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Events for April – August 2016

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Apr. 01, 2016 1:10pm (ECCR 1…	Kempner Giulia Sacca (Stony Brook) X There are very few known examples of compact Kahler manifolds that admit a holomorphic symplectic form. Two series of examples were discovered by Beauville in the 80's and then two sporadic (or exceptional) examples of dimension 6 and 10 were found by O'Grady. In this talk I will review the construction of these examples and describe some recent work describing the geometry of the two exceptional examples. One way to achieve this is to use Lagrangian fibrations. I will review what a Lagrangian fibration is as well as its basic properties, and then show how this structure allows one to study or construct examples of holomorphic symplectic manifolds. Holomorphic symplectic manifolds and Lagrangian fibrations Sponsored by the Meyer Fund
Apr. 01, 2016 3pm (ECCR 265)	Kempner John Hunter (University of California Davis) X Surface plasmons (SPs) are electromagnetic surface waves that propagate on the interface between an insulator and a conductor, such as air and gold. At short wavelengths, the frequency of SPs approaches a nonzero constant independent of their wavenumber, and we consider the effect of nonlinearity on SPs in this nondispersive limit. We show that the amplitude of weakly nonlinear SPs satisfies a cubically nonlinear, nonlocal asymptotic equation. We discuss the existence of smooth and weak solutions of this equation, and the effects of nonlinearity on the focusing and possible formation of singularities in SPs. This is joint work with Ryan Halabi. Nonlinear surface plasmons
Apr. 05, 2016 1pm (MATH 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) The ultimate generalization of Gaussian elimination to solve CSPs, Part 3
Apr. 05, 2016 1pm (Math 350)	Functional Analysis Sooran Kang (University of Colorado, Boulder) X We first discuss the original construction of the QHMs by Rieffel as generalized fixed point algebras of certain crossed product -algebras in some detail. Then we describe the QHM as a crossed product by a Hilbert -bimodule, , where is the fixed point algebra of the QHM and is the first spectral subspace of the QHM. Quantum Heisenberg manifolds as crossed products by Hilbert -bimodules. Sponsored by the Meyer Fund
Apr. 05, 2016 4pm (Math220)	Katharine Adamyk (CU Boulder) X In part II of this talk, we will examine several other definitions for Catalan numbers, along with the various explicit formulas for Cn that these definitions lead to. In addition, we will discuss some applications, generalizations, and interesting properties of Catalan numbers. Even More Things to Count with Catalan Numbers
Apr. 08, 2016 4pm (MATH 100)	Kempner R. Tyrell Rockafellar (University of Washington) X Uncertainty over the future, even if well supported by statistical information, causes a fundamental difficulty in optimization. Decisions can't pin future "costs" to specific values and are instead limited only to shaping their probability distributions as random variables. Hard upper bounds on "costs" may then be impossible or too expensive to enforce. On the other hand, traditional approaches in such a contest in terms of safety margins built around standard deviation, or probability of failure based on quantiles, can have serious mathematical shortcomings. It is essential to build optimization models in which cost variables are kept "adequately" below some threshold. Adequacy can be expressed by a choice of how risk is quantified, but that choice should fit with axioms of "coherency". Questions arise then on the statistical side as well. In estimating the consequences of decisions, through regression or some other means, the handling of databases may need to interact with the particular approach taken to risk. Risk, Optimization and Statistics
Apr. 12, 2016 11am (MATH 350)	Number Theory James Ricci (Daemen College) X One of the many interesting algebraic objects associated to a given elliptic curve defined over the rational numbers, E / Q, is its full-torsion representation ρ_E:Gal( Q /Q) → GL₂(Ẑ). Generalizing this idea, one can create another full-torsion Galois representation, ρ_(E₁,E₂): Gal( Q /Q)→(GL₂(Ẑ))² associated to a pair (E₁,E₂) of elliptic curves defined over Q. The goal of this talk is to provide an infinite number of concrete examples of pairs of elliptic curves whose associated full-torsion Galois representation ρ_(E₁,E₂) has maximal image. Elliptic Curves with Maximally Disjoint Division Fields
Apr. 12, 2016 12:10pm (MATH …	Kempner Sandra Laursen (Ethnography & Evaluation Research (E&ER) at CU Boulder) X Mathematics serves as students’ gateway to majors and careers in STEM fields (science, technology, engineering, mathematics), but lack of success in their initial college math courses deters many STEM-intending students from progressing in their desired pathways. While studies across the STEM disciplines support the use of student-centered teaching and learning to increase student learning and persistence, there has been less previous research on these approaches in college mathematics. Our group’s study of inquiry-based learning (IBL) as implemented in over 100 course sections at four universities shows that student-centered approaches can succeed when implemented on a departmental scale (http://www.colorado.edu/eer/research/steminquiry.html ). Despite variation in how IBL was implemented, student outcomes are improved in IBL courses relative to traditionally taught courses, as assessed by general measures that apply across course types. Particularly striking, the use of IBL eliminates a sizable gender gap that disfavors women students in lecture-based courses. Other evidence suggests particular benefits to initially lower-achieving students, and to pre-service K-12 teachers. The study supports IBL teaching as a means to improve learning outcomes and, potentially, student retention in college mathematics. In this session, I will highlight key results from our work that mathematics instructors may find useful to consider and to share with students and colleagues. Student outcomes of inquiry-based learning (IBL) in undergraduate mathematics: Key findings from education research
Apr. 12, 2016 1pm (MATH 220)	Grad Algebra/Logic Nik Ruskuc (University of St Andrews) X Let V be a class of algebras (usually a variety), and let P be an algebraic property. Is it true that the direct product AxB of two algebras from V satisfies the property P if and only if A and B satisfy P? If not, what are the conditions on A and B that make AxB satisfy P? For instance, in the group case the first question has a positive answer for most standard properties P (such as, say, finite generation, finite presentability, residual finiteness, etc.) Peter Mayr and I have been pursuing a project of investigating these questions within the context of Universal Algebra. It turns out that group theoretic results often generalise to much broader classes of algebra, e.g. Malcev varieties and such like. However, it would appear that semigroups by and large tend to remain outside the scope of our general theorems. My aim for this talk is to present it as a kind of 'companion' to our work, in which I will describe the curious landscape for semigroups, and present some remaining open problems. Finiteness properties for direct products of semigroups Sponsored by the Meyer Fund
Apr. 12, 2016 1pm (Math 350)	Functional Analysis Sooran Kang (University of Colorado, Boulder) X As promised in the first lecture, we conclude this series of Ulam lectures by describing the quantum Heisenberg manifolds as twisted groupoid -algebras. First we discuss the definition of a twist groupoid over a groupoid associated to a local homeomorphism and a locally compact Hausdorff space, and discuss the result by Deaconu-Kumjian-Muhly, which says that the twisted groupoid -algebra, , is isomorphic to a Cuntz-Pimsner algebra , where is a Hilbert -correspondence. Combining this result with the result of the isomorphism classes of the QHMs, we show that the QHM can be realized as a twisted groupoid -algebra. Quantum Heisenberg manifolds as twisted groupoid -algebras Sponsored by the Meyer Fund
Apr. 12, 2016 2pm (MATH 220)	Nik Ruskuc (University of St Andrews) X Magnus' solution for the word problem (1932) for one-relator groups is one of the first results in the then-emerging field of combinatorial group theory, and the corresponding problem for one-relation semigroups is perhaps the most easily stated and longest-standing open problem in combinatorial algebra. This talk will be mostly historical in nature, with the aim of outlining the relationship between the two problems, and the present state of knowledge about one-relation semigroups. I will also indicate some possible ways forward, which I am currently pursuing in a project with Robert Gray. One relation groups and semigroups Sponsored by the Meyer Fund
Apr. 12, 2016 4pm (Math220)	Megan Ly (CU Boulder) X Elementary particles such as electrons have an intrinsic angular momentum called "spin". In order to account for spin, we must use the double cover SU(2) of the three dimensional rotation group SO(3) to describe the rotations of fundamental quantum systems. In this talk we construct the irreducible representations of using first and second order differential operators to classify different spin states. Spin and irreducible representations of
Apr. 13, 2016 4pm (MATH 350)	Grad Student Seminar Leo Herr X What is a union? Marital, setwise, or otherwise, they're all types of colimits. Enter the hermeneutic circle hand-in-hand with Yoneda, learn a family recipe for triangles, and discover crossword puzzles are colimits, too! We'll be extremely concrete in discussions of unions, coequalizers, colimits, the Yoneda Lemma, Simplicial objects, sheaves, and ultimately topoi. Colimits Aren't Stupid -- A Geometric Realization
Apr. 14, 2016 12:10pm (MATH …	Kempner Paul Baum (Penn State University) X Topics in the talk are : #1. The basic definition of K-theory #2. A brief history of K-theory #3. Algebraic versus topological K-theory #4. The unity of K-theory This is an expository talk intended for a general mathematical audience. What is K-theory and what is it good for?
Apr. 14, 2016 1pm (Math 350)	Noncomm Geometry Paul Baum (Penn State University) X Topics in the lecture are : #1. Two low-dimensional examples #2. Dirac Operator #3. Spin-c manifolds #4. Proof of the Dirac case #5. The Dirac case implies the general case The aim of this expository talk is to give a clear and simple proof of Atiyah-Singer as a corollary of Bott periodicity. It is my intention that this proof should be understandable to graduate students. ATIYAH-SINGER REVISITED Sponsored by the Meyer Fund
Apr. 19, 2016 1pm (MATH 220)	Grad Algebra/Logic Jakub Bulin (CU Boulder) Finitely related algebras, Part 1
Apr. 19, 2016 3pm (Math 350)	Laura Scull (Fort Lewis College) X Effective orbifolds were originally defined via charts and atlases, analogous to manifolds. In the current literature, they are often defined instead via certain topological groupoids, which gives a more convenient category for studying them. These two approaches are known to be equivalent and produce the same objects. More recently, applications arising in geometry and physics have led to an interest in more general ineffective orbifolds. However, the current generalizations of the atlas definition do not correspond to the objects created by ineffective topological groupoids. I will discuss a project, joint with D. Pronk and M. Tommasini, where we develop an alternate atlas definition for ineffective orbifolds. Our atlases generalize the existing effective atlas definition, and our definition leads to the same objects as those defined by ineffective topological groupoids. Atlases for Ineffective Orbifolds Sponsored by the Meyer Fund
Apr. 20, 2016 4pm (MATH 350)	Grad Student Seminar Hanson Smith X Quadratic reciprocity was first proved by Gauss, and in Disquisitions Arithmeticae he calls it the "Fundamental Theorem." In this talk, we will motivate quadratic reciprocity with some of the ideas and conjectures of Fermat, Euler, and Legendre. I will present a proof that is relatively short and requires only a familiarity with groups, rings, and fields to understand. The goal of this talk is to give an appreciation for a useful and historically significant theorem, so I aim for it to be very accessible. A Proof of Quadratic Reciprocity
Apr. 21, 2016 3pm (MATH 350)	Probability Diane Holcomb (University of Arizona) X The Gaussian Unitary/Orthogonal Ensembles (GUE/GOE) are some of the most studied Hermitian random matrix models. They may be generalized to a one parameter family of point processes indexed by beta called the beta-Hermite ensembles. When appropriately rescaled, the eigenvalues in the interior of the spectrum converge to a translation invariant limiting point process called the Sine_beta process. One expects the Sine_beta process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). We find the asymptotic probability of two rare events. The first is a large deviation problem for the density of points in a large interval. The second is the asymptotic probability of overcrowding in a fixed interval. Rare events for the Sine_beta process
Apr. 26, 2016 12:10pm (MATH …	Kempner David Aristoff (Colorado State University) X We discuss Markov chain sampling problems arising in computational chemistry. In particular, we show how to sample functions of paths, especially hitting times, in the presence of practical difficulties like metastability and slow convergence. We show how a mathematical framework for these techniques allows for quantification of convergence, opening the way for improvement in efficiency and minimization of error. Sampling distributions of Markov chains
Apr. 26, 2016 1pm (MATH 220)	Grad Algebra/Logic Jakub Bulin (CU Boulder) Finitely related algebras, Part 2
Apr. 26, 2016 1pm (Math 350)	Functional Analysis Nadia Larsen (University of Oslo) X In recent work with Laca, Neshveyev, Sims and Webster we investigate the factor types of the extremal KMS states for the preferred dynamics on the -algebra of a strongly connected finite k-graph. At inverse temperature 1, the prevalent outcome is a type III factor. We compute its Connes invariant in terms of the spectral radii of the coordinate matrices and the degree of cycles in the graph. Our analysis has two main steps: one is to analyze the von Neumann algebra of the equivalence relation of the -graph groupoid, the other is to formulate and prove a Frobenius-type result for the family of matrices associated to the -graph. Von Neumann algebras of strongly connected higher-rank graphs Sponsored by the Meyer Fund
Apr. 26, 2016 3pm (Math 350)	Dorette Pronk (Dalhousie University) X For this talk we will consider orbifolds as Lie groupoids with certain properties with generalized maps between them. The generalized maps are obtained by taking the bicategory of fractions of the 2-category of orbifold groupoids with groupoid homomorphisms. The fractions are needed because we want to consider Morita equivalences as invertible We will first consider what generalized maps between orbifolds are and how we can think of them in terms of a generalization of an atlas refinement together with a map on the refinement. We will then discuss the structure of the 2-cells between them (the arrows of the mapping groupoid) and find that this is simpler than what one would expect for an arbitrary bicategory of fractions, and we will give a nice geometric interpretation. We will end by calculating various examples, and in particular showing that the two ineffective groupoids that Laura Scull presented last week are indeed not Morita equivalent. Mapping Spaces for Orbifolds Sponsored by the Meyer Fund
Apr. 27, 2016 4pm (MATH 350)	Grad Student Seminar Elizabeth Parsons X Digital signal processing (DSP) is the numerical manipulation of continuous time, continuous amplitude sinusoidal functions, usually with the intention of discretizing analog signals. In this talk, we will discuss the importance of the discrete Fourier transform, short time Fourier transform and wavelet transform in DSP. Our focus will be on the time-frequency representations obtained from the short time Fourier transform and wavelet transform. We will find hidden images within audio files which can only be obtained using time-frequency representations of signals. Digital Signal Processing: Using Math to Turn Music into Pictures
May. 17, 2016 11am (MATH 350)	Number Theory Amanda Tucker (University of Rochester) X What's the difference between 3, 1/3, the sum from n=1 to infinity of 1/n^3, and pi? We will start with a discussion of what it means for a number to be integral, rational, algebraic, or transcendental. We will talk about the history of the multiple zeta values (certain real numbers) and some open problems about them. You will leave this talk knowing more about the real numbers than you thought was possible, but also with more questions about the real numbers than you thought was possible! Multiple zeta values
May. 17, 2016 1pm (MATH 350)	Algebraic Geometry Amanda Tucker (University of Rochester) X We will introduce the area of arithmetic statistics and the Cohen- Lenstra heuristics. We will give some background about the work of Bhargava- Shankar-Tsimerman and Taniguchi-Thorne on the Davenport-Heilbronn theorems. We will then explain our recent proof that 96.2% of totally real cubic fields have genus number one and, if time permits, talk about some applications. This represents joint work with Kevin McGown. The arithmetic statistics of number fields
Jun. 01, 2016 2pm (Math 350)	Functional Analysis Sarah Reznikoff (Kansas State University) X In 1948 Naimark showed that the C-algebra of compact operators, K(H), has a unique irreducible representation and asked whether the converse is true; that is, must any C-algebra with this property be isomorphic to K(H)? We will out line the proof of Akemann and Weaver (2003) that the existence of a counterexample to Naimark's Problem is consistent with the axioms of set theory ZFC. Notes on the relative consistency of Naimark's Problem Sponsored by the Meyer Fund
Aug. 24, 2016 4pm (MATH 350)	Grad Student Seminar Boramey Chhay X So it's been a while since Calculus 3 and now you need to be able to answer student questions about it in the Help Lab. Not a problem! This talk will be a review of stuff from Calculus 3 that you may or may not have forgotten. All the Calc 3 you need for Help Lab
Aug. 30, 2016 12:10pm (MATH …	Kempner Janos Englander (CU Boulder) X Given a sequence of numbers in , consider the following experiment. First, we flip a fair coin and then, at step , we turn the coin over to the other side with probability , . What can we say about the distribution of the empirical frequency of heads as ? We show that a number of phase transitions take place as the turning gets slower (i.e. is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is . Among the scaling limits, we obtain some well known special (Uniform, Gaussian, Semicircle and Arcsine) laws. The talk is intended to a general audience and no expertise in probability is assumed! This is joint work with S. Volkov (Lund, Sweden) Turning a coin instead of tossing it
Aug. 30, 2016 1pm (MATH 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) X The Algebraic Dichotomy Conjecture predicts that the constraint satisfaction problem CSP(A) with a fixed finite template A is in P if the algebra associated to A has a Taylor term. (Otherwise, CSP(A) is known to be NP-complete.) The goal of this sequence of talks is to discuss a recent theorem of M. Olsak (Charles University) which greatly simplifies the condition "an algebra has a Taylor term". Namely, Olsak proved that if an algebra (finite or infinite) has a Taylor term, then it has a specific Taylor term in 12 variables. Taylor Terms, Part 1: Introduction

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