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Events for November 2016 – March 2017

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Nov. 01, 2016 1pm (MATH 220)	Hubie Chen (San Sebastian) X The constraint satisfaction problem (CSP) involves deciding, given a set of variables and a set of constraints on the variables, whether or not there is an assignment to the variables satisfying all of the constraints. One formulation of the CSP is as the problem of deciding, given a pair (G,H) of relational structures, whether or not there is a homomorphism from the first structure to the second structure. The CSP is in general NP-hard; a common way to restrict this problem is to fix the second structure H, so that each structure H gives rise to a problem CSP(H). The problem family CSP(H) has been studied using an algebraic approach, which links the algorithmic and complexity properties of each problem CSP(H) to a set of operations, the so-called polymorphisms of H. Certain types of polymorphisms are known to imply the polynomial-time tractability of CSP(H), and others are conjectured to do so. In a recent joint article with B. Larose, we systematically study---for various classes of polymorphisms---the computational complexity of deciding whether or not a given structure H admits a polymorphism from the class. Among other results, we prove the NP-completeness of deciding a condition conjectured to characterize the tractable problems CSP(H), as well as the NP-completeness of deciding if CSP(H) has bounded width. In this talk, I will describe some of the main results from this article. This talk is based on the article available at https://arxiv.org/abs/1604.00932. Asking the metaquestions in constraint tractability Sponsored by the Meyer Fund
Nov. 01, 2016 2pm (MATH 220)	Hubie Chen (San Sebastian) X The quantified constraint satisfaction problem (QCSP) is the computational problem of deciding, given a structure and a first-order prenex sentence whose quantifier-free part is the conjunction of atoms, whether or not the sentence holds on the structure. One obtains a family of problems by defining, for each structure B, the problem QCSP(B) to be the QCSP where the structure is fixed to be B. We will discuss the research program of trying to classify the complexity of QCSP(B) for each finite structure B. It is possible to associate an algebra to each finite structure in such a way that studying the algebra of a structure gives insight into the QCSP complexity of the structure. We will overview the use of universal-algebraic notions and techniques in this research program. In particular, there is a close connection between QCSP complexity and the growth rate of the number of elements needed to generate the powers A^1, A^2, ... of an algebra A: showing an at-most polynomial growth rate can (essentially) be translated to a complexity upper bound for a structure with algebra A. This research area gives (we believe) a fascinating interplay between computational complexity, universal algebra, and logic. This talk will partially be based on the overview article available at http://arxiv.org/abs/1201.6306. Meditations on Quantified Constraint Satisfaction Sponsored by the Meyer Fund
Nov. 01, 2016 3pm (MATH 350)	Topology David Gepner (Purdue U) X In this talk we will give a gentle introduction to chromatic homotopy theory with emphasis on the close connection between algebraic topology and algebraic geometry which arises from the theory of formal groups. We will then consider how this relationship can be made even closer by enlarging our scope to include spaces with group actions, and what this amounts to in the interesting case of elliptic cohomology. Cohomology theories associated to elliptic curves
Nov. 08, 2016 2pm (MATH 220)	Mark Pullins (CU) Bad groups, 2
Nov. 08, 2016 2pm (MATH 350)	Lie Theory Shawn Burkett (CU) X It is well known that two elements of $G=GLn(Fq)$ are conjugate if and only if they have the same rational canonical form. This leads to a parameterization of the conjugacy classes of $G$ by partition valued functions from the set of monic irreducible polynomials over $𝔽_{q}$ . It is also true that the complex irreducible characters of $G$ are parameterized by partition valued functions. In this talk, we will discuss these parameterizations and then describe a method of constructing supercharacter theories of $G=GLn(Fq)$ from supercharacter theories of $Z(G)≃ G/[G,G]$ which can be described combinatorially using these parameterizations. A new supercharacter theory construction for $GLn(Fq)$
Nov. 08, 2016 3pm (MATH 350)	Topology Jordan Watts (CU Boulder) Intersection Homology
Nov. 15, 2016 10am (MATH 220)	Number Theory Robert Hines (University of Colorado, Boulder) X We consider a pair of dynamical systems on the complex plane inspired by the action of the super-Apollonian group, realized here as reflections in the sides of a right-angled ideal hyperbolic octahedron. These systems are “reflective” versions of Asmus Schmidt’s continued fraction algorithms over Q(i). We study aspects of the dynamics including the natural extension, invariant measure, action on rational points, and restriction to the real line. Super-Apollonian Continued Fractions
Nov. 15, 2016 10am (MATH 350)	Geometry/Analysis Taylor Joseph Klotz (CU Boulder) X The famous Inverse Scattering Transform (IST) is a powerful method for finding solutions to so called Integrable Systems. It is considered to be a nonlinear analog of the Fourier Transform technique. By looking at the "scattering data" for two associated linear problems, we can reconstruct the solution to the nonlinear problem in question. We'll focus on what this method is, its inspiration, and work through an explicit example. Either the classic KdV (Korteweg de-Vries) equation or the NLS (nonlinear Schrodinger) equation. Focusing on the Inverse Scattering Transform
Nov. 15, 2016 12:10pm (MATH …	Kempner Youri Davydov (Université des Sciences et Technologies de Lille) X It is well known that the transformations of symmetrization introduced by Steiner for the proof of Isoperimetric Theorem play an important role in geometry and its applications. Another important transformation is the Minkowski (sometimes called Blaschke) symmetrization. In the talk, the asymptotic behavior of random sequences of successive Steiner and Minkowski symmetrizations is investigated. In the case of independent directions, we prove the almost sure convergence of successive symmetrizations at exponential rate for Minkowski, and at rate $-c n$ , with $c > 0$ , for Steiner. Random symmetrizations of convex bodies
Nov. 15, 2016 1pm (MATH 220)	Zak Mesyan (CU Colorado Springs) X Let V be an arbitrary vector space over a field K, and let End(V) be the ring of all K-linear transformations of V. We characterize the diagonalizable linear transformations in End(V), as well as the (simultaneously) diagonalizable subalgebras of End(V), generalizing results from classical finite-dimensional linear algebra. These characterizations are formulated in terms of a natural topology on End(V), which reduces to the discrete topology when V is finite-dimensional. (Based on joint work with Miodrag Iovanov and Manuel Reyes.) Infinite-Dimensional Diagonalization
Nov. 15, 2016 2pm (MATH 350)	Lie Theory Mandi Schaeffer-Fry (Metro State) X As a consequence to his Galois-McKay conjecture, G. Navarro has conjectured a necessary and sufficient condition for a finite group $G$ to have a self-normalising Sylow $2$ -subgroup, which is given in terms of the ordinary irreducible characters of $G$ . I will discuss this conjecture and my reduction to (quasi) simple groups. Then I’ll discuss the specific case of Type A. Galois Automorphisms and Self-Normalizing Sylow $2$ Subgroups
Nov. 15, 2016 2pm (MATH 220)	Mark Pulllins (CU) Bad groups, 3
Nov. 15, 2016 3pm (MATH 350)	Topology Leo Herr (CU Boulder) Grothendieck's Six Operations
Nov. 17, 2016 2pm (Math 350)	Functional Analysis Sarah Reznikoff (Kansas State University) X We will give a brief survey of results in operator algebras that have been proven with strong set-theoretic hypotheses, describing these assumptions in detail. Some of these theorems are in fact independence proofs; we will go over what this means and describe the major strategies for producing independence proofs. Finally, we outline how the method of set-theoretic forcing can be used in the study of graph algebras. Note that no special background in either operator algebras or set theory will be required to gain some appreciation for this area of research from the talk. Applications of Set-Theoretic Axioms and Techniques to Operator Algebras Sponsored by the Meyer Fund
Nov. 29, 2016 12:10pm (MATH …	Kempner David Roberts (University of Adelaide) X Historically, bundles in geometry arose from several sources: covering spaces, tangent bundles and homogeneous bundles. Only with this rich supply of examples could the theory of bundles (vector bundles, principal bundles and fibre bundles more generally) arise. It was a marvellous convergence when it was discovered that physical theories such as electromagnetism and electroweak theory, and even general relativity, were also described by the data of connections on such geometric structures. Contemporary physical(-ish) field theories that arise from, say, string theory require a much richer setting, and the geometry behind this setting has become known as "higher geometry". Not surprisingly the geometry has its roots dating back to none other than the differential geometer and category theorist Charles Ehresmann. It is the marriage of geometry and (higher) category theory as exemplified in Lie groupoids that provides this richer setting that physics requires. However, the kind of examples that drove the theory of bundles in the first half of the 20th century have been lacking until recently. By going back to those old sources it is in hindsight obvious how to generate many examples in higher geometry, and this talk will give a flavour of how one does this. No prior knowledge of Lie groupoids or higher category theory will be necessary. Low-dimensional higher geometry by examples
Nov. 29, 2016 1pm (MATH 220)	Ralph McKenzie (Vanderbilt University) Strong solvability in finitely decidable varieties, I
Nov. 29, 2016 2pm (MATH 220)	Ralph McKenzie (Vanderbilt University) Strong solvability in finitely decidable varieties, II
Nov. 29, 2016 2pm (MATH 350)	Lie Theory Farid Aliniaeifard (York) X Normal supercharacter theory is a mechanism to substitute conjugacy classes by normal subgroups in such a way that this coarsening simulates some useful features of irreducible characters. We construct a normal supercharacter theory for the group of unipotent upper-triangular matrices. The supercharacters in this supercharacter theory are indexed by Dyck paths. We show that this construction is identical with Scott Andrews' construction after gluing the superclasses and supercharacters by the action of torus group. In the end, we build up some Hopf structures base on these supercharacters. Normal supercharacter theory, Dyck paths, and Hopf Structures Sponsored by the Meyer Fund
Nov. 29, 2016 3pm (MATH 350)	Topology Lucas Simon (CU Boulder) Perverse Sheaves - Part I
Dec. 01, 2016 2pm (Math 350)	Functional Analysis David Roberts (University of Adelaide, Australia) X The concept of the space of all maps of a particular kind between a pair of manifolds is an old one, and it is the subject of global analysis to study the structure of such spaces and when one can do differential geometry on them. The nicest possible example of such a space is the (infinite-dimensional) Fréchet manifold of smooth loops in a smooth manifold $M$ . More generally, given a compact smooth manifold $K$ and a smooth manifold $M$ , one can define the Fréchet manifold of smooth maps from $K$ to $M$ . The assumption of compactness of $K$ is not necessary, but then the topology gets a lot more wild. Moving just beyond manifolds, one can consider the space of smooth loops in an orbifold, which is important to understanding the string topology of orbifolds, or more generally, what a smooth mapping space from a compact manifold to an orbifold looks like. Such a thing will not necessarily be a priori a smooth manifold. Despite several attempts in the literature, the constructions have been extremely messy, and at least one case has counterexamples to some of its claims. The most mature approach to orbifolds treats them as differentiable stacks with the property that they are presented by a proper étale Lie groupoid. This talk will outline a recent construction of a presentation by a Lie groupoid of the smooth mapping stack of maps from a compact manifold (possibly with corners) to a differentiable stack $X$ . In the special case that $X$ is an orbifold, the mapping stack is then presented by a proper étale Fréchet-Lie groupoid, a candidate for an infinite-dimensional smooth orbifold. Smooth Mapping Stacks of Differentiable Stacks and Orbifolds Sponsored by the Meyer Fund
Dec. 06, 2016 2pm (MATH 220)	* We skip this week
Dec. 06, 2016 3pm (MATH 350)	Topology Lucas Simon (CU Boulder) Perverse Sheaves - Part II
Dec. 08, 2016 2pm (Math 350)	Functional Analysis Dr. Trubee Davison (Boulder, Colorado) X Given an iterated function system (IFS) on a complete and separable metric space Y, there exists a unique compact subset X contained in Y satisfying a fixed point relation with respect to this IFS. This subset is called the attractor set, or fractal set, associated to this IFS. The attractor set supports a specific Borel probability measure, called the Hutchinson measure, which itself satisfies a fixed point relation. A particular instance of an attractor set is the infinite string space on a finite alphabet, with the members of the IFS being the shift maps. There is known relationship between the Hutchinson measure on the infinite string space, and the Hutchinson measure on an arbitrary attractor set, via the so called coding map. In this talk, we will generalize this relationship to the functional analytic setting, building off the work of Palle Jorgensen (U of Iowa). This generalized relationship will be an example of Naimark's dilation theorem for positive operator-valued measures. Operator-Valued Measures Associated to Iterated Function Systems
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 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. 24, 2017 3pm (MATH 350)	Topology Sebastian Bozlee and Cherry Ng (CU Boulder) Cohomology Theories and Brown Representability
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 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. 07, 2017 3pm (MATH 350)	Topology Matthew Pierson (CU Boulder) The Stable Homotopy Category as a Triangulated Symmetric Monoidal Category
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 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
Feb. 28, 2017 3pm (MATH 350)	Topology Cliff Blakestad (CU Boulder) Quillen's Theorem: MU and Formal Group Laws
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 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 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 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

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