from Cech $1$ -cocycles. This is joint work with Marius Ionescu. Obstructions to Lifting Cocycles on Groupoids and the Associated $C^{*}$ -Algebras (Part II)
Feb. 14, 2017 3pm (MATH 350)	Topology Jordan Watts (CU Boulder) Vector Bundles and the Thom Isomorphism Theorem
Feb. 15, 2017 5pm (Math 350)	MathClub Nat Thiem (CU Boulder) X Unipotent polytopes are a family of geometric shapes that arise in a fundamental combinatorial puzzle. Such problems in turn have a linear algebraic interpretation in terms of matrix ranks. This talk introduces these three perspectives to the same problem and explores some of the open questions I'd like answered. Unipotent polytopes
Feb. 21, 2017 11am (MATH 350)	Number Theory Arthur Baragar (University of Nevada Las Vegas) X The Apollonian circle and sphere packings are well known classical objects. The four dimensional analog, though, has long been thought to not exist, owing to a perceived obstruction. In this talk, we describe the four dimensional analog and the obstruction. Most of the talk will be geometrical, requiring little background, and with only a little emphasis on the connection to number theory. Near the end of the talk, if time permits, we will describe how all these objects can be thought of as ample cones for elliptic K3 surfaces. Apollonian packings in higher dimensions and ample cones on K3 surfaces
Feb. 21, 2017 12:10pm (Math …	Kempner Peter Elliott (CU Boulder) X A short history of key theorems and their methodology will be given , leading to more recent results and an open ended problem that appears disarmingly simple. Asymptotic estimates for mean values of multiplicative functions, then and now
Feb. 21, 2017 1pm (MATH 220)	Jeffrey Shriner (CU Boulder) X For a fixed finite algebra A, the subpower membership problem is the decision problem which asks, given tuples a_1,...,a_n,b of A^m, is b in the subalgebra generated by a_1,...,a_n? We are interested in classifying the complexity of this problem, which is dependent on the algebra A. Much work has been done towards this goal, though there are still large unanswered questions, particularly within congruence modular varieties. We present an example of an algebra with a Taylor term whose subpower membership problem is NP-hard, and show how this example extends to other examples of interest within congruence modular varieties. NP-hardness Results for the Subpower Membership Problem, Part 1
Feb. 21, 2017 2pm (MATH 220)	Jakub Bulin (CU) X The aim of this talk series is to present Zhuk's technique of untangling primitive positive definitions of essential relations into formulas of tree shape. This technique was successfully applied to obtain several important results about finitely related algebras. Most recently, Barto's proof of the Valeriote conjecture: In a congruence modular variety, all finitely related algebras have few subpowers. Blockers, loop lemmata and tree formulas, 1
Feb. 21, 2017 3pm (Math 220)	Functional Analysis Alex Kumjian (University of Nevada, Reno) X Let $G$ be an amenable locally compact groupoid, and let $A$ be a closed subgroup of a locally compact abelian group $B$ . Given a $B/A$ -valued $1$ -cocycle $ϕ$ on $G$ , there is a central extension $Σ_{ϕ}$ of $G$ by $A$ that is trivial iff $ϕ$ lifts to a $B$ -valued cocycle. We prove that $C(Σφ)$ is isomorphic to the induced algebra of the natural action of $B/A$ on $C(G)$ . We also consider a simple class of examples arising from Cech $1$ -cocycles. This is joint work with Marius Ionescu. Obstructions to Lifting Cocycles on Groupoids and the Associated $C^{*}$ -Algebras (Part III)
Feb. 21, 2017 3pm (Math 350)	Algebraic Geometry Graeme Wilkin (NUS) X Title: The reverse Yang-Mills-Higgs flow in the neighbourhood of a critical point Abstract: The Yang-Mills-Higgs flow was originally studied by Simpson in the context of constructing Hermitian-Einstein metrics on Higgs bundles. Methods of Donaldson and Simpson show that the Yang-Mills-Higgs flow resembles a nonlinear heat equation on the space of Hermitian metrics on the bundle and therefore the downwards flow is well-behaved with respect to existence and uniqueness, and it also has nice smoothing properties. On the other hand, the reverse flow is ill- posed and therefore even existence of solutions is problematic. In this talk I will describe a new method to construct long-time solutions to the reverse Yang-Mills-Higgs flow that converge to a given critical point. The methods naturally lead to an algebraic criterion for two critical points to be connected by a flow line and a geometric condition to distinguish between broken and unbroken flow lines in terms of secant varieties of the underlying curve. If time permits I will talk about work in progress to describe the compactification of the space of flow lines in terms of the geometry of secant varieties. The reverse Yang-Mills-Higgs flow in the neighbourhood of a critical point Sponsored by the Meyer Fund
Feb. 21, 2017 4pm (MATH 350)	Topology Andy Riddle (CU Boulder) Thom Spectra and the Pontryagin-Thom Theorem
Feb. 28, 2017 11am (MATH 350)	Number Theory Pedro Berrizbeitia (ULAM Visitor) X Let $x$ and $n$ be integers, $n >1$ . If $xn-1$ is prime, then it is a Mersenne Prime! The 6 largest known primes are Mersenne primes, and so are 10 of the 11 largest primes. We start the talk reviewing the properties of Mersenne Numbers which contribute to explain this fact. The desire of finding other sequences of natural number with similar properties led eventually to the notion of ``Algebraic Mersenne Primes," which I define as prime numbers that are the absolute norm of $αn -1$ for some algebraic integer $α$ and some $n >2$ . Simple necessary conditions on $α$ and $n$ are given without which the sequence $Mα,n = \| NQ(α)/ Q(αn -1)\|$ contains no prime. The first interesting examples of sequence we studied are the Gaussian-Mersenne Primes, $M1-i,p= 2p ± 2p+1/2+1$ and the Eisenstein-Primes, $M1-ω,p=3p ± 3p+1/2 +1$ . The use of the Biquadratic Reciprocity Law and the Cubic Reciprocity Law are the key ingredients for the primality test for this sequence obtained in a joint paper with Boris Iskra. We then will show the results obtained by Miguel Rodriguez in his master thesis in 2004. He studied the sequence $M2-(ζ5 + ζ5-1), p =5p - Fp 5p+1/2 + 1$ , where $Fp = p$ -th Fibonnaci number, for which the Proth Primality test can be used. In 2006 V\'{\i}ctor Ram\'{\i}rez gave a deterministic primality test based on variants of AKS, for an infinite set of sequences, which we named Cyclotomic Mersenne Integers, $M1+ζm,p$ , $m ≥ 5$ . For the test, he made use of a reciprocity Law due to Hasse and extended my variant of AKS for its purpose. We also present recent results obtained by Euro Lucena in his mater's thesis in 2015. In his thesis he studies the properties of $M3+3,p$ and uses the Reciprocity Law of order 6 to give the primality test. He also searches for primes of the form $αn + 1$ and obtains interesting new results, one of which is a primality test based on the cubic test! The talk will have a historic flavor and we will include related results from Lehmer and collaborators and recent results of Palimar and Shankra on ``Mersenne primes in Real Quadratic Fields". Algebraic Mersenne Numbers
Feb. 28, 2017 12:10pm (Math …	Kempner Timothy Trujillo (Colorado School of Mines) X The two-dimensional Ramsey theorem states that if the edges of a complete graph with countably infinitely many vertices are colored with two colors then there is a complete subgraph with countably infinitely many vertices all whose edges are monochromatic. The theorem can be extended to all finite dimensions. However, the naive infinite-dimensional Ramsey conjecture is false in ZFC. Nonetheless the collection of all sets that satisfy the conclusion of the infinite-dimensional Ramsey theorem form a sigma-algebra of sets. The Ellentuck theorem characterizes this sigma-algebra as the sigma-algebra of sets with the Baire property ( that is, a symmetric difference of an open set and a meager set ) with respect to a topology that extends the natural metric topology on the collection of infinite subsets of the natural numbers. We will discuss the historical development of the Ellentuck theorem and use it to motivate and introduce topological Ramsey theory and the abstract Ellentuck theorem. Time permitting, we will discuss some new examples of topological Ramsey spaces and their applications. Topological Ramsey theory
Feb. 28, 2017 1pm (MATH 220)	Jeffrey Shriner (CU Boulder) NP-hardness Results for the Subpower Membership Problem, Part 2
Feb. 28, 2017 2pm (Math 220)	Jakub Bulin (CU) X Continuation. Blockers, loop lemmata and tree formulas, 2
Feb. 28, 2017 3pm (MATH 350)	Topology Cliff Blakestad (CU Boulder) Quillen's Theorem: MU and Formal Group Laws
Feb. 28, 2017 3pm (Math 220)	Functional Analysis Leonard Huang (CU Boulder) X The uniqueness of the Heisenberg Commutation Relations is usually stated in the language of unitary operators on a Hilbert space. This is done by first using the Baker-Campbell-Hausdorff Formula to integrate the Heisenberg Commutation Relations to the Weyl Commutation Relations and then applying the Stone-von Neumann Theorem. The question thus remains whether or not the uniqueness result can be stated purely in the original language of self-adjoint unbounded operators on a Hilbert space. Using the concept of analytic vectors created by Edward Nelson in the 1950s and a result about such vectors due to Moshé Flato and his collaborators, we attempt to answer the question in the affirmative. Uniqueness of the Heisenberg Commutation Relations from the Viewpoint of Unbounded Operator Theory
Mar. 01, 2017 4:03pm (MATH 3…	Grad Student Seminar Sarah Salmon X Coxeter groups are defined by generators (all order two) and relations (called braids). In fact, some of your favorite groups are actually Coxeter groups. When we work with Coxeter groups, we get some nice tools like a well-defined length, root systems, a partial ordering, and more! In this talk, we'll go through some definitions, build geometric intuition, and (as always) draw some pictures! An introduction to groups you don't know you've met
Mar. 07, 2017 11am (MATH 350)	Number Theory Beth Malmskog (Villanova University) X Picard curves are genus 3 curves of the form y^3=f(x), where f(x) is a polynomial of degree 4. They are the simplest non-hyperelliptic curves. This talk will discuss work with Chris Rasmussen, in which we found all Picard curves defined over the rationals with good reduction at all primes except p=3. Our work was inspired by Nigel Smart's enumeration of genus 2 curves with good reduction at all primes except p=2. As part of this project, we wrote code in SageMath to solve the equation x+y=1 over the S-units of some number fields. In collaboration with Alejandra Alvarado, Angelos Koutsianas, Christelle Vincent, and Mckenzie West, we recently generalized the implementation and have submitted our functions for release in upcoming releases of Sage. The S-unit equation and Picard curves good away from p=3
Mar. 07, 2017 3pm (MATh 350)	Topology Markus Pflaum (CU Boulder) K-Theory
Mar. 07, 2017 3pm (Math 220)	Functional Analysis Timothy Rainone (Arizona State University) X Blackadar and Kirchberg introduced the notion of a matricial field (MF) $C^{}$ -algebra; a $C^{}$ -algebra is MF if it can be embedded into a corona of matrix algebras. They asked whether every stably finite $C^{}$ -algebra satisfies the MF property; this is the $C^{}$ -analogue of the Connes Embedding Problem. I will discuss this problem in the context of $C^{}$ -dynamical systems and group $C^{}$ -algebras. We show that both stable finiteness and the MF property admit equivalent $K$ -theoretic properties. We use deep results from the classification literature to lift $K$ -theoretic dynamical information and thus answer the BK question in the affirmative for crossed products of certain classifiable $C^{}$ -algebras by free groups. We will also see that semi-direct product groups of the form $G ⋊ 𝔽_{r}$ , where $G$ is amenable, admit MF reduced group $C^{}$ -algebras. This is joint work with Chris Schafhauser. A $K$ -Theoretic Approach to the Blackadar-Kirchberg Question Sponsored by the Meyer Fund
Mar. 08, 2017 4:03pm (MATH 3…	Grad Student Seminar Michael Wheeler X Nonstandard analysis was originally used to answer the longstanding controversy in calculus as to whether the use of infinitesimals could be given a rigorous foundation. However, the nonstandard methods have since been applied to several other fields (notably infinite Ramsey theory and topological dynamics) to significantly reduce the complexity of certain (rather long) proofs. While we will not cover these examples explicitly, the goal will be to provide enough of the machinery of nonstandard analysis to be able to see how nonstandard methods and interpretations can be applied to a variety of disciplines. In particular, we will describe what a nonstandard framework is, discuss some properties of frameworks that are enlargements, and give a quick proof of the Stone representation theorem for Boolean algebras using nonstandard methods. Time permitting, we will finish with a brief listing of some interesting properties of the hyperreals, including how to think about germs of functions using monads. Nonstandard Analysis: A Talk about Almost Nothing
Mar. 14, 2017 12:10pm (MATH …	Kempner Mark Behrens (Notre Dame) X In physics, the detection of certain particles has required working at increasingly high energy. These accelerator experiments have required increasingly elaborate detectors and data analysis to reconstruct events from the production of dozens, sometimes hundreds of resulting particles. I will use this analogy to discuss the increasingly elaborate "detectors" used in homotopy theory to probe the stable homotopy groups of spheres in increasingly large dimensions. These "detectors" are versions of the Adams spectral sequence. To maximize the accessibility of this talk, I will avoid using the language of spectral sequences. Detectors in homotopy theory
Mar. 14, 2017 2pm (Math 220)	Peter Jipsen (Chapman University) X The study of Boolean algebras with operators dates back to Jónsson and Tarski’s seminal 1951-52 papers, and has many applications in the theory of relation algebras, modal logic and computer science. It was generalized to distributive lattices with operators by Gehrke and Jonsson in 1994, and to lattices with operators by Gehrke and Harding in 2001. In this talk we consider Heyting algebras with operators, with applications to residuated Heyting algebras and intuitionistic relation algebras. We define weakening relation algebras as division-conuclear images of relation algebras, and show that they are distributive involutive residuated lattices with a Heyting implication. The class of representable weakening relation algebras is not finitely based, but all members are downset-algebras of partially ordered Brandt groupoids. Furthermore we prove that downset algebras of po-groupoids are cyclic involutive residuated Heyting algebras, and we calculate the Gumm-Ursini ideal terms that describe congruences of residuated Heyting algebras. This is joint research with Nick Galatos from the University of Denver. Residuated Heyting algebras and intuitionistic relation algebras
Mar. 14, 2017 2pm (MATH 350)	Lie Theory Sarah Wolff (Denison University) X The (classical) Fast Fourier Transform remains among the most important algorithms in information processing. It is in fact not one algorithm, but a family of algorithms that together effect the efficient computation of the discrete Fourier transform (DFT). We take an algebraic point of view and cast the DFT as a change of basis in the complex group algebra of the cyclic group of order n. This in turn gives rise to a more general notion of the DFT on an arbitrary finite group or semisimple algebra, with concomitant questions of algorithmic efficiency. We will present a diagrammatic approach using Bratteli diagrams to construct efficient algorithms for computing these transforms, and will then discuss a random walk application Fourier Analysis on Groups and Algebras: an Algorithm and a Walk Sponsored by the Meyer Fund
Mar. 14, 2017 3pm (MATH 350)	Topology Mark Behrens (University of Notre Dame) X Quillen-Sullivan theory encodes a rational homotopy type in either a differential graded rational Lie algebra, or a differential graded rational commutative algebra. The first encodes the rational homotopy groups, with its Whitehead products, and the second encodes the rational cohomology groups, with its cup products. I will describe a generalization of this theory, joint with Charles Rezk, where "rational homotopy theory" is replaced with "unstable v_n-periodic homotopy theory. A generalization of Quillen-Sullivan rational homotopy theory Sponsored by the Meyer Fund
Mar. 14, 2017 4pm (MATH 350)	Topology Xiang Tang (Washington University at St. Louis) X A G-gerbe is a principal BG bundle, where BG is the classifying space of G. In this talk, we will explain what they are, and introduce a Leray-Hirsch principle to study the geometry of G gerbes on orbifolds. The key ingredient in our development is the Mackey machine for the restriction of a representation. We will aim to make the talk accessible to graduate students. A Leray-Hirsch Principle for Gerbes on Orbifolds Sponsored by the Meyer Fund
Mar. 21, 2017 1pm (MATH 220)	Miklos Maroti (Vanderbilt University) X The set Quo(A) of compatible quasiorders (reflexive and transitive relations) of an algebra A forms a lattice under inclusion, and the lattice Con(A) of congruences of A is a sublattice of Quo(A). We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice M3. We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices. The shape of quasiorder lattices of varieties Sponsored by the Meyer Fund
Mar. 21, 2017 2pm (MATH 220)	Joshua Wiscons (California State University, Sacramento) X In this talk, I will discuss a handful of old and new applications of the geometry of involutions to the study of groups of small Morley rank. I will also present some early meditations on the role that generically defined geometries may play in analyzing simple groups of finite Morley rank with a small Prufer 2-rank. Geometries and small groups of finite Morley rank Sponsored by the Meyer Fund
Mar. 21, 2017 3pm (MATH 350)	Topology Agnès Beaudry (CU Boulder) S-Modules
Mar. 21, 2017 4pm (MATH 220)	Geometry/Analysis Chi Hin Chan (National Chiao Tung University) X This talk is dedicated to the proof of the Holder continuity regularization effects of Hamilton- Jacobi equations. The proof is based on the De Giorgi method. The regularization is independent of the regularity of the Hamiltonian. The Holder Regularity of Solutions to Hamilton-Jacobi Equations
Mar. 23, 2017 12:10pm (MATH …	Kempner Elizabeth Gillaspy (University of Munster) X The space $E∞$ of infinite paths in a directed graph E is naturally a Cantor set; Julien and Savinien (building on work by Pearson and Bellissard) have studied spectral triples for such Cantor sets, and in particular shown that the Laplace-Beltrami operators $Δss ∈ R$ of these spectral triples all have the same eigenspaces. In joint work with Farsi, Julien, Kang, and Packer, we have discovered that this eigenspace decomposition agrees with a wavelet-type orthogonal decomposition of $L2(E∞, M)$ first introduced by Marcolli and Paolucci. Moreover, the Dixmier trace measure associated to the Pearson-Bellissard spectral triple agrees with the Hausdorff measure on the Cantor set $E∞$ . In my talk, I will sketch the proofs of the above results; no prior knowledge of spectral triples, wavelets, or Hausdorff measure will be assumed. Wavelets, spectral triples, and Hausdorff measure for directed graphs Sponsored by the Meyer Fund
Mar. 23, 2017 3pm (MATH 350)	Probability Nathaniel Eldredge (University of Northern Colorado) X Hypercontractivity is a property of Gaussian measure which says, roughly, that the transition semigroup of the Ornstein-Uhleneck process improves the integrability of functions as time passes, at a certain characteristic rate. It is intimately connected to another famous property, the logarithmic Sobolev inequality, which relates the integrability of a function and its derivative. If we work in a complex setting and restrict our attention to analytic functions, a stronger version of hypercontractivity turns out to hold; integrability improves faster. I'll discuss how these topics fit together in the classical setting of Euclidean space with Gaussian measure, then introduce some recent work extending these results to the context of stratified Lie groups with hypoelliptic heat kernel measure. This talk includes joint work with Leonard Gross and Laurent Saloff-Coste. Strong hypercontractivity on stratified Lie groups
Apr. 04, 2017 12:10pm (Math …	Kempner Andrew Linshaw (University of Denver) X Vertex algebras are a class of noncommutative, nonassociative algebras that arose out of conformal field theory in the 1980s, and were axiomatized by Borcherds in his proof of the Moonshine Conjecture. They have applications in many areas of mathematics including representation theory, combinatorics, finite group theory, number theory, and algebraic geometry. I will give an introduction to the subject and will discuss some recent progress on a certain classification problem. Introduction to vertex algebras
Apr. 04, 2017 2pm (MATH 350)	Lie Theory Megan Ly and Shawn Burkett (CU) X Classical Schur-Weyl duality relates the irreducible characters of the symmetric group $S_{n}$ to the irreducible characters of the general linear group $GLn(C)$ via their commuting actions on tensor space. We investigate the analog of this result for the group of unipotent upper triangular matrices $UTn(Fq)$ . In this case the character theory of $UTn(Fq)$ is unattainable, so we must employ supercharacter theory, creating a striking variation. AND The normal subgroups of a finite group $G$ can be realized as intersections of stabilizers of the irreducible $G$ -modules. A supercharacter theory of $G$ is an analogue to the representation theory of $G$ where the theory is built from nearly irreducible modules. Via stabilizers, each such theory gives a sublattice of the lattice of normal subgroups, along with similar theories on each subquotient of the lattice. Now given a sublattice of the lattice of normal subgroups of a finite group $G$ and supercharacter theories on the covering relations, it is natural to ask under what conditions a supercharacter theory of $G$ can be built that respects the imposed covering relations. Recently, Aliniaeifard gave a construction which allows one to build from a sublattice $ℒ$ a supercharacter theory with $ℒ$ as its associated sublattice. However, this construction does not allow for the choice of supercharacter theories on each subquotient. In this talk, we will discuss Aliniaeifard's construction as well as some methods being developed to refine it to account for more general covering relations. Centralizer Algebras of Unipotent Upper Triangular Matrices AND Constructing supercharacter theories from lattices of normal subgroups
Apr. 04, 2017 3pm (Math 350)	Algebraic Geometry Dhruv Ranganathan (MIT / IAS) X The Brill-Noether varieties of Riemann surface C are objects of classical interest. They parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension, i.e. equations for the curve. When C is general, these varieties are well understood: they are smooth, irreducible, and have the "expected" dimension. As one ventures deeper into the moduli space, past the general locus, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill--Noether theorem, which determines the dimensions of the Brill-Noether varieties on a general curve of fixed gonality, i.e. "general inside a chosen special locus”. In the first hour, I will give an introduction to the tropical linear series techniques that allow combinatorial tools to be used to attack such problems. In the second hour, I will explain some of the new ideas used, which introduce logarithmic geometry as a new tool in Brill-Noether theory. This is joint work with Dave Jensen. The space of equations for an algebraic curve Sponsored by the Meyer Fund
Apr. 06, 2017 1pm (Math 350)	Noncomm Geometry Jacek Brodzki (University of Southampton) X In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. There are applications of the theory surrounding the operator to $C*$ -algebra $K$ -theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of $p$ -adic groups. This talk will present an extension of the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces which relies on the beautiful geometry of nonpositively curved spaces. The utility of this construction is demonstrated through a new proof of $K$ -amenability for groups that act properly on bounded-geometry CAT(0)-cubical spaces. This talk is based on joint work with Nigel Higson and Erik Guentner. A differential complex for CAT(0) spaces.
Apr. 06, 2017 3pm (MATH 350)	Probability Boguslaw Zegarlinski (Imperial College London) X I will review recent results and open problems concerning constructions and ergodicity of dissipative dynamics for classical large interacting systems. In particular I will talk about dynamics defined by formally hypoelliptic generators and some generalised Dunkl type extensions, and how to control their long time behaviour (by use of coercive inequalities or generalised gradient type bounds). Dissipative Dynamics for Large Interacting Systems
Apr. 11, 2017 2pm (MATH 350)	Lie Theory Monica Vazirani (UC Davis) X The BMW algebra is a deformation of the Brauer algebra, and has the Hecke algebra of type A as a quotient. Its specializations play a role in types B, C, D akin to that of the symmetric group in Schur-Weyl duality. One can enlarge these algebras by a commutative subalgebra $X$ to an affine, or annular, version. Unlike the affine Hecke algebra, the affine BMW algebra is not of finite rank as a right $X$ -module, so induction functors are ill-behaved, and many of the classical Hecke-theoretic constructions of simple modules fail. However, the affine BMW algebra still has a nice class of $X$ -semisimple, or calibrated, representations, that don't necessarily factor through the affine Hecke algebra. I will discuss Walker's TQFT-motivated 1-handle construction of the $X$ -semisimple, or calibrated, representations of the affine BMW algebra. While the construction is topological, the resulting representation has a straightforward combinatorial description in terms of Young diagrams. This is joint work with Kevin Walker. Representations of the affine BMW algebra Sponsored by the Meyer Fund
Apr. 11, 2017 2pm (MATH 220)	Stan Wagon (Macalester College) X Ever since the Banach-Tarski Paradox was discovered, people have wondered about eliminating the Axiom of Choice in favor of a system in which the paradox evaporates. But, as discovered by Mycielski and Sierpinski, the most common way of doing this -- taking all sets to be Lebesgue measurable -- leads to a different sort of paradox that is just as disturbing. I will also present a construction that shows how congruent regular tetrahedra can be chained together face-to-face so as to make a loop that is closed to within any preassigned error. This settles a question first raised and studied 60 years ago by Steinhaus and Swierczkowski. A Paradox Due to the Absence of the Axiom of Choice; and a Proof of the Tetrahedral Loop Conjecture
Apr. 11, 2017 3pm (Math 220)	Functional Analysis Marty Walter (CU Boulder) X An interesting connection between functional analysis and number theory. Abelian Groups Revisited
Apr. 11, 2017 3pm (MATH 350)	Topology Paul Lessard (CU Boulder) Symmetric Spectra
Apr. 12, 2017 4:03pm (MATH 3…	Grad Student Seminar Jordan DuBeau X Induction and recursion can be collectively thought of as the hammer in our mathematical toolkit. They are simple, versatile, and ultimately indispensable in constructing our proofs. In this talk, we will present transfinite induction and recursion, which allow us to use the same proof ideas on larger and more complicated sets. This powered-up version of the hammer is the sledgehammer of mathematics. Join us as we use the sledgehammer to prove the existence of a two-point set: a set of points in the plane that intersects every straight line exactly twice. Two-Point Sets: Applying the Mathematical Sledgehammer
Apr. 13, 2017 3pm (MATH 350)	Probability Rinaldo Schinazi (University of Colorado, Colorado Springs) TALK CANCELED!
Apr. 18, 2017 10am (MATH 350)	Geometry/Analysis Peter Vassiliou (Australian National University) X Control systems, typically nonlinear, abound in our daily experience of the world. Mathematically, they can be described by differential equations endowed with functional parameters. The prototypical example is balancing a broomstick on the tip of your finger. A basic problem in control theory is the following. Suppose a control system $ω$ is in state $A$ and you want to ``steer" it to state $B$ . How do you choose the (functional) parameters in the system -- the controls -- in order to achieve this? One oft-used technique is to find a special type of transformation, $T$ , that maps $ω$ to a certain type of universal linear control system called a Brunovsky normal form, $β$ . The trajectories of $β$ can easily be written down in terms of arbitrary functions of time and their time derivatives after which $T-1$ maps the trajectories of $β$ to those of $ω$ , from which the steering problem can be solved. If we can construct such a solution, $s$ of $ω$ , we will say that it is {\em explicitly integrable}. This settles the steering problem for linearizable control systems which form a small subset of control systems of interest. But while every linearizable system is explicitly integrable, the converse is false and for the past few decades there has been a research program underway to characterize the phenomenon of explicit integrability beyond the linearizable case. However it turns out that the Brunovsky normal forms are familiar objects in differential geometry. They correspond to the contact systems on a special class of jet spaces over the real line. In this talk we will explore this geometry to show how to establish explicit integrability for control systems which are intrinsically nonlinear - those that have no linearizing diffeomorphisms. A key ingredient is the invariance of $ω$ under the free and regular action of a Lie group. Symmetry & Geometric Control Theory Sponsored by the Meyer Fund
CANCELED Apr. 18, 2017	Number Theory Alan Reid (University of Texas) TBA
Apr. 18, 2017 12:10pm (Math …	Kempner Rafael Frongillo (CU Boulder) X Symbolic dynamics (in one dimension) is the study of subshifts: subsets of bi-infinite sequences of symbols from a finite alphabet which are invariant under a shift to the left. We say two subshifts are conjugate (isomorphic) if there is a homeomorphism between the underlying subsets which commutes with the shift map. In particular, we will focus on vertex shifts, which are given by bi-infinite walks on a directed graph. A decomposition theorem due to Williams states that every conjugacy between vertex shifts can be written as a sequence of state (vertex) splittings followed by state amalgamations. In this talk, I will introduce the basic definitions of subshifts, and discuss a fascinating problem which has been open for some time: the decidability of conjugacy between vertex shifts. Barring an extreme and unprecedented stroke of insight, I will not solve this problem, but instead focus on a simpler version where only state amalgamations are allowed. We will see that this state amalgamation problem is NP-hard, which has implications for the simplification of Markov shifts. Complexity of conjugacy in 1-dimensional symbolic dynamics
Apr. 18, 2017 1pm (MATH 220)	Arash Rafiey (Computer Science, Indiana State University) X We consider the problem of finding a homomorphism from an input digraph G to a fixed digraph H (HOM(H)). A homomorphism from G to H is a mapping from vertices in G to H that preserve the adjacency (a generalization of coloring problem). A polymorphism of H is just a homomorphism from H^k (for some integer k>1) to H. The algebraic work of several researchers such as Bulatov, Jeavons, Krokhin, Maroti and McKenzie led to discovery of weak-near-unanimity polymorphism and the proof of NP-completeness of HOM(H) when H does not admit any weak-near-unanimity polymorphism. We show that if H admits a weak-near-unanimity polymorphism $ϕ$ then HOM(H) is polynomial time solvable. Our algorithm implies the validity of the dichotomy conjecture due to Feder and Vardi (1993). We transform the problem into an instance of the list homomorphism problem where initially all the lists are full (contain all the vertices of H). We use the polymorphism $ϕ$ as a guide to reduce the lists to singleton lists, which yields a homomorphism if one exists. This is based on joint work with Tomas Feder and Jeff Kinne and many useful discussions with Victor Dalmau and Pavol Hell. Dichotomy for Digraph Homomorphism Problems, Part 1 Sponsored by the Meyer Fund
Apr. 18, 2017 2pm (MATH 350)	Lie Theory James Wilson (Colorado State University) X Data arranged as a high-dimensional grid of numbers is a "tensor". Questions from Hilbert to modern day P vs. NP have been cast as asking: what properties of a tensor are unchanged when we change a basis? We explore tensors with an underlying Lie algebra that upends a century of convention of tensor products with rings. The resulting tensor spaces are highly compressed, display nuanced structure, and some can even be classified. I will show the problems, the method, and the many things we don't yet understand but might soon. Tools to tame tensors Sponsored by the Meyer Fund
Apr. 18, 2017 2pm (MATH 220)	Arash Rafiey (Computer Science, Indiana State University) Dichotomy for Digraph Homomorphism Problems, Part 2 Sponsored by the Meyer Fund
Apr. 18, 2017 3pm (Math 220)	Functional Analysis Konrad Aguilar (University of Denver) X We provide conditions for when quotients of AF algebras are quasi-Leibniz quantum compact metric spaces, building on our previous work with F. Latrémolière. Given a $C^{*}$ -algebra, its ideal space may be equipped with natural topologies. Next, we impart criteria for when convergence of ideals in the Fell topology of an AF algebra can provide convergence of quotients in quantum propinquity. We then apply these findings to a certain class of ideals of the Boca-Mundici AF algebra by providing a continuous map from this class of ideals equipped with various topologies, including the Jacobson and Fell topologies, to the space of quotients with the quantum propinquity topology. Convergence of Quotients of AF Algebras in Quantum Propinquity by Convergence of Ideals Sponsored by the Meyer Fund
Apr. 18, 2017 3pm (MATH 350)	Topology Paul Lessard (CU Boulder) Symmetric Spectra Part II
Apr. 19, 2017 4:03pm (MATH 3…	Grad Student Seminar Matthew Pierson X TBA TBA
Apr. 20, 2017 10am (ECCR 110)	Geometry/Analysis Benjamin Dodson (Johns Hopkins) X In this talk we will discuss global well-posedness and scattering for the defocusing, cubic 3D problem with radial data. To prove this it is useful to use hyperbolic coordinates. Global well-posedness and scattering for the 3D cubic nonlinear wave equation Sponsored by the Meyer Fund
Apr. 20, 2017 2pm (Math 220)	Lie Theory Steve Butler (Iowa State) X We give an introduction to parking functions, give basic counts, and discuss a generalization to trees. In this latter setting we look at using generating functions to help efficiently count the number of these parking functions. Parking functions and parking functions on trees
Apr. 20, 2017 3pm (MATH 220)	Probability Miklos Z. Racz (Microsoft Research, Redmond) X If you add an edge to a random graph, do its conductance properties almost always get better? Perhaps surprisingly the answer is no, and I will discuss how this is connected to delocalization properties of eigenvectors. This is based on joint work with Ronen Eldan and Tselil Schramm. Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors
Apr. 25, 2017 11am (MATH 350)	Number Theory Hershey Kisilevsky (Concordia University) X The (open) Inverse Problem of Galois Theory asks if an arbitrary finite group G can be realized as the Galois group of a finite extension K of the rational field Q. Among the solutions K/Q with Galois group G, the minimally ramified realizations are those with the least number of primes of Q ramified in K. We study minimally ramified extensions of Q for various groups G, and consider the possible decomposition configurations of the primes of Q ramified in such extensions. In the process we are led to consider some novel combinatorial problems. Decomposition of Ramified primes in Minimally Ramified extensions
Apr. 25, 2017 12:10pm (MATH …	Kempner Boris Hanin (MIT) X Consider a polynomial p_N(z) in one complex variable. The Gauss-Lucas Theorem says that the critical points of p_N lie inside the convex hull of its zeros. But how are critical points actually distributed inside the convex hull if p_N is chosen at random? The purpose of this talk is to explain that in fact each critical point of p_N typically comes paired with a single zero. The distance between a critical point and its paired zero is on the order of N^{-1}, which is much smaller than the typical N^{-1/2} spacing between order of N independently selected points on the sphere. In the first part of my talk, I will give a heuristic interpretation for this pairing by relating zeros and critical points to electrostatics on the Riemann sphere. In the second part, I explain what rigorous theorems are now available and state a few open problems. Pairing between zeros and critical points of random polynomials
Apr. 25, 2017 1pm (MATH 220)	Trevor Jack (CU Boulder) X An element x in a semigroup S is regular if there exists some y in S such that xyx = x. We show that given a semigroup S of transformations by its generators, checking whether an element in S is regular is PSPACE-complete. Checking regularity in semigroups
Apr. 25, 2017 2pm (Math 220)	Steve Weinell (CU) Jonsson lattices
Apr. 25, 2017 2pm (MATH 350)	Lie Theory Marty Walter (CU) X We redevelop our previous results in a way which is in some ways more complicated, and in other ways, simpler. The goal was/is to solve a problem, which was/is successful. Abelian Groups Revisited
Apr. 25, 2017 3pm (MATH 350)	Topology Jonathan Wise (CU Boulder) Infinity Categories
Apr. 25, 2017 3pm (Math 220)	Functional Analysis Nathan Davidoff (CU Boulder) X Hawkins et al. showed how to produce spectral triples on the Bunce-Deddens algebra arising from the odometer action on the Cantor set. We show how the Bunce-Deddens algebra arises, up to strong Morita equivalence, from a $ℤ$ -action on a certain rooted tree. Does the construction of spectral triples on graph algebras and crossed products then yield new spectral triples on the Bunce-Deddens algebra? Good question. Spectral Triples and the Bunce-Deddens Algebra
Apr. 27, 2017 3pm (MATH 350)	Probability Boris Hanin (MIT) X Random Hermite functions are eigenfunctions of fixed energy for the isotropic harmonic oscillator in R^d. They therefore transition from highly oscillatory to exponentially damped at the caustic, the analog of the edge of the spectrum in random matrix theory. The purpose of this talk is to explain how the Airy kernel and its relatives appear in the scaling limit of random Hermite functions around the caustic as Planck's constant h goes to 0. Rather than being the kernel of the determinantal process, the Airy kernel will appear as the covariance function of a limiting Gaussian field. This is joint work with Steve Zelditch and Peng Zhou. Airy Scaling of Random Hermite Functions at the Caustic
May. 02, 2017 10am (MATH 350)	Geometry/Analysis Robert Bryant (Duke University) X A Riemannian manifold $M,g)$ is said to be {\it curvature-homogeneous} if it is homogeneous to second order, i.e., if, for any two points in $M$ the Riemannian curvature tensors are equivalent under some isometry of the two tangent spaces. Of course, a locally homogeneous metric is curvature-homogeneous, but the converse is not true in dimensions greater than $2$ . (For a surface (i.e., in dimension $2$ ) curvature-homogeneity is equivalent to having constant Gaussian curvature, and such metrics are, of course, all classified locally and they are locally homogeneous.) Already in dimension $3$ , there are many questions about the existence and generality of curvature homogeneous metrics, even locally. In this case, curvature-homogeneity is equivalent to having the eigenvalues of the Ricci curvature be constant, which is a system of partial differential equations on the metric. In this talk, I will review what is known about such metrics in dimension $3$ , particularly the work of O. Kowalski and his collaborators during the 1990s. I will show that, for certain values of the eigenvalues of the Ricci tensor, these partial differential equations are integrable by the Darboux’ method, which yields some surprising relations with classical subjects, such as the theory of holomorphic curves in the complex projective plane and the geometry of contact curves in the $3$ -sphere. Curvature-homogeneous metrics in dimension 3 Sponsored by the Meyer Fund
May. 02, 2017 11am (MATH 350)	Number Theory Cam Stewart (University of Waterloo) X Let F be a binary form with integer coefficients, non-zero discriminant and degree d with d at least 3.. We shall discuss the problem of estimating the number of square free and more generally k-free integers of absolute value at most Z represented by F. This is joint work with Stanley Xiao. On the representation of k-free integers by binary forns
May. 02, 2017 1pm (MATH 220)	Peter Mayr (CU Boulder) A loop that is not supernilpotent but finitely based
May. 02, 2017 3pm (Math 220)	Functional Analysis Leonard Huang (CU Boulder) X If $f$ is a complex-valued continuous function on the unit circle $𝕋$ that is even (i.e., $f(ei θ) = f(e- i θ)$ for all $θ ∈ R$ ), and all we know about $f$ is its average value on the $n$ -th roots of unity for each $n ∈ N$ , then is it possible to recover $f$ itself from this information? By applying basic Fourier analysis and exploiting properties of the Möbius function and the divisor function, we show that the question has an affirmative answer if $f$ is assumed to be $C2$ on $𝕋$ and to have mean $0$ (i.e., $∫02 π f(ei θ) ~ d/θ = 0$ ). When $f$ is not even, we describe how $f$ may be recovered from the average values of appropriate phase shifts of $f$ on roots of unity. Number-Theoretical Aspects of Averaging a Function on the Unit Circle on Roots of Unity
May. 02, 2017 3pm (Math 350)	Algebraic Geometry Philip Engel (Harvard University) X Philip Engel: The Hurwitz theory of elliptic orbifolds Consider a tiling of a compact, oriented surface by triangles, squares, or hexagons. Enumeration of such tilings can be phrased in the language of Hurwitz theory of an elliptic orbifold. There is a combinatorial curvature associated to each vertex. We will outline a proof that the generating series for the number of tilings with a given number of tiles and specified nonzero curvatures lies in a ring of quasi-modular forms. The asymptotics as the number of tiles goes to infinity allows one to derive formulas for the volumes of strata of cubic, quartic, and sextic differentials---in particular, these volumes are a power of pi times a rational number. This generalizes work of Eskin and Okounkov in the case of abelian and quadratic differentials The Hurwitz theory of elliptic orbifolds Sponsored by the Meyer Fund
May. 03, 2017 4:03pm (MATH 3…	Grad Student Seminar Robert Hines X What do you get when you take something combinatorial (the free group on two generators), cram it any which way into something geometric (SO(3) acting on the sphere), and take for granted the ability to pick apart the resulting mess (the axiom of choice)? Banach-Tarski is what. We'll prove that one can "double the ball," i.e. there is a bijection B\to B\cup B that is defined piecewise by isometries on a finite partition of B. Free bonus "joke" - Q: What's an anagram of "Banach-Tarski"? A: "Banach-Tarski, Banach-Tarski" The Banach-Tarski paradox
May. 04, 2017 3pm (MATH 350)	Probability Yuval Peres (Microsoft Research) X Given $n$ uniform points on the surface of a sphere, how can we partition the sphere fairly among them in an equivariant way? "Fairly" means that each region has the same area. "Equivariant" means that if we rotate the sphere, the solution rotates along with the points. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition—with exactly equal areas, no matter how the points are distributed. Moreover, this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of $n$ blue uniform points to $n$ red uniform points on the sphere. This is joint work with Nina Holden and Alex Zhai. Gravitational allocation to uniform points on the Sphere

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