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Events for October 2016 – February 2017

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Oct. 04, 2016 10am (MATH 350)	Geometry/Analysis Jae Min Lee (CUNY Graduate Center) X In this talk, we will study the local well-posedness of the Camassa-Holm equation on the real line. That is, we will show the local existence and uniqueness of the solution, and the continuous dependence of the solution on the initial data. The main idea is to use the Lagrangian flow to rewrite the PDE as a smooth ODE on a Banach Space. We will also show that the group of diffeomorphisms is a topological group, which is necessary to prove the smoothness of the equation and continuous dependence. Local Well-posedness of the Camassa-Holm equation on the real line
Oct. 04, 2016 10am (MATH 220)	Number Theory Stefan Erickson (Colorado College) X Leavitt path algebras are algebraic structures associated with a given directed graph. A natural question is to study the structure of these algebras, including the Grothendieck group generated by the Leavitt path algebras of certain Cayley graphs. As finite abelian groups, they have a canonical cyclic group decomposition. Studying the cyclic components leads to interesting divisibility relations between certain sequences of numbers. After a brief introduction to Leavitt path algebras, we will present some number theoretic results related to groups arising from a class of "almost" cyclic graphs. This is ongoing work with Gene Abrams at UCCS and Cristobal Gil at University of Malaga. Divisibility Sequences Arising from Leavitt Path Algebras
Oct. 04, 2016 12:10pm (MATH …	Kempner Joe Auslander (University of Maryland) X A flow $(X, T)$ is a jointly continuous action of a topological group $T$ on a compact Hausdorff space $X$ . The flow is minimal if every orbit is dense. The flow $(X, T)$ is said to be equicontinuous if the maps defined by $T$ form an equicontinuous family, in which case the action can be embedded in a compact group. Distality is a generalization of equicontinuity, and the structure of distal minimal flows is provided by a deep theorem of Hillel Furstenberg which says that an arbitrary distal minimal flow can be obtained from the trivial one point flow by a (possibly transfinite) sequence of “equicontinuous extensions”. Given an arbitrary flow, there are distal and equicontinuous structure relations–closed $T$ invariant equivalence relations such that the factor flow is, respectively distal or equicontinuous. These relations are generated, respectively, by the proximal and regionally proximal relations. For a minimal flow in many cases (including when $T$ is abelian) surprisingly the latter coincides with a smaller relation due to Veech. Structure and homomorphisms of minimal flows
Oct. 04, 2016 2pm (MATH 220)	Mark Pullins (CU) X Morley rank is a model-theoretic generalization of dimension, and interest in groups of finite Morley rank traces its origins to a conjecture due to Gregory Cherlin (1979) and Boris Zilber (1977) that all infinite $ω$ -stable simple groups are algebraic groups over algebraically closed fields. This has not yet been proven even for groups of finite Morley rank, which is a subclass of the class of $ω$ -stable groups. A purely-model theoretic approach appears insufficient to answer this question and, over the years, techniques inspired by finite group theory and intuition from algebraic geometry have played a role in the study of groups of finite Morley rank. In this talk, we will introduce an axiomatic approach to Morley rank and examine the basic properties of rank. From there, we can begin to discuss the structure of groups of finite Morley rank. An Introduction to Groups of Finite Morley Rank, 1
Oct. 04, 2016 2pm (MATH 350)	Lie Theory Shawn Burkett (CU) X Normal subgroups play a central role in the study of finite groups and their irreducible characters. Many times, one may stitch together information about a normal subgroup and its quotient to learn a tremendous amount about about the original group and its representation theory. For example, if a group is a semidirect product $G=N H$ the structure of the $G$ is determined by $N$ and $G/N≃ H$ and the way they interact. In certain situations (i.e. direct products), the irreducible characters of $G$ can be completely determined by knowing the irreducible characters of both $N$ and its quotient $H$ . Not surprisingly, normal subgroups also play a central role in the study of supercharacter theories of a finite group. In this talk, we will explore some of the ways that normal subgroups can be used to understand the supercharacter theories of a finite group. Normal Subgroups and Supercharacter Theory
Oct. 04, 2016 3pm (MATH 350)	Topology Sarah Yeakel (U Maryland) X Goodwillie's calculus of functors is a method of approximating a homotopy functor with easier "polynomial" functors, building a Taylor tower where the homogeneous pieces are classified by certain spectra, called the derivatives of the functor. It is known that the derivatives of the identity functor of spaces form an operad and composable functors have a chain rule. We will discuss a general framework for these results and the extra structure that the derivatives possess. During the introductory talk, we will review the theory of functor calculus and why operads show up. Derivatives in Goodwillie Calculus
Oct. 06, 2016 11am (Math 350)	Noncomm Geometry John Lott (University of California, Berkeley) X It is an invariant of smooth manifolds that combines differential forms with K-theory. I'll give background material and describe some models for differential K-theory. One of them, based on Hilbert bundles, is joint work with Alexander Gorokhovsky. What is differential K-theory?" Sponsored by the Meyer Fund
Oct. 06, 2016 2pm (Math 350)	Functional Analysis Elizabeth Gillaspy (University of Münster, Germany) X Between 2012 and 2015, Kumjian, Pask and Sims introduced cohomology theory for higher-rank graphs ( $k$ -graphs). Higher-rank graphs provide concrete examples of $C$ -algebras that are relatively easy to compute and to study, in particular because one can also view these $C$ -algebras as arising from groupoids. Group cohomology theories often generalize to groupoids, leading one to ask how the cohomology of a $k$ -graph relates to the cohomology of its associated groupoid. This natural question was only answered in a few special cases by Kumjian, Pask and Sims, using ad hoc arguments that seemed unlikely to generalize. In joint work with Alex Kumjian, we have found a new way to describe the cohomology of a $k$ -graph, which is equivalent to the original definition of Kumjian, Pask and Sims, but which enables us to describe a homomorphism from $k$ -graph cohomology to groupoid cohomology in full generality. Along the way, we discovered that for $k$ -graph groupoids $G$ (and in fact, for a much more general class of groupoids, namely those associated to actions of directed semigroups), the groupoid cohomology can be computed from a relatively small, friendly subset of $G$ : the Renault-Deaconu semigroupoid of $G$ . These insights all rely on recasting the cohomology of a $k$ -graph, and of its associated groupoid, in category-theoretic terms. In this talk, we will begin by presenting the category-theoretic definition of groupoid cohomology, which we relied on for results mentioned above. After showing that this perspective is equivalent to the usual definition of the continuous cocycle cohomology, we observe that the new definition generalizes easily to the setting of $k$ -graphs and of Renault-Deaconu semigroupoids, and again matches with the more familiar definitions of cohomology in the literature. These preliminary results, together with an inductive limit functor connecting $k$ -graph modules and groupoid sheaves, combine to prove our main results: the homomorphism from $k$ -graph cohomology to groupoid cohomology, and the isomorphism between groupoid cohomology and Renault-Deaconu-semigroupoid cohomology. Cohomology for $k$ -Graphs, Groupoids, and Renault-Deaconu Semigroupoids Sponsored by the Meyer Fund
Oct. 06, 2016 3pm (MATH 350)	Probability Vladimir Rotar (San Diego State University) X The talk presents a survey and some remarks regarding the notion of pathwise asymptotic optimality in infinite horizon stochastic optimization problems: setups, approaches, results. Retrospective Remarks on Pathwise Asymptotic Optimality with an infinite horizon
Oct. 11, 2016 10am (MATH 220)	Number Theory Xin Zhang (University of Illinois Urbana-Champaign) X Let $Λ < S L (2, ℤ)$ be a finitely generated, non-elementary Fuchsian group of the second kind, and $v, w$ be two primitive vectors in $Z2-0$ . We consider the set $S=⟨ vγ,w⟩R2:γ∈Λ$ , where $⟨⋅,⋅⟩R2$ is the standard inner product in $R2$ . Using Hardy-Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich and Sarnak, together with Gamburd's 5/6 spectral gap, we show that if $Λ$ has parabolic elements, and the critical exponent $δ$ of $Λ$ exceeds 0.995371, then a density-one subset of all admissible integers (i.e. integers passing all local obstructions) are actually in $𝒮$ , with a power savings on the size of the exceptional set (i.e. the set of admissible integers failing to appear in $𝒮$ ). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when $Λ$ is free, finitely generated, has no parabolics and has critical exponent $δ>0.999950$ . Finding integers from orbits of thin subgroups of SL(2, Z)
Oct. 11, 2016 1pm (MATH 220)	Agnes Szendrei (CU Boulder) Taylor Terms, Part 3: Olsak's Theorem
Oct. 11, 2016 2pm (MATH 220)	Mark Pullins (CU) An Introduction to Groups of Finite Morley Rank, 2
Oct. 11, 2016 2pm (MATH 350)	Lie Theory Scott Andrews (Boise State) X The symmetric group is sometimes referred to as "the general linear group over the field with one element." Although this statement is not mathematically precise, it motivates us to extend results about the symmetric group to the general linear groups over finite fields. I will present a standard construction of the irreducible modules of the symmetric group and show how it can be extended in a natural way to construct the unipotent modules of the finite general linear groups. I will also discuss related open problems. Irreducible modules of symmetric groups and finite general linear groups Sponsored by the Meyer Fund
Oct. 11, 2016 3pm (MATH 350)	Topology Agnes Beaudry (CU Boulder) Derived Categories
Oct. 12, 2016 4pm (MATH 350)	Grad Student Seminar Keli Parker A few dual graphs and the moduli space of curves
Oct. 13, 2016 2pm (Math 350)	Functional Analysis Kathy Merrill (Colorado College) X We present some results of MacArthur and Taylor, who have produced Haar-type wavelets for the action of wallpaper symmetry groups on $L2(R2)$ . Following work on composite dilations by Guo, Labate, Lim, Weiss and Wilson, we also give a method for constructing wavelet set wavelets for the action of these groups. Simple examples of both types will be given. Wavelets for Crystallographic Groups Sponsored by the Meyer Fund
Oct. 18, 2016 10am (MATH 220)	Number Theory Arseniy Sheydvasser (Yale University) X Studying circle/sphere packings that arise as limit sets of discrete subgroups of the hyperbolic isometric group is a long studied problem, generalizing the familiar case of the Apollonian gasket. Far less is known about integral packings, where the curvature of each sphere is required to be an integer---even producing explicit examples is a non-trivial task. In this talk, I will discuss a new method of explicitly constructing such sphere packings by considering special types of orders in rational quaternion algebras. Quaternion Algebras and Integral Sphere Packings
Oct. 18, 2016 10am (Math 350)	Geometry/Analysis Mason Thomas Pelfrey (CU Boulder) X Cartan's method of equivalence is a technique for determining when two geometric objects are equivalent up to diffeomorphism by finding invariants of the objects under some group action. A control system is an under determined set of ODEs which show up most often when describing mechanical systems which are "steerable" in some sense. In this talk, we will discuss what it means for two control systems to be equivalent and how Cartan's Method can be used to determine if an equivalence exists between two control systems. Control Systems and Cartan's Method of Equivalence
Oct. 18, 2016 12:10pm (MATH …	Kempner Laurentiu Maxim (University of Wisconsin, Madison) X By analogy with knot theory, complex hypersurfaces can be studied via Alexander-type invariants of their complements. I will discuss old and new results concerning rigidity properties of such invariants, including (twisted) Alexander polynomials, $L2$ -betti numbers and Novikov homology of hypersurface complements. The first part of the talk will be an overview of the subject, addressed to a general audience. In the second part I will show how intersection homology and perverse sheaves can be used to relate the local and global topology of such hypersurface complements. Alexander-type invariants of complex hypersurface complements
Oct. 18, 2016 1pm (MATH 220)	Agnes Szendrei (CU Boulder) Taylor Terms, Part 4: Olsak's Theorem
Oct. 18, 2016 2pm (MATH 220)	Mark Pullins (CU) Bad groups, 1
Oct. 18, 2016 2pm (MATH 350)	Lie Theory JP Cossey (University of Akron) X Let $G$ be a finite group and p a prime. A block of a finite group $G$ is a certain subset of the characters of $G$ that are in a sense related when “reduced modulo $p$ ”. There are certain particularly nice “small” blocks, called the nilpotent blocks of $G$ , but there is no easily checked sufficient condition for a block $B$ to be nilpotent. In this talk we will discuss one proposed sufficient condition, and introduce enough algebraic combinatorics to show that the conjecture holds for the symmetric groups. The conjecture is also known to hold for solvable groups and some other types of blocks of arbitrary finite groups. Small blocks of finite groups Sponsored by the Meyer Fund
Oct. 18, 2016 3pm (MATH 350)	Topology Laurentiu Maxim (U Wisconsin Madison) X By analogy with knot theory, complex hypersurfaces can be studied via Alexander-type invariants of their complements. I will discuss old and new results concerning rigidity properties of such invariants, including (twisted) Alexander polynomials, L2-betti numbers and Novikov homology of hypersurface complements. The first part of the talk will be an overview of the subject, addressed to a general audience. In the second part I will show how intersection homology and perverse sheaves can be used to relate the local and global topology of such hypersurface complements. Alexander-type invariants of complex hypersurface complements
Oct. 18, 2016 4pm (Math 220)	Algebraic Geometry Steffen Marcus (TCNJ) X Logarithmic structures provide geometers a systematic way of picking out, locally, a finite collection of functions and their powers on algebraic varieties. The combinatorial structure of these local choices comes along for the ride. In this talk I will report on work in progress with Alana Huszar, Matthew Satriano, and Martin Ulirsch to develop a couple of ways of altering logarithmic structures to highlight, combinatorially, slightly more of the geometry of the underlying space. I hope to also discuss potential applications to tropical geometry and Gromov-Witten theory. Pointed and twisted logarithmic structures Sponsored by the Meyer Fund
Oct. 19, 2016 4pm (MATH 350)	Grad Student Seminar Leo Herr Rings are Spaces
Oct. 20, 2016 2pm (Math 350)	Functional Analysis Ian Long (CU Boulder) X A measure is said to be spectral if its $L2$ -space has an orthonormal basis of exponential functions. A class of “fractal measures” discovered by Hutchinson in 1981 has been shown to contain many members which are spectral. More recently, it has been discovered that some of these measures in fact have infinitely many spectra, which relate in a surprisingly nice way. These numerous spectra give rise to unitary operators, which themselves have been shown to have interesting and fractal-like properties. In this talk, I will discuss these phenomena while giving details of my recent work in exploring them in the case of a relatively unexplored subclass of Hutchinson’s fractal measures. 3×3 Hadamard Matrices and Commuting Operators
Oct. 25, 2016 10am (MATH 350)	Geometry/Analysis Jeanne Clelland (CU Boulder) X A Beltrami field on an open set U in R^3 is a vector field u on U satisfying the conditions curl(u) = f*u, div(u) = 0, where f is a smooth function on U. When f is constant, it is straightforward to show via Cartan-Kahler that local, real analytic solutions to these equations depend on 2 arbitrary functions of 2 variables; in particular, when f=0, solutions have the form u = grad(h), where h is an arbitrary harmonic function on U. When f is non-constant, the story is more complicated. Encisco and Peralta-Salas have shown that "Beltrami fields with a non-constant proportionality factor are rare," in fact, there are no solutions at all unless f satisfies an explicit (but very complicated) differential equation. In this work (joint with Stephen Preston and Taylor Klotz), we seek to gain a better understanding of Beltrami fields with non-constant proportionality factor. Using a moving frames approach, we prove the following results regarding local solutions in a neighborhood of any point in U where f and grad(f) are both nonzero: (1) If the level surfaces of f are totally umbilic (i.e., open subsets of planes or spheres), then there are no local solutions unless the level sets of f are contained in either parallel planes or concentric spheres, in which case local solutions depend on 2 functions of 1 variable. (2) If the level surfaces of f contain no umbilic points, then there is at most a 3-dimensional space of local solutions. We also give an explicit example with a 2-dimensional space of local solutions. Unfortunately, the question of precisely which functions f admit solutions remains computationally intractable. A Cartan-Kahler argument suggests that the functions f admitting solutions should depend on 3 functions of 2 variables, so the tantalizing question remains: which functions f admit solutions, and how many solutions does each such function admit? Beltrami fields with non-constant proportionality factor via moving frames
Oct. 25, 2016 1pm (MATH 220)	James Mitchell (St Andrews, UK) X In this talk I will outline recent progress in the application of computational group theory to the problem of effectively computing a finite semigroup. Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which thJaese results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and $0$ -matrix semigroups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Green's relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate its ideal structure, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Green's classes of an element of such a subsemigroup without determining the global structure of the semigroup. Computing with finite semigroups Sponsored by the Meyer Fund
Oct. 25, 2016 2pm (MATH 220)	James Mitchell (St Andrews, UK) X In this talk I will discuss the problem of finding minimally generated dense subgroups of the groups of automorphisms of some interesting combinatorial structures, such as the natural numbers, the random graph, the rational numbers with the usual order, and the so-called countable ultrahomogeneous graphs. Approximating permutations and automorphisms Sponsored by the Meyer Fund
Oct. 25, 2016 3pm (MATH 350)	Topology Jordan Watts (CU Boulder) Stratified Spaces
Oct. 26, 2016 4pm (MATH 350)	Grad Student Seminar Michael Wheeler X Everyone likes playing games. But, did you know that certain types of games can be used to study the properties of sets? We will discuss finite and infinite two-player games with perfect information and no draws or randomness, and analyze whether these games are determined (whether one of the players must have a winning strategy). We will end with a discussion of some examples of infinite games that give us information about the pay-off set when either player 1 or player 2 have winning strategies. Fun (and Infinite) Games
Oct. 27, 2016 1pm (MATH 350)	Probability Ricardas Zitikis (University of Western Ontario, Canada) X “The dose makes the poison,” said Paracelsus, the father of toxicology. Thinking of risk, how much of it is good or bad? Despite all the illuminating and far-reaching results available in the literature on the topic, I will go back to the very basics: first, I will briefly touch upon risk preferences, then look at some elusive behavior of larger risks, such as insurance losses above deductibles, and finally venture into extremes. Though somewhat philosophical in nature, the talk is based on very recent results published by the speaker and his co-authors in journals such as the Annals of Actuarial Science (2015, 9(1), 58-71), Insurance: Mathematics and Economics (2016, 69(July), 97-10), and ASTIN Bulletin - The Journal of the International Actuarial Association (2015, 45(3), 661-678). Measuring risks – a tricky task with a lot of scientific fun
Oct. 27, 2016 2pm (Math 350)	Functional Analysis Arlan Ramsay (CU Boulder) X Robert Richtmyer, a former member of this department, thought that it would be good to teach axiomatic Euclidean and hyperbolic geometry using axioms that treated the real numbers as given and used some of their properties as tools for proving the theorems. He made notes for his classes, and I assisted him in turning the notes into a book. I want to outline some of the basic results and discuss how facts about continuity from undergraduate real analysis can be used to prove existence and uniqueness of Euclidean and hyperbolic planes. This is in the spirit of differential geometry, but with minimal use of derivatives. We do manage to avoid some of the very clever purely geometric arguments found in most other treatments of these topics. Using Continuity Arguments in Plane Geometry
Oct. 27, 2016 3pm (MATH 350)	Probability Ricardas Zitikis (University of Western Ontario, Canada) X The notion of monotone rearrangement of functions has been known for a long time and can be traced back to the classical book on “Inequalities” by G. H. Hardy, J. E. Littlewood, and G. Pólya. The notion of convex rearrangement of functions originates from a work of Yu. Davydov published several decades ago. To illustrate, the quantile function in Statistics is a monotone rearrangement, whereas the Lorenz curve in Econometrics is a convex rearrangement. In this talk I will discuss an index of monotonicity that has naturally arisen from these rearrangements and provided a useful tool for comparing functions, stochastic processes, and other objects according to their monotonicity properties. I will hint at several problems in Insurance and Finance, and in particular in Education, where such comparisons are of interest. Monotone and convex rearrangements: mathematics, probability, and applications
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 220)	Mark Pulllins (CU) Bad groups, 3
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 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 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 2pm (MATH 220)	Ralph McKenzie (Vanderbilt University) Strong solvability in finitely decidable varieties, II
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 350)	Topology Cliff Blakestad (CU Boulder) Quillen's Theorem: MU and Formal Group Laws
Feb. 28, 2017 3pm (Math 220)	Functional Analysis Leonard Huang (CU Boulder) X The uniqueness of the Heisenberg Commutation Relations is usually stated in the language of unitary operators on a Hilbert space. This is done by first using the Baker-Campbell-Hausdorff Formula to integrate the Heisenberg Commutation Relations to the Weyl Commutation Relations and then applying the Stone-von Neumann Theorem. The question thus remains whether or not the uniqueness result can be stated purely in the original language of self-adjoint unbounded operators on a Hilbert space. Using the concept of analytic vectors created by Edward Nelson in the 1950s and a result about such vectors due to Moshé Flato and his collaborators, we attempt to answer the question in the affirmative. Uniqueness of the Heisenberg Commutation Relations from the Viewpoint of Unbounded Operator Theory

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