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Events for April – August 2017

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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 220)	Arash Rafiey (Computer Science, Indiana State University) Dichotomy for Digraph Homomorphism Problems, Part 2 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 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 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 2pm (Math 220)	Steve Weinell (CU) Jonsson lattices
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 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. 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. 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
Jun. 15, 2017 12pm (MATH 350)	Number Theory Elena Fuchs (University of California Davis) X In the early 70’s, the concept of “expander graphs” was introduced as an optimal model for networks of arbitrary size. Back then, it was not even known whether such graphs exist or not. Today, we not only know that they exist, but have seen their importance come up in a great spectrum of fields in mathematics and computer science. In this talk, we will explore what these graphs are and give an idea about how they have recently made a splash on the number theory scene. What network optimization has to do with number theory
Aug. 04, 2017 1pm (MATH 350)	Probability Jason Fulman (USC) X Many casinos routinely use mechanical card shuffling machines. We were asked to evaluate a new product, a shelf shuffler. This leads to new probability, new combinatorics and to some practical advice which was adopted by the manufacturer. The interplay between theory, computing, and real-world application is developed. This work is joint with Persi Diaconis and Susan Holmes Analysis of casino shelf shuffling machines
Aug. 30, 2017 4:03pm (MATH 3…	Grad Student Seminar Sebastian Bozlee X Generating functions are a tool that allow one to study sequences with power series. I will define the ordinary generating function of a sequence, then apply these generating functions to a few examples, including the Fibonacci numbers and the general solution of a linear homogeneous recurrence relation. A bit about generating functions

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