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Events for November 2017 – March 2018

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Nov. 02, 2017 11am (MATH 350)	Geometry/Analysis Congming Li (CU Boulder, Applied Math) X First, we give a brief introduction on the central role played by a priori estimates in PDE analysis. We then present some Liouville type theorems via blowing-up arguments to establish some important a priori estimates. We also plan to present some related maximum principle type theorems. A priori estimates--the core of PDE analysis
Nov. 07, 2017 10am (Math 350)	Functional Analysis Robin Deeley (CU Boulder) X The stable $C^{}$ -algebra of a Smale space is obtained from the étale groupoid associated to the space’s stable equivalence relation. When the original system is mixing, the stable $C^{}$ -algebra is simple, separable, nuclear and stably finite. I will outline the construction of an explicit inductive-limit decomposition of the stable $C^{}$ -algebra when the stable sets are totally disconnected. From this, one can often compute the $K$ -theory of the stable $C^{}$ -algebra. The talk will be example-based. In particular, no knowledge of Smale spaces is required. This talk is based on joint work with Allan Yashinski. The Structure of the Stable $C^{*}$ -Algebra of a Class of Smale Spaces
Nov. 07, 2017 11am	Number Theory Amy Feaver (King's University) X (Joint with Anna Puskas) In this talk, we determine all of the imaginary $n$ -quadratic fields with class number dividing 32, $n>1$ ; results for $n=1$ were previously computed by Watkins. Our results provide techniques to find complete lists of imaginary $n$ -quadratic fields of class number $2m$ for nonnegative integers $m$ . Further, given a fixed $m>0$ we find a bound $B(m)$ on $n$ for which there are no imaginary $n$ -quadratic fields of class number $2m$ whenever $n>B(m)$ . Imaginary Multiquadratic Fields of Class Number Dividing 32
Nov. 07, 2017 1pm (MATH 220)	Peter Mayr (CU Boulder) Promises and constraints 4
Nov. 07, 2017 3pm (Math 350)	Kempner Kathryn Hess (EPFL) X I will describe results obtained in collaboration with the Blue Brain Project on the topological analysis of the structure and function of digitally reconstructed microcircuits of neurons in the rat cortex. The topology of the microconnectome
Nov. 07, 2017 4pm (MATH 350)	Topology Kathryn Hess (EPFL) X I will explain the construction of a new model for the configuration space of a product of two closed manifolds in terms of the configuration spaces of each factor separately. The key to the construction is the lifted Boardman-Vogt tensor product of modules over operads, developed earlier in joint work with Dwyer. (Joint work with Bill Dwyer and Ben Knudsen.) Configuration spaces of products Sponsored by the Meyer Fund
Nov. 08, 2017 4:03pm (MATH 3…	Grad Student Seminar Sarah Arpin X TBD TBD
Nov. 08, 2017 5pm (Math 350)	MathClub Keith Kearnes (CU Boulder) X I will talk about how algebra can be used to classify the hardness of complexity problems. Algebra and Algorithms
Nov. 14, 2017 10am (Math 350)	Functional Analysis Leonard Huang (CU Boulder) X In a 1999 paper, Alex Kumjian and David Pask proved that if a discrete group $G$ acts freely on a directed graph $F$ , and if $α$ denotes the induced action of $G$ on the $C^{}$ -algebra $C(F)$ of the graph, then the reduced crossed-product $Cr(G,C(F),α)$ is strongly Morita-equivalent to the $C^{}$ -algebra $C(F/G)$ of the quotient graph. Very recently, Valentin Deaconu has been studying actions of discrete groupoids on Cuntz-algebra graphs, where the vertex set of such a graph is precisely the unit space of the acting groupoid. In this talk, I shall explain my observation that Jonathan Brown’s theory of proper groupoid dynamical systems may be used to prove an analog of the Kumjian-Pask Theorem for free actions of discrete groupoids on Cuntz-algebra graphs (in the sense of Deaconu). A Kumjian-Pask Theorem for Free Actions of Discrete Groupoids on Cuntz-Algebra Graphs
Nov. 14, 2017 11am (MATH 350)	Number Theory Scott Ahlgren (UIUC) X Kloosterman sums are ubiquitous in number theory. We estimate sums of Kloosterman sums of half-integral weight. Our estimates are uniform in all parameters in analogy with Sarnak and Tsimerman’s improvement of Kuznetsov's bound for the ordinary Kloosterman sums. As an application, we obtain an improved estimate for the classical problem of estimating the size of the error term in Rademacher's famous formula for the partition function. This is joint work with Nick Andersen (UCLA) Kloosterman sums, Maass cusp forms, and Rademacher’s formula
Nov. 14, 2017 12:10pm (Math …	Kempner Josh Grochow (CU Boulder) X Wildness is a phenomenon that grew up in representation theory, but has broad implications for classification problems, and turns out to be at the mathematical core of several major open problems in computational complexity. To give the flavor: Drozd's Dichotomy Theorem says that any finite-dimensional associative algebra over a field either has: (a) only finitely many indecomposable representations (e.g., finite groups in characteristic zero), (b) only finitely many 1-parameter families of indecomposable representations of each dimension (e.g., the polynomial ring k[x], where the 1-parameter families are given by choosing an eigenvalue for a Jordan block), or (c) is so complicated that it "contains" the representations of all other finite-dimensional algebras. The latter are called wild. This talk will provide a relatively gentle* introduction to wildness, and a few of the many places it arises in mathematics and computational complexity, including algorithms for testing isomorphism of finite groups, algorithms for multiplying matrices, and the only current approach to resolving major open questions such as P versus NP. * - In particular, by "gentle" I mean that I will not assume any background in representation theory. Wildness & geometry in representation theory & computational complexity
Nov. 14, 2017 1pm (MATH 220)	Athena Sparks (CU Boulder) Polynomial-time algorithm for PCSPs with majority polymorphisms
Nov. 14, 2017 2pm (MATH 350)	Lie Theory Farid Aliniaeifard (CU) X We introduce a new class of polytopes generalizing chain polytopes of posets introduced by Stanley. More precisely, we define a polytope for each (non-reflexive) relation R on a finite set X. The vertices of the polytopes are indexed by the independent sets for the relation, which are natural generalizations of the notion of an antichain in a poset. Part of the motivation for studying this class of polytopes is the variety of polytopes that it encompasses. These include polytopes whose vertices are indexed by partitions, noncrossing partitions, nonnesting partitions. We describe the 1-skeletons of these polytopes and show that the 2-faces are either triangles or quadrilaterals. In the end, we define the Birkhoff R-polytope and describe its 1skeletons. R-Polytopes
Nov. 14, 2017 2pm (MATH 220)	Peter Mayr (CU Boulder) Promises and constraints 5
Nov. 14, 2017 3pm (MATH 350)	Topology Andy Riddle X Abstract: In the 1960’s Quillen developed model categories as a way to axiomatize homotopy theory. In this setup, a Quillen model category is a category with three special classes of morphism (weak equivalences, fibrations, and cofibrations) which satisfy a few axioms. These axioms then give a framework in which it is possible to setup the machinery of homotopy theory in a wide variety of contexts. In this talk I will give a gentle introduction to the definition of a model category and consider some of the model category structures which occur most frequently in algebraic topology. Introduction to Model Categories
Nov. 15, 2017 4:03pm (MATH 3…	Grad Student Seminar Sebastian Bozlee X Solutions to polynomial equations are most easily found in affine space, $\C^n$, but a better place to study them is projective space. We will use the problem of where parallel lines intersect to arrive at a definition of projective space, then take a look around this new world with the help of a pair of computer programs written by the speaker. A look into projective space
Nov. 16, 2017 3pm (MATH 350)	Probability Yu-Jui Huang (CU Boulder) X For time-inconsistent control/stopping problems, it is known that one should employ an equilibrium strategy, formulated in an intertemporal game between current and future selves. Such strategies, however, are not unique. This gives rise to two unsettled problems: (i) How do we find all equilibria? (ii) If we found all equilibria, how do we select the appropriate one to use? For stopping problems under non-exponential discounting, we develop a new method, called the iterative approach, to resolve both (i) and (ii). First, we formulate equilibria as fixed points of an operator, which represents strategic reasoning that takes into account future selves’ behaviors. Under appropriate regularity conditions, every equilibrium can be found through a fixed-point iteration. When the state process is one-dimensional, we further establish the existence of an optimal equilibrium, which generates larger value than any other equilibrium does at all times. To the best of our knowledge, this is the first time a dominating subgame perfect Nash equilibrium is shown to exist in the literature of time-inconsistency. Our theory is illustrated explicitly in several real options models. Optimal Equilibrium for Time-inconsistent Stopping Problems
Nov. 28, 2017 10am (Math 350)	Functional Analysis Elmar Wagner (Instituto de Física y Matemáticas Universidad Michoacana de San Nico…) X One of the most frequent relations appearing in quantum-group theory is $x y = q y x$ , where $q$ denotes a real deformation parameter. Defining a $$ -structure by $x := y$ , one obtains the relation for the quantum complex plane: $x x = q x x$ . Operator theorists call Hilbert-space operators satisfying this equation “ $q$ -normal operators”. It is known that $q$ -normal operators are never bounded. To study such quantum spaces in the $C^{}$ -algebraic setting, one may apply Woronowicz’s theory of $C^{}$ -algebras generated by unbounded elements. It turns out that these quantum spaces can be described by subalgebras of certain crossed-product $C^{}$ -algebras. In this talk, I will give a complete topological description of the $C^{}$ -algebras generated by $q$ -normal operators, including their $K$ -theory, $K$ -homology and index pairings. Among all these $C^{}$ -algebras, there is only one that has the same $K$ -theoretical behavior as its classical counterpart. This happens if and only if the spectrum of $\|x\|$ is the entire positive half-line. (Joint work with Ismael Cohen.) $C^{*}$ -Algebras Generated by the Quantum Complex Plane Sponsored by the Meyer Fund
Nov. 28, 2017 11am (MATH 350)	Number Theory Robert A. Hines (CU) X We will recall two characterizations of badly approximable real numbers (via continued fractions and geodesic trajectories on the modular surface) and discuss real quadratic irrationals from both perspectives. We will then generalize these characterizations to imaginary quadratic fields (using continued fractions when applicable and geodesic trajectories in the Bianchi orbifolds more generally). Finally we will use these characterizations to produce examples of badly approximable numbers (those lying on anisotropic rational circles). These examples include algebraic numbers of every even degree over the rationals. Badly approximable numbers over imaginary quadratic fields
Nov. 28, 2017 12:10pm (MATH …	Kempner Alexander Hulpke (CSU Fort Collins) X The group $SLn(Z)$ (as well as many of its friends) is ubiquitous in Algebra, Number Theory or Geometry. For $n≥ 3$ it has the congriuence subgroup property, meaning that any finite index subgroup $U$ contains the kernel of a ``reduction by m’’ homomorphism $colon SLn(Z)→ SLn(Z/mZ)$ ’’. The smallest $m$ with this property is called the level of $U$ . For a subgroup $U$ , given by generators, we want to compute its level (which then allows to compute index and test membership efficiently). This task consists of two parts: Determining the primes dividing $m$ and then the power of each prime. Using the structure of SL over residue class rings $Z/kZ$ , the latter can be computed in finite congruence images. For the former we utilize the generic description of maximal subgroups of SL according to Aschbacher classes and develop effective tests for determining those primes, for which the congruence image lies in a particular congruence image. This is joint work with Alla Detinko (St Andrews) and Dane Flannery (Galway) Finding the level of arithmetic subgroups of SL_n
Nov. 28, 2017 1pm (MATH 220)	Nik Ruskuc (St Andrews) X It has become a folk-lore in group theory that subgroups of finite index share many basic properties with their parent group. For instance, Schreier's Theorem asserts that a subgroup H of finite index in a group G is finitely generated if and only if G is finitely generated. One can replace 'finitely generated' by 'finitely presented' (Reidemeister-Schreier Theorem), 'residually finite', etc. Motivated by the generality, ubiquity and importance of these types of results, one may be tempted to look for analogues in other algebraic systems. In this talk I will concentrate on such analogues for the Schreier and Reidemeister-Schreier theorems in rings and semigroups. It turns out that for rings both theorems carry over, and generalise to K-algebras over noetherian commutative rings. However, the proof exhibits some significant differences from the original group result. For semigroups, even the meaning of 'index' comes into question, but then it turns out that one can prove several different types of the above results, e.g. for subgroups of semigroups, or for Schutzenberger groups, or for subsemigroups of finite complement. I will conclude with some open problems arising from the comparison between rings and semigroups. The ring version of the Reidemeister-Schreier Theorem is recent joint work with Peter Mayr. Finite index and finiteness conditions: Groups and Semigroups Sponsored by the Meyer Fund
Nov. 28, 2017 2pm (MATH 220)	Nik Ruskuc (St Andrews) X It has become a folk-lore in group theory that subgroups of finite index share many basic properties with their parent group. For instance, Schreier's Theorem asserts that a subgroup H of finite index in a group G is finitely generated if and only if G is finitely generated. One can replace 'finitely generated' by 'finitely presented' (Reidemeister-Schreier Theorem), 'residually finite', etc. Motivated by the generality, ubiquity and importance of these types of results, one may be tempted to look for analogues in other algebraic systems. In this talk I will concentrate on such analogues for the Schreier and Reidemeister-Schreier theorems in rings and semigroups. It turns out that for rings both theorems carry over, and generalise to K-algebras over noetherian commutative rings. However, the proof exhibits some significant differences from the original group result. For semigroups, even the meaning of 'index' comes into question, but then it turns out that one can prove several different types of the above results, e.g. for subgroups of semigroups, or for Schutzenberger groups, or for subsemigroups of finite complement. I will conclude with some open problems arising from the comparison between rings and semigroups. The ring version of the Reidemeister-Schreier Theorem is recent joint work with Peter Mayr. Finite index and finiteness conditions: Rings Sponsored by the Meyer Fund
Nov. 28, 2017 2pm (MATH 350)	Lie Theory Jon Lamar (CU) X TBD Two Hopf subalgebras from restriction supercharacter theories
Nov. 28, 2017 3pm (MATH 350)	Topology Michael Fried (UC Irvine) X This seminar consists of two consecutive talks (3-4 pm and 4-5 pm). First talk 3-4 pm Title: What Gauss Told Riemann About Abel's Theorem, Part 1: Tools from understanding rational functions in 1 complex variable Abstract: There are two exceedingly famous Abel Theorems. The more well-known, and easier, describes the "abelian (analytic) functions" on a compact complex 1-dimensional torus. The other bundles together all those complex tori, with their prime degree isogenies. Riemann's generalization of the first features his famous theta functions. His deepest work aimed at generalizing Abel's second theorem. He died too young to fulfill the whole vision. We can see how such a generalization works, and its relation to modern problems, by considering how Galois used Abel's spaces to produce simple groups (the opposite of solvable) from algebraic functions. In this first lecture, I introduce tools using just rational functions in 1-variable (like those that appeared in high school algebra). I will conclude with a Proposition, that introduces Braids for a formula to calculate the genus, in particular, of Abel's spaces. Second talk 4-5 pm Title: What Gauss Told Riemann About Abel's Theorem, Part 2: Abel's spaces in towers, and how those towers generalize Abstract: Abel's spaces are level 0 of a tower of spaces we now label as {X_0(pk+1)}k=0^{infty} (modular curves). I will use rational functions to produce this tower. Then, almost without lifting my chalk from the blackboard, I will give such towers using (essentially) any finite group G that isn't nilpotent (product of its p-Sylows) and has r = 4 generating conjugacy classes. Like modular curves, we see each tower level is an upper half-plane quotient, ramifying at the points (0, 1, infty) over the famous copy of Riemann's sphere, the j-line. The braid formula applies to computing their genuses, too. Yet, except for special groups G, the tower levels are not modular curves. I will do one example, G = A5 (the alternating group of degree 5), where we clearly see these are not modular curves. The same construction applies for any value of r. For all r, these spaces have properties that generalize famous results about modular curves, and those interpret as statements about The Inverse Galois Problem. These lectures introduce a book I am completing: Monodromy, l-adic Representations and The Regular Inverse Galois Problem. What Gauss told Riemann about Abel's theorem
Nov. 28, 2017 3pm (MATH 350)	Number Theory Michael Fried (UC Irvine) X This seminar consists of two consecutive talks (3-4 pm and 4-5 pm). First talk 3-4 pm Title: What Gauss Told Riemann About Abel's Theorem, Part 1: Tools from understanding rational functions in 1 complex variable Abstract: There are two exceedingly famous Abel Theorems. The more well-known, and easier, describes the "abelian (analytic) functions" on a compact complex 1-dimensional torus. The other bundles together all those complex tori, with their prime degree isogenies. Riemann's generalization of the first features his famous theta functions. His deepest work aimed at generalizing Abel's second theorem. He died too young to fulfill the whole vision. We can see how such a generalization works, and its relation to modern problems, by considering how Galois used Abel's spaces to produce simple groups (the opposite of solvable) from algebraic functions. In this first lecture, I introduce tools using just rational functions in 1-variable (like those that appeared in high school algebra). I will conclude with a Proposition, that introduces Braids for a formula to calculate the genus, in particular, of Abel's spaces. Second talk 4-5 pm Title: What Gauss Told Riemann About Abel's Theorem, Part 2: Abel's spaces in towers, and how those towers generalize Abstract: Abel's spaces are level 0 of a tower of spaces we now label as {X_0(pk+1)}k=0^{infty} (modular curves). I will use rational functions to produce this tower. Then, almost without lifting my chalk from the blackboard, I will give such towers using (essentially) any finite group G that isn't nilpotent (product of its p-Sylows) and has r = 4 generating conjugacy classes. Like modular curves, we see each tower level is an upper half-plane quotient, ramifying at the points (0, 1, infty) over the famous copy of Riemann's sphere, the j-line. The braid formula applies to computing their genuses, too. Yet, except for special groups G, the tower levels are not modular curves. I will do one example, G = A5 (the alternating group of degree 5), where we clearly see these are not modular curves. The same construction applies for any value of r. For all r, these spaces have properties that generalize famous results about modular curves, and those interpret as statements about The Inverse Galois Problem. These lectures introduce a book I am completing: Monodromy, l-adic Representations and The Regular Inverse Galois Problem. What Gauss told Riemann about Abel's theorem
Nov. 28, 2017 3pm (Math 350)	Algebraic Geometry Michael Fried (UC Irvine) X This seminar consists of two consecutive talks (3-4 pm and 4-5 pm). First talk 3-4 pm Title: What Gauss Told Riemann About Abel's Theorem, Part 1: Tools from understanding rational functions in 1 complex variable Abstract: There are two exceedingly famous Abel Theorems. The more well-known, and easier, describes the "abelian (analytic) functions" on a compact complex 1-dimensional torus. The other bundles together all those complex tori, with their prime degree isogenies. Riemann's generalization of the first features his famous theta functions. His deepest work aimed at generalizing Abel's second theorem. He died too young to fulfill the whole vision. We can see how such a generalization works, and its relation to modern problems, by considering how Galois used Abel's spaces to produce simple groups (the opposite of solvable) from algebraic functions. In this first lecture, I introduce tools using just rational functions in 1-variable (like those that appeared in high school algebra). I will conclude with a Proposition, that introduces Braids for a formula to calculate the genus, in particular, of Abel's spaces. Second talk 4-5 pm Title: What Gauss Told Riemann About Abel's Theorem, Part 2: Abel's spaces in towers, and how those towers generalize Abstract: Abel's spaces are level 0 of a tower of spaces we now label as {X_0(pk+1)}k=0^{infty} (modular curves). I will use rational functions to produce this tower. Then, almost without lifting my chalk from the blackboard, I will give such towers using (essentially) any finite group G that isn't nilpotent (product of its p-Sylows) and has r = 4 generating conjugacy classes. Like modular curves, we see each tower level is an upper half-plane quotient, ramifying at the points (0, 1, infty) over the famous copy of Riemann's sphere, the j-line. The braid formula applies to computing their genuses, too. Yet, except for special groups G, the tower levels are not modular curves. I will do one example, G = A5 (the alternating group of degree 5), where we clearly see these are not modular curves. The same construction applies for any value of r. For all r, these spaces have properties that generalize famous results about modular curves, and those interpret as statements about The Inverse Galois Problem. These lectures introduce a book I am completing: Monodromy, l-adic Representations and The Regular Inverse Galois Problem. What Gauss Told Riemann About Abel's Theorem
Nov. 30, 2017 11am (MATH 350)	Geometry/Analysis Dana Mendelson (Chicago) X We consider the (de)focusing cubic Gross-Pitaevskii (GP) hierarchy on R, which is an infinite hierarchy of coupled linear non-homogeneous PDE which appears in the derivation of the cubic nonlinear Schrodinger (NLS) equation from quantum many-particle systems. Motivated by the fact that the cubic NLS on R is an integrable equation which admits infinitely many conserved quantities, we exhibit an infinite sequence of operators which generate analogous conserved quantities for the GP hierarchy. This is joint work with Andrea Nahmod, Natasa Pavlovic, and Gigliola Staffilani. An infinite sequence of conserved quantities for the cubic Gross-Pitaevskii hierarchy on R Sponsored by the Meyer Fund
Nov. 30, 2017 3pm (MATH 350)	Probability Rafael Frongillo (CU Boulder) X In statistics, the term "elicitation" refers to the relationship between a loss function and its (expected) minimizer. The term arises from economics, where the loss is taken to be a monetary penalty, and one wishes to incentivize someone to truthfully report a desired statistic of some eventually observed random variable. Interestingly, not all statistics are elicitable, including simple measures of risk or deviation like the variance of a random variable. I will give a brief overview of elicitation, and mention a few interesting open questions in elicitation complexity, where we seek to indirectly elicit otherwise non-elicitable statistics. I will conclude with a connection to the notion of random individual sequences. Elicitation Complexity and Individual Sequences
Dec. 05, 2017 1pm (MATH 220)	Jeff Shriner (CU Boulder) X The Subpower Membership Problem for a fixed finite algebra A (SMP(A)) is a combinatorial decision problem which asks, given a set of generators a_1,...,a_n and a distinguished tuple b in a direct power of A, is b in the subalgebra generated by a_1,...,a_n? In this talk, we discuss a construction which begins with any finite algebra A, and ends with a structurally 'nicer' algebra B for which SMP(B) is at least as hard as SMP(A). Given a specific finite algebra A, it is then natural to ask if the complexity of the associated problem SMP(B) is equal to or harder than SMP(A). We will look at a known example of a finite semigroup S for which SMP(S) is NP-complete, and show that the subpower membership problem for the associated 'nicer' algebra is also NP-complete. An NP-Complete Example for the Subpower Membership Problem
Dec. 05, 2017 2pm (MATH 350)	Lie Theory Josh Grochow (CU) X How hard it is to multiply matrices is a central open question in computational complexity, with ties to algebraic geometry and representation theory. In this talk I'll survey some recent work on matrix multiplication which brings together algorithms, representation theory, and additive combinatorics. Highlights include a transparent proof (joint with C. Moore) of Strassen's original 1969 algorithm for matrix multiplication - which was the first to beat $O(n3)$ steps - and using ideas from the resolution of the Cap Set Conjecture to deepen our understanding of the Cohn-Umans group-theoretic approach to matrix multiplication (joint with J. Blasiak, T. Church, H. Cohn, E. Naslund, W. F. Sawin, and C. Umans). Representation theory and additive combinatorics in algorithms for matrix multiplication
Dec. 05, 2017 2pm (MATH 220)	Trevor Jack (CU Boulder) X A semigroup S is nilpotent if S^n = 0 for some n. We describe a polynomial-time algorithm for determining whether a semigroup of transformations given by generators is nilpotent. Checking nilpotence for semigroups
Dec. 05, 2017 3pm (MATH 350)	Topology Nikki Sanderson X Time series analysis traditionally relies upon statistics and frequency analyses that make restrictive assumptions about the data - i.e. nonlinearity, non-stationarity. We believe topological data analysis (TDA) can be of benefit in these situations. The process of delay coordinate reconstruction ``unfolds” a scalar time-series into a point cloud in Rm. We can then compute the persistent homology of the reconstructed data to obtain a topological signature. With the ultimate goal of regime shift detection in mind, we choose to use the witness complex - a sparse simplicial complex - for these computations. Topologically accurate delay reconstruction requires appropriate choices for the dimension m and time delay. We introduce novel witness relations that incorporate time and improve the robustness of the resulting homology with respect to choice of delay. The new relations seek to inhibit data points from witnessing landmarks traveling in dissimilar directions, as these can create false connections. We explore how these relations can ameliorate additional challenges that arise when dealing with nonuniform samples of strange attractors. Witness Complexes for Time Series Analysis
Dec. 06, 2017 4:03pm (MATH 3…	Grad Student Seminar Ruofan Li X Chebyshev polynomials (of the first kind) are one of the simplest family of maps that appear in the study of dynamical system. They can also be found in many branches of mathematics: complex analysis, differential equation, number theory and etc.. This talk will cover the definition of Chebyshev polynomials and some basic properties of them. Introduction to Chebyshev polynomials
Dec. 06, 2017 5pm (Math 350)	MathClub Faan Tone Liu (CU Boulder) X In this meeting, we will learn how to fold two particular units in modular origami. We will use some basic geometry theorems to calculate the angles within the units, and then construct a cube and a dodecahedron from the folded units. As time permits we will look at other Platonic and Archimedean solids and investigate how they might be constructed from modular origami units. Solids, Geometry and Folding
Dec. 07, 2017 12:10pm (MATH …	Kempner Geoffrey Grimmett (Cambridge) X The problem of self-avoiding walks (SAWs) arose in statistical mechanics in the 1940s, and has connections to probability, combinatorics, and the geometry of groups. The basic question is to count SAWs. The so-called 'connective constant' is the exponential growth rate of the number of n-step SAWs. We summarise recent joint work with Zhongyang Li concerned with the question of how the connective constant depends on the choice of graph. This work includes equalities and inequalities for connective constants, and a partial answer to the so-called 'locality problem' for graphs and particularly Cayley graphs. Self-avoiding walks on graphs and groups
Dec. 07, 2017 3pm (MATH 350)	Probability Manuel Lladser (Applied Math, CU Boulder) X In this talk, I will present work in progress to approximate the distribution of so-called linear functionals of paths in Markov chains. Archetypical examples of these are sojourn-times, such as those encountered in genomic sequence analyses, telecommunication protocols, and inventory models. Approximating the distribution of linear functionals is a challenging problem in transient regimes because exact computations, as well as stationarity assumptions, are typically infeasible in such regimes. Instead, using low-rank operators and probabilistic arguments, we will see how to approximate the distribution of these functionals in total variation distance - provided the existence of a so-called “regeneration” set; in particular, this setting encompasses Harris-recurrent Markov chains. This work is in collaboration with Dr. Barrera from Universidad Adolfo Ibañez in Chile. Non-asymptotic approximation of Markovian functionals in transient regimes
Dec. 12, 2017 10am (Math 350)	Functional Analysis Jonathan Brown (University of Dayton) X Let $G$ be an étale groupoid. In joint work with Nagy, Reznikoff, Sims and Williams, we study a subgroupoid $G'$ of $G$ consisting of isotropy, in an effort to determine when a representation of $C(G)$ is injective. We show that a representation is injective on $C(G)$ if and only if its restriction to $C(G )$ is, and that when $G'$ is closed, then $C(G )$ is a Cartan subalgebra of $C(G)$ . Now, Renault in 2008 showed that given a Cartan subalgebra $D$ of a $C^{}$ -algebra $A$ , one can construct a twist $T$ so that $C(T)$ is isomorphic to $A$ . In this talk, I will first describe my joint results with Nagy et al. and then show how the work of Muhly, Renault and Williams can be used to explicitly construct a twist associated to the inclusion of $C(G )$ into $C(G)$ when $G'$ is all of the isotropy. I will then point out how this fails more generally. Canonical Cartan Subalgebras of Étale Groupoid $C^{}$ -Algebras and the Construction of Twists Sponsored by the Meyer Fund
Dec. 12, 2017 12:10pm (MATH …	Kempner Kenneth McLaughlin (CSU) X Some surprising questions in analysis arise in the interconnections of the topics in the title. We will encounter zeros of the Taylor approximants of exp(z), and other analytic functions. We will consider questions of support of equilibrium charge distributions in the plane. Semi-classical analysis of d-bar problems will provide merriment along the way. Random matrices, d-bar problems, and approximation theory
Jan. 16, 2018 11am (MATH 350)	Number Theory Andrew Sutherland (MIT) X Number fields with the same Dedekind zeta function are said to be arithmetically equivalent. Such number fields necessarily have the same degree, discriminant, signature, Galois closure, and isomorphic unit groups, but may have different regulators, class groups, rings of adeles, and idele class groups. Motivated by a recent result of Prasad, I will discuss three stronger notions of arithmetic equivalence that force isomorphism of some or all of these invariants without forcing an isomorphism of number fields, along with explicit examples. These results also have applications to the construction of curves with isomorphic Jacobians (due to Prasad), isospectral Riemannian manifolds (due to Sunada), and isospectral graphs (due to Halbeisen and Hungerbuhler). Strong Arithmetic Equivalence
Jan. 23, 2018 11am (MATH 350)	Number Theory Nikolay Moshchevitin (Lomonosov Moscow State University) X Let $α$ be a real irrational number. Almost all information about the rational approximation to $α$ may be found in the irrationality measure function $ψα(t) = minq ∈ Z+, q ≤ t \|\|qα\|\|$ (here \|\| · \|\| stands for the distance to the nearest integer). By Lagrange theorem we know that the jumps of the function $ψα(t)$ correspond to the convergents of the continued fraction expansion of $α$ . However, there are various different "irrationality measure functions". We will discuss properties some of these functions, Diophantine spectra and relatively new results on indeterminacy and quantization. Continued fractions, irrationality measures and oscillation theorems
Jan. 23, 2018 1pm (MATH 220)	Peter Mayr (CU Boulder) X Commutators and derived notions like nilpotence, solvability ... have been generalized from groups to universal algebras in the 1970s. The more recently developed higher commutators yield stronger concepts like supernilpotence. We compare these two versions of nilpotence and pose some open questions about them. Supernilpotent algebras 1
Jan. 23, 2018 2pm (MATH 220)	William DeMeo (CU Boulder) X The first in a two-part series on the Lean proof assistant, this lecture introduces the mathematics of functional programming, namely, the lambda calculus. The second lecture will give a basic introduction to Lean, a relatively new addition to the growing family of functional programming languages. Lean improves upon earlier proof assistants by making it more natural and less painful to formalize mathematical proofs. A Tutorial Introduction to the Lean Prover: Part 1
Jan. 23, 2018 3pm (MATH 350)	Topology Craig Westerland (University of Minnesota) X Topological T-duality -- first investigated by Bouwknegt-Evslin-Mathai, Bunke-Schick, and others -- concerns a surprising duality isomorphism between twisted cohomology theories on different spaces. It has traditionally been studied for twisted K-theory and deRham cohomology, owing to a physical origin where these objects can be considered as spaces of fields in a certain field theory. In joint work with John Lind and Hisham Sati, we have established a general context in which duality theorems of this sort can be proven. This context is naturally related to Cartier duality of finite Hopf algebras, and gives a host of new examples, particularly arising from chromatic homotopy theory. Cartier duality and topological T-duality Sponsored by the Meyer Fund
Jan. 23, 2018 4pm (MATH 350)	Topology Hans-Christian Herbig (Federal University of Rio de Janeiro) X In the study of quadratic moment maps there appears to be a dichotomy between large and small representations. In the large case the moment map exhibits some features of regularity. This enables one to make qualitative and quantitative statements about the symplectic quotients. In the talk I will report on recent joint work with Gerald Schwarz and Christopher Seaton which proves that for 2-large representations the appropriately defined complex symplectic quotient has symplectic singularities. This in particular entails that the symplectic quotient is graded Gorenstein domain and is a normal variety with rational singularities. It is expected that this result generalizes to small representations as well, even though here the moment map can have all sorts of pathologies. For this conjecture it is crucial to use the definition of the symplectic quotient that involves the real radical of the ideal generated by the moment map. Moment map phenomenology Sponsored by the Meyer Fund
Jan. 24, 2018 4:02pm (MATH 3…	Grad Student Seminar Paul Lessard X We will begin with an exploration of colimits in the category of sets, a categorical notion characterizing compatible maps out of a diagram, and use a universal construction for them to relate colimits and equivalence relations. We'll then define the notion of a congruence, an equivalence relation compatible with further structure on those sets, e.g. group, ring, etc, and ask: "from which shapes emerge such equivalence relations?" The answer to this question will be sifted diagrams. We'll then examine other properties that sifted colimits have, and stumble upon, albeit in the direction counter to history, Lawvere's algebraic theories, the first way that universal algebra was instantiated category theorically. This talk is intended to be accessible. Very little will be assumed as is hopefully conveyed by the abstract. Of Colimits, Sifted and Not, and the Naturalness of Algebraic Theories
Jan. 30, 2018 11am (MATH 350)	Number Theory David McKinnon (University of Waterloo, Canada) X Of all the algebraic curves around, the rational ones have the most rational points. (And yes, the terminology is mostly accidental!) So maybe if an algebraic variety of higher dimension has lots of rational points, then it ought to have lots of rational curves too? The answer is "probably yes", but is complicated enough that I can spin it out to a whole seminar's worth of talking. Vojta's Conjectures, the Batyrev-Manin Conjectures, and Diophantine approximation are but three of the threads that I will weave into the narrative. Rational curves and rational points
Jan. 30, 2018 12:10pm (MATH …	Kempner Jesse Wolfson (UC Irvine) X Algebraic topology was invented by Poincare in 1895 to study the behavior of algebraic functions. In his seminal ICM address 5 years later, Hilbert posed a fundamental challenge to the field: find a topological obstruction to reducing the solution of the general degree 7 polynomial to an expression in functions of two or fewer variables. In this talk, I'll review some of the beautiful history of algebraic topology and algebraic functions, discuss Hilbert's problem(s), and outline ongoing work in applying the topology of braids and algebraic functions to this problem. This is joint work with Benson Farb. Algebraic Topology and Hilbert's 13th Problem(s)
Jan. 30, 2018 1pm (MATH 220)	Peter Mayr (CU Boulder) Supernilpotent algebras 2
Jan. 30, 2018 2pm (MATH 350)	Lie Theory Farid Aliniaeifard (CU Boulder) X By using the action of a Galois group on complex irreducible characters and conjugacy classes, we define the Galois characters and Galois classes. We will introduce a set of Galois characters, called Galois irreducible characters, that each Galois character is a positive linear combination of the Galois irreducible characters. For some family of groups, we show that whenever the complex characters of the groups produce a positive self-dual Hopf algebra (PSH), Galois characters of the groups also produce a PSH. We will classify the Galois characters and Galois classes of the general linear groups over finite fields. In the end, we will precisely indicate an isomorphism between the PSH of Galois characters and a certain tensor product of Hopf algebra of symmetric functions. Galois Character Theory
Jan. 30, 2018 2pm (MATH 220)	William DeMeo (CU Boulder) A Tutorial Introduction to the Lean Prover: Part 2
Jan. 30, 2018 3pm (MATH 350)	Topology Jesse Wolfson (University of California Irvine) X Abstract: Basic questions in analytic number theory concern the density of one set in another (e.g. square-free integers in all integers). Motivated by Weil's number field/function field dictionary, we introduce a topological analogue measuring the “homological density” of one space in another. In arithmetic, Euler products can be used to show that many seemingly different densities coincide in the limit. By combining methods from manifold topology and algebraic combinatorics, we discover analogous coincidences for limiting homological densities arising from spaces of 0-cycles (e.g. configuration spaces of points) on smooth manifolds and complex varieties. We do not yet understand why these topological coincidences occur. This is joint work with Benson Farb and Melanie Wood. Title: Coincidences of homological densities, predicted by arithmetic
Jan. 31, 2018 4:03pm (MATH 3…	Grad Student Seminar Taylor Klotz X ODEs (ordinary differential equations) and PDEs (partial differential equations) are frequently viewed as being a rather “applied” subject in mathematics. In this talk I’ll give some basic background on differential equations and then move onto how differential equations have either helped solved problems in other pure fields of math or how the study of particular types of differential equations have helped kickstart seemingly unrelated areas of research. Specific areas I plan on mentioning will be number theory, lie algebra, algebraic geometry, topology, differential geometry and possibly others! Differential Equations in Funny Places
Jan. 31, 2018 5pm (Math 350)	MathClub Joe Timmer (CU Boulder) X From Xeno's Paradox about Achilles and the Tortoise to the Banach-Tarski Paradox, the concept of "infinity" has raised some serious problems in mathematics and philosophy. We will explore the different concepts of "infinity" in mathematics and philosophy through examples and the mind-bending implications these have. The Incongruities of Infinity
Feb. 01, 2018 11am (MATH 350)	Mike Fried (CSU and University of California, Irvine) X Arithmetic Geometry is algebraic geometry with the goal of finding how the absolute Galois group of Q acts on geometric objects defining solutions to problems. In this particular seminar the geometric objects will be spaces that define solutions to several well-known problems, especially including the Inverse Galois problem. One example of those moduli spaces are the modular curves studied by Mazur, Wiles, and many others. The introduction of these spaces requires some cohomology of finite groups. The first part of the talk may be well-known to those who have had some cohomology. Yet, the treatment is different. It will allow introducing groups and equations that put some of those famous problems in a new context. Group cohomology and Arithmetic Geometry
Feb. 06, 2018 11am (MATH 350)	Number Theory Number Theory Lab X Our number theory seminar slot is sometimes unoccupied (the next such date is January 23rd). We will begin filling these empty slots with "Number Theory Lab Meeting". This is aimed at creating a research community, especially for the benefit of our graduate students, but it will be of interest to all the number theorists and aspiring or closet number theorists in the department and so I wanted to make an announcement to the whole department. Young students who are considering number theory as their future track are especially invited, of course. You'll see it listed in the seminar calendar under the number theory seminar, and more information follows below. The basic design is modelled after CSU's "lab meeting". It's fluid and adaptable, but the basic structure, at least to start, is: 11-11:10 - tea and informal chatter 11:10 - progress reports (we'll go around the room and everyone can give an update on what they are doing, mostly research, but also relevant may be pedagogy, job applications, etc.) after that - a totally informal 20 minute chalk talk by one member of the group, about their research interests (background, for now, later after we are all more familiar, maybe recent results). There are enough of us that I expect we may all get to speak approximately once per semester. We'll ask for volunteers as we rotate through so you have some warning/choice about when your turn is coming. This is friendly, and not meant to add pressure. When interest dictates, the time can be used for other things, like career discussions, practice talks, trying out a cool number theory outreach activity, learning Sage together, whatever comes up. I'll (Kate Stange) happily take the first slot and give some background on a problem I'm interested in on January 23rd. Your job is to show up and ask a ton of questions. Number Theory Lab Meeting
Feb. 06, 2018 1pm (MATH 220)	Peter Mayr (CU Boulder) Supernilpotent algebras 3
Feb. 06, 2018 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X The partition algebra is a classical diagram algebra arising out of Schur—Weyl duality in representation theory. While there are numerous approaches to constructing a $q$ -deformation of the partition algebra, they each seem to have had different obstructions. This talk will review the partition algebra, its possible generalizations, and then focus on one particular version that has seen some recent progress. We recover some possibly new combinatorial objects called splatters, which are a generalization of set partitions. This work is joint work with Tom Halverson. A $q$ -partition algebra and splatter combinatorics
Feb. 06, 2018 3pm (Math 350)	Algebraic Geometry Mark Shoemaker (CSU) X Gromov-Witten theory gives a way of defining enumerative invariants of a smooth complex variety X by, roughly, counting the number of curves on X satisfying various constraints. If two varieties X and Y have similar geometries, say they are birational, it is natural to wonder how their Gromov-Witten invariants relate to one another. Many conjectures and results along these lines have dealt with the case where X and Y are K-equivalent or related by crepant transformation. In joint work with Acosta, we extend these results, in the toric case, to more general birational transformations. Given two complete toric orbifolds X and Y related by wall crossing under variation of GIT, we prove that certain Gromov-Witten generating functions are related by linear transformation and asymptotic expansion. This generalizes the relationship described by the crepant transformation conjecture to the non-crepant setting. Gromov-Witten theory of toric birational transformations
Feb. 07, 2018 4:03pm (MATH 3…	Grad Student Seminar Natalie Coston Eigenvalues, outliers, and where to find them
Feb. 08, 2018 11am (MATH 350)	Mike Fried (University of California at Irvine) X The groups in question are what we referred to last week as "extreme" extensions of finite groups. They are Frattini covers: special homomorphisms of one finite group to another that are quotients of a certain universal extension (of any finite group). The towers of varieties they bind include classical towers of what are called modular curves. By generalizing the properties of those modular curve towers we connect two books of J.P. Serre: "Abelian l-adic Representations" (from elliptic curves) and "Topics in Galois Theory" (about the regular Inverse Galois Problem). These books were written over 25 years apart. We will construct these group covers from two viewpoints, structurally and homologically. This will allow understanding the structure of the spaces they produce, for which our examples will be upper half plane quotients by finite index subgroups of PSL_2(Z). A very small example of such a cover is the Spin cover of Alternating groups. Sequences of finite group extensions that bind algebraic varieties into a tower
Feb. 08, 2018 2pm (MATH 350)	Functional Analysis Leonard Huang (CU Boulder) X In this talk, I will first introduce Jonathan Rosenberg’s notion of a Levi-Civita connection for a Riemannian metric on a rank- $n$ free Hilbert module over a ‘generically transcendental’ quantum torus of dimension $n$ . I will then describe how to use Rosenberg’s Levi-Civita connections to construct metrized quantum vector bundles over ‘generically transcendental’ quantum tori, thus providing another class of examples for Frédéric Latrémolière’s work on the modular Gromov-Hausdorff propinquity. A technical tool that appears in the construction is the Minkowski gauge functional, which was used by Marc Rieffel in his work on Leibniz seminorms. (The new examples presented in this talk constitute the first part of ongoing joint work with Konrad Aguilar.) Constructing Metrized Quantum Vector Bundles over Quantum Tori from Rosenberg’s Levi-Civita Connections
Feb. 13, 2018 11am (MATH 350)	Number Theory Number Theorists of Boulder X Number Theory Lab is for number theorists of all types to form community and discuss their current research in an informal setting. Agenda: tea + progress reports + a small informal talk by Rob. Number Theory Lab Meeting
Feb. 13, 2018 12:10pm (MATH …	Kempner Nik Ruskuc (St Andrews) X Let S denote the set of all finite permutations, considered as rearrangements of {1,...,n}, n=1,2,3,... Pattern involvement is a partial order on S, and it arises naturally in the analysis of sorting mechanisms, such as stacks and queues, and also from a model-theoretic viewpoint, when permutations are considered as relational structures with two linear orders. Of particular interest are downward closed sets under this ordering, usually called pattern classes. Such classes can be defined by specifying minimal forbidden permutations; we write Av(pi_1,pi_2,...) to denote the pattern class consisting of all permutations avoiding pi_1,pi_2,... Thus, for instance: - Av(21) consists of all increasing permutations 12...n, n in N; - Av(321) consists of all permutations with no descending subsequences of length 3; - Av(231) consists of all permutations that can be sorted by a stack. In this talk I will discuss the subclasses contained in Av(321) and Av(231). I will set the scene by outlining some superficial similarities: e.g. they are both enumerated by the Catalan numbers. Then I will point out some radical differences: e.g. Av(231) is well quasi ordered (it has no infinite antichains), while Av(321) is not. Most of the talk will be devoted to an explication of some less obvious, but deeper, similarities between the subclasses, to do with their enumeration sequences. The notion of well quasi order and encodings by regular languages play a major background role in this development. The new results concerning the subclasses of Av(321) are a joint work with M. Albert, R. Brignall and V. Vatter. On classes of permutations avoiding 231 or 321
Feb. 13, 2018 2pm (MATH 220)	Nik Ruskuc (St Andrews) X A subdirect product of algebras A_1,...,A_n of the same type is any subalgebra C of their direct product A_1 x... x A_n which projects onto each A_i. In my ongoing work with Peter Mayr, we are considering the conditions under which such a subdirect product is: (a) finitely generated; (b) finitely presented. In this lecture I will give an introduction into this problem, by comparing subdirect products with what known for direct products. I will then present a construction of fiber products which completely characterises subdirect products of two factors in congruence permutable varieties (Fleischer's Lemma), and revisit the basic questions in this context. I will conclude with a summary of our main results. Finiteness properties of subdirect products Sponsored by the Meyer Fund
Feb. 14, 2018 4:03pm (MATH 3…	Grad Student Seminar Mark Pullins X When you're juggling at home or the office, it's natural to wonder how many different ways you can juggle. It turns out there's a mathematical approach to this. In this talk, we will focus primarily on simple juggling, which restricts the number of balls being thrown at a time. We will introduce notation that is used by both jugglers and mathematicians to describe juggling sequences, and then develop some tools for checking whether a sequence can be juggled and how to generate new juggling sequences. If time allows, we may venture deeper into the mathematics and connect juggling to the affine Weyl group and braids. This talk should be extremely accessible since I only have a basic knowledge of the material being covered. Obviously there will be juggling. Mathematical Juggling
Feb. 14, 2018 5pm (Math 350)	MathClub Jonathan Wise (CU Boulder) X A moduli space is a geometric space whose points have some intrinsic meaning. In this talk, we will examine the moduli space of plane curves to see how thinking about a space of all curves allows us to prove some interesting geometric theorems. We will also examine some moduli spaces from Euclidean geometry to get a sense of the peculiar geometry that can occur in moduli spaces. What is a moduli space?
Feb. 15, 2018 11am (MATH 350)	Geometry/Analysis Jeanne Clelland (University of Colorado, Boulder) X In this talk, I will give an outline of our recent proof for the local existence of a smooth isometric embedding of a smooth 3-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into 6-dimensional Euclidean space. Our proof avoids the sophisticated microlocal analysis used in earlier proofs by Bryant-Griffiths-Yang and Nakamura-Maeda; instead, it is based on a new local existence theorem for a class of nonlinear, first-order PDE systems that we call "strongly symmetric positive." These are a subclass of the symmetric positive systems, which were introduced by Friedrichs in order to study certain PDE systems that do not fall under one of the standard types (elliptic, hyperbolic, and parabolic). As in earlier proofs, we construct solutions via the Nash-Moser implicit function theorem, which requires showing that the linearization of the isometric embedding PDE system near an approximate embedding has a smooth solution that satisfies "smooth tame estimates." We accomplish this in two steps: (1) Show that the approximate embedding can be chosen so that the reduced linearized system becomes strongly symmetric positive after a carefully chosen change of variables. (2) Show that any such system has local solutions that satisfy smooth tame estimates. The main advantage of our approach is that step (2) is much more straightforward than similar results for other classes of PDE systems used in prior proofs, while step (1) requires only linear algebra. The talk will focus on the main ideas of the proof; technical details will be kept to a minimum. This is joint work with Gui-Qiang Chen, Marshall Slemrod, Dehua Wang, and Deane Yang. Isometric embedding via strongly symmetric positive systems
Feb. 15, 2018 11am (Math 220)	Mike Fried (University of California at Irvine) X Reminder: This week we will be meeting in Room 220, instead of Room 350. Also, next week the seminar will meet on Tuesday, in the Number Theory time at 11AM. I will give an exposition on how the construction this week gives tools tying together three sets of famous problems. Last week we used the phrase "extreme" extensions of a group $G$ . This name for Frattini covers, was to contrast them with split extensions of the group. We then produced the universal object which covers any particular case of such covers. It is projective in the category of profinite groups. Also, it decomposes as the fiber product over $G$ of covers $G_{ℓ} \to G$ , with kernel a pro-free $ℓ$ -adic group, one for each prime $ℓ$ dividing the order of $G$ . a seminar participant asked how we characterize these extensions cohomologically. That answer appears as the 4th among Propositions labeled " $ℓ$ -pieces, Parts 1, 2, 3 and 4" for the properties we need for the groups $G_{ℓ}$ . Part 1 is above. We do Part 2 this week: The explicit construction of $G_{ℓ}$ when the $ℓ$ -Sylow in $G$ is normal. We will also construct the quotients from these groups that will bind the spaces that turn the Regular Inverse Galois problem into a pure diophantine problem. The Main Conjecture, when $G$ is a dihedral group, is equivalent to an unsolved problem on certain torsion points on hyperelliptic Jacobians of a fixed dimension $d$ . This is known only for $d=1$ , where the spaces of hyperelliptic jacobians are modular curves. The Main Conjecture has been shown for the case corresponding to $d=1$ for any group $G$ . Sequences of Frattini covers of any finite group
Feb. 15, 2018 2pm (MATH 350)	Functional Analysis Sooran Kang (Chung-Ang University, Korea) X We first briefly discuss the (quantum) Yang-Mills theory on noncommutative spaces (in particular noncommutative tori) developed by Connes and Rieffel, and then we review the Yang-Mills functional and Laplace’s equation on the quantum Heisenberg manifolds $D_{μ, ν}^{c}$ . In particular, we review the Grassmannian connection and the minimizing connection on the finitely generated projective bimodule $Ξ$ over $E_{μ, ν}^{c}$ and $D_{μ, ν}^{c}$ , where $E_{μ, ν}^{c}$ is a generalized fixed-point algebra of a certain crossed product $C^{*}$ -algebra that also can be realized as a quantum Heisenberg manifold with different parameters. Then we give various examples of compatible linear connections on $Ξ$ , on its submodule $Ξ'$ associated to the projection $P$ whose trace is $2 ν$ , and on the tensor product $P Ecμ,ν ⊗ Ξ$ , and we discuss how the Yang-Mills theory of $D_{μ, ν}^{c}$ is different from that of the noncommutative tori. Linear Connections on Quantum Heisenberg Manifolds
Feb. 20, 2018 11am (MATH 350)	Number Theory Michael Fried (UC Irvine, visiting U Colorado) X I. Schur's 1921 Problem for a polyomial f with integer coefficients was to describe all such $f$ that mapped one-one on the integers mod $p$ for almost all primes $p$ . Example: $w → w5$ does if $p$ -1 is prime to 5. Schur's problem was solved in 1969 over all number fields $F$ . I will explain why the analogue for rational functions had more major consequences, by showing aspects of the solution for $f$ of two types: Of prime degree $ℓ$ ; and of degree $ℓ2$ . The principal point will be its relation with Serre's Open Image Theorem, and in particular, with the theory of Complex multiplication. In both cases, the problem is to describe which rational functions have a particular relation between the Galois groups of the splitting fields of $f(w)-z$ over $F(z)$ and over $Fcl(z)$ , $F^{c l}$ an algebraic closure of $F$ . This problem raises a question on realizing dihedral groups as Galois groups; in turn an unsolved mystery about hyperelliptic jacobians. Then, I will point out that the same question has a formulation to every finite group $G$ with its order divisible by a prime $ℓ$ for which it has no Z/ $ℓ$ quotient. That relates generalizing Serre's Open Image Theorem to the Regular Version of the Inverse Galois Problem, the topic of a book I am writing on while I am here. Galois groups and Complex multiplication
Feb. 20, 2018 12:10pm (MATH …	Kempner John Francis X The Ran space Ran(X) is the space of finite subsets of X, topologized so that points can collide. Ran spaces have been studied in diverse works from Borsuk–Ulam and Bott, to Beilinson–Drinfeld, Gaitsgory–Lurie and others. The alpha form of factorization homology takes as input a manifold or variety X together with a suitable algebraic coefficient system A, and it outputs the sheaf homology of Ran(X) with coefficients defined by A. Factorization homology simultaneously generalizes singular homology, Hochschild homology, and conformal blocks or observables in conformal field theory. I'll discuss applications of this alpha form of factorization homology in the study of mapping spaces in algebraic topology, bundles on algebraic curves, and perturbative quantum field theory. Factorization homology and the Ran space
Feb. 20, 2018 1pm (MATH 220)	Peter Mayr (CU Boulder) X We give a description of subdirect products of several factors in congruence permutable varieties using Fleischer's Lemma and higher commutators. As an application we will obtain a sufficient condition for such a subdirect product to be finitely generated. This is joint work N. Ruskuc. Subdirect products of multiple factors 1
Feb. 20, 2018 2pm (MATH 350)	Lie Theory Shawn Burkett (CU) X Since its introduction, supercharacter theory has been used to study a wide variety of problems. However, the structure of supercharacter theories themselves remains mysterious. In this talk, we will discuss supercharacter theories that are able to detect nilpotence, in some sense. By defining analogs of the center and commutator subgroup for a given supercharacter theory $S$ of $G$ , one may use these to define a coarser version of nilpotence, which we call $S$ - nilpotence. The supercharacter theories $S$ of a nilpotent group $G$ for which $G$ is $S$ -nilpotent will be classified, with particular emphasis on $p$ -groups. Then some potential applications and further generalizations will be discussed, as time permits. A generalized version of nilpotence arising from supercharacter theory
Feb. 20, 2018 3pm (MATH 350)	Topology John Francis (Northwestern University) X I'll also describe a beta form of factorization homology, where one replaces the Ran space with a moduli space of stratifications, designed to overcome certain strict limitations of the alpha form. The key notion is that of a constructible bundle: a K-point of this moduli space classifies a constructible bundle over K. One application is to proving the Cobordism Hypothesis, after Baez–Dolan, Costello, Hopkins–Lurie, and Lurie. This is joint work with David Ayala. Factorization homology and the moduli space of stratifications Sponsored by the Meyer Fund
Feb. 20, 2018 4pm (MATH 350)	Topology Greg Friedman (Texas Christian University) X Intersection homology was developed by Goresky and MacPherson to extend Poincare duality and related invariants such as signatures from manifold theory to "singular spaces" such as non-smooth algebraic varieties. We'll provide an introduction to these topics, including an overview of some early applications and more recent contributions by the speaker and collaborators. Stratified Spaces and Intersection Homology
Feb. 21, 2018 4:03pm (MATH 3…	Grad Student Seminar Leo Herr X Combinatorics and algebraic geometry seem to be an unlikely combination. However, various surprising encounters have inaugurated a field of research in between the two. We'll see what some of the fuss is about (most likely Stanley's g-theorem) after lightly dipping our toes into the background as into a cold bath. Expect to see fans, polytopes, and examples. How many sides are on a triangle?
Feb. 22, 2018 10am (Math 220)	Cliff Blakestad (University of Colorado) X An elliptic curve $E$ over the complex numbers can be expressed as $C$ modulo a lattice $Λ$ . The problem of finding meromorphic functions on $E$ becomes the same as looking for meromorphic functions on $C$ which are periodic with respect to $Λ$ . We will outline these ideas emphasizing the roles of the Weierstrass special functions $ζ$ , $σ$ , the Riemann $θ$ function and the Weierstrass $p$ -function. We will then introduce how this story ports to the $p$ -adic setting, due to Tate. Correction: The uniformization theory of elliptic curves seminar is Thursday, Feb 22nd
Feb. 22, 2018 11am (MATH 350)	Geometry/Analysis Tam Nguyen Phan (Max Planck) X Aspherical manifolds are manifolds that have contractible universal covers, or equivalently, those that have trivial higher homotopy groups. The homotopy types of such manifolds are determined by their fundamental groups. Examples include nonpositively curved manifolds (by Cartan-Hadamard), and in particular, the rich class of locally symmetric spaces of non compact type. However, it is in general not easy to construct aspherical manifolds. e.g. the obvious connect sum operation usually ruins asphericity. One way to construct aspherical manifolds is to use the "reflection group trick". I will describe how to construct examples of aspherical manifolds by applying the reflection group trick to the Borel-Serre compactifications of locally symmetric spaces. This will be an example-oriented talk and everything will be explained via examples. Examples of aspherical manifolds, the reflection group trick and Borel-Serre compactifications Sponsored by the Meyer Fund
Feb. 22, 2018 2pm (Math 350)	Functional Analysis Dana Williams (Dartmouth College) X An important way to construct examples of $C^{}$ -algebras is to form a groupoid $C^{}$ -algebra from a locally compact groupoid, in analogy to the classical group- $C^{*}$ -algebra construction. To do this, we need an analogue of a Haar measure called a “Haar system”. In contrast to the group case, there are a number of interesting open questions surrounding the existence and uniqueness of Haar systems. I’ll briefly discuss these and focus on the relation to the important notion of “groupoid equivalence”. In particular, I’ll talk about a recent result showing that if a groupoid is equivalent to one with a Haar system, then it also possesses a Haar system. Haar Systems on Equivalent Groupoids Sponsored by the Meyer Fund
Feb. 22, 2018 3pm (MATH 350)	Probability Koushik Ramachandran (Oklahoma State University) X A polynomial lemniscate is a curve in the complex plane defined by $z ∈ C : \|p(z)\| = t$ . Erdos, Herzog, and Piranian posed the extremal problem of determining the maximum length of a lemniscate $Λ = z ∈ C : \|p(z)\| = 1$ when p is a monic polynomial of degree n. In this talk, we will look at the length and topology of a random lemniscate whose defining polynomial has independent Gaussian coefficients. When the polynomial is sampled from the Kac ensemble we show that the expected length approaches a nonzero constant as $n → ∞$ . Concerning the connected components of a random lemniscate, we prove that the average number of them is asymptotically n and that there is a positive probability (independent of n) of a giant component. Based on joint work with Erik Lundberg. Random Lemniscates: Geometry and topology
Feb. 27, 2018 11am (MATH 350)	Number Theory Number Theorists of Boulder X This week, Daniel will give us a brief overview of what he's up to. Number Theory Lab Meeting
Feb. 27, 2018 12pm (MATH 350)	Geometry/Analysis Marcelo Disconzi (Vanderbilt) This is a Colloquium that should be of interest to anyone usually attending the seminar.
Feb. 27, 2018 12:10pm (Math …	Kempner Marcelo Disconzi (Vanderbilt) X We will discuss some recent results concerning mathematical aspects of relativistic fluids, focusing on the evolution problem for Einstein’s equations coupled to the equations of relativistic fluid dynamics. After introducing some background, we will present two alternative formulations of the relativistic Euler equations. The first one is inspired by work of H. Friedrich and employs a frame formalism to recast the Einstein-Euler system as a first order symmetric hyperbolic system. We will give a motivation behind this approach and establish an existence and uniqueness result. The second formulation reveals a remarkable null-structure of the relativistic Euler equations. Such structure is “hidden,” i.e., it is not apparent in the equation’s usual form, and has important consequences for the study of the nature of shock formation. We will then turn our attention to the problem of relativistic fluids with viscosity. The question of how to formulate viscous fluids within relativity theory goes back to Eckart in 1940 and remains largely open. After discussing the history of the problem and reasons for considering relativistic viscous fluids, we will present a recent formulation that provides the first example in the literature of a theory of relativistic fluids with viscosity that admits existence and uniqueness of solutions when coupled to Einstein’s equations, and satisfies further requirements intrinsic to relativistic theories. Some of the results I will discuss are joint with F. Bemfica, M. Czubak, J. Noronha, and J. Speck. Recent developments in the theory of relativistic fluids
Feb. 27, 2018 1pm (MATH 220)	Peter Mayr (CU Boulder) X We give a description of subdirect products of several factors in congruence permutable varieties using Fleischer's Lemma and higher commutators. As an application we will obtain a sufficient condition for such a subdirect product to be finitely generated. This is joint work N. Ruskuc. Subdirect products of multiple factors 2
Feb. 27, 2018 3pm (MATH 350)	Algebraic Geometry Ronny Wells (Jacobs University Bremen and University of Colorado) X Every Riemann surface admits nonconstant meromorphic functions, but this is not the case for higher dimensional complex manifolds, as Siegel pointed out with an example in 1949 (a complex two torus). Kodaira showed in 1964 that any compact complex surface with no nonconstant meromorphic functions must be a torus, a type of Hopf surface or a K3 surface. In this lecture I want to show that such surfaces with no nonconstant meromorphic functions are dense in the natural modulli spaces for these three classes of surfaces. Compact complex surfaces with no nonconstant meromorphic functions
Feb. 27, 2018 4pm (MATH 350)	Topology Agnes Beaudry (CU) X In this talk, I will give an introduction to equivariant stable homotopy theory and some of the tools used in this field. Some familiarity with the classical stable homotopy category will be assumed. An introduction to Equivariant Homotopy Theory
Feb. 28, 2018 4:03pm (MATH 3…	Grad Student Seminar Katharine Adamyk X Spend any amount of time around a topologist and you’re likely to hear the words “nice topological space”. It’s a certainly a handy phrase to exclude counterexamples that might break your proof, but what properties make a space “nice”? How “nice” does a space have to be to be a “nice space”? We will begin with some motivating examples of spaces that are definitely not nice, and then consider different ways of ruling out these pathologies. Eventually, we will introduce CW complexes and talk about their role in topology, especially homotopy theory. (The basics of homotopy will be defined.) If time permits, we’ll talk about what makes a "nice" category of spaces, but no categorical background knowledge is needed. How (Not) to Be Nice: Some Pathological Spaces and an Introduction to CW-Complexes
Feb. 28, 2018 5pm (Math 350)	MathClub Josh Grochow (CU Boulder: Department of Computer Science) X Although computational complexity was motivated by the rise of digital computers, it turns out to have deep relationships with both pure mathematics and the natural sciences. The computational complexity properties of a mathematical object - how computationally difficult it is to compute various functions on the object, or to answer various questions about it - can be as fundamental as its other properties, such as the measure of a set, the homology of a topological space, or the dimension of a ring or group. Computational complexity has also revealed unexpected interactions between seemingly disparate mathematical areas: for example, there's a precise sense in which questions of propositional logic, knot theory, systems of integer (Diophantine) equations, and graph coloring are all isomorphic. Computational complexity can also shed light on some of the central questions in mathematics, for example, about classification of various objects such as groups, topological spaces, or metrics on a given manifold. This talk will start from the beginning (what is computational complexity?) and take us on a brief tour of some of these beautiful and surprising connections. - And if you've seen NP-completeness before, it's not just that these problems are all NP-complete! The really is a much stronger notion of isomorphism between them. Computational Complexity and Mathematics
Mar. 01, 2018 11am (MATH 350)	Geometry/Analysis Nathaniel Eldredge (Northern Colorado) X I will present a recent result, joint with Maria Gordina and Laurent Saloff-Coste, showing that there is a uniform volume doubling constant for all left-invariant Riemannian geometries on the compact Lie group SU(2). This implies uniformity in a wide variety of functional inequalities for this family of geometries, including eigenvalue bounds for the Laplacian, heat kernel estimates, and more. Classical estimates based on curvature are not available in this setting, since there is no uniform lower bound on the Ricci curvatures of these metrics. A key idea in the proof is to study the size and shape of metric balls by a careful analysis of the Campbell--Baker--Hausdorff--Dynkin--Strichartz formula. We conjecture that the same uniform doubling result holds for every compact connected Lie group. A preprint is available at https://arxiv.org/abs/1708.03021 Uniform volume doubling for left-invariant geometries on SU(2) Sponsored by the Meyer Fund
Mar. 01, 2018 2pm (MATH 350)	Functional Analysis Karen Strung (Radboud Universiteit, Netherlands) X Results from the Elliott classification program can be used to translate theorems of optimal transport into calculations of the distance between unitary orbits of normal elements in well-behaved $C^{}$ -algebras. In particular, in certain simple Jiang-Su stable $C^{}$ -algebras with real rank zero and trivial $K1$ , the distance between full-spectrum unitaries can be computed in terms of spectral data. This talk is based on joint work with Bhishan Jacelon and Alessandro Vignati. Unitary Orbits via Transportation Theory Sponsored by the Meyer Fund
Mar. 06, 2018 11am (MATH 350)	Algebraic Geometry Luca Battistella (Imperial College) X Moduli spaces of stable maps to projective space are not equidimensional in higher genus. Work of J. Li, R. Vakil and A. Zinger led to a desingularization of the main component (the generic element of which represents a map from a smooth elliptic curve) and the introduction of reduced invariants for complete intersections. For threefolds these are related to ordinary Gromov-Witten invariants by the Li-Zinger's formula; they have a better enumerative meaning in the sense that they discard a degenerate contribution from rational curves. More recently M. Viscardi has introduced smaller compactifications by allowing maps from Smyth's singularities, e.g. cusps. In a joint work with F. Carocci and C. Manolache we show that reduced and cuspidal invariants coincide for the quintic threefold. We use p-fields and local equations in order to split the intrinsic cone, adapting techniques of H-L. Chang, Y. Hu and J. Li. g=1 reduced invariants of the quintic threefold from maps with cusps Sponsored by the Meyer Fund
Mar. 06, 2018 1pm (MATH 170)	Number Theory Alex Kontorovich (Rutgers) X We define a general class of sphere packings, and study the subclass of "superintegral" such. By connecting these to the theory of hyperbolic arithmetic reflection lattices, we prove that they only exist in finitely many dimensions, and in fact in finitely many commensurability classes, in principle allowing for a complete classification. Sphere Packings and Arithmetic - note unusual time and place!
Mar. 06, 2018 2pm (MATH 350)	Lie Theory Spencer Gerhardt (CSU) X Let $C1,…, Ce$ be conjugacy classes of the algebraic group $G=SLn(k)$ defined over a sufficiently large field $k$ . We consider the problem of determining necessary and sufficient conditions for the existence of $xi∈ Ci$ such that $⟨ x1,… ,xe⟩$ is Zariski dense in $G$ . Related questions are considered for other simple algebraic groups, and applications to generic stabilizers in linear representations of algebraic groups and random generation of finite groups of Lie type are discussed. Topological generation of algebraic groups and applications Sponsored by the Meyer Fund
Mar. 08, 2018 2pm (MATH 350)	Functional Analysis Leonard Huang (CU Boulder) X Given a $C^{}$ -dynamical system $(G,A,α)$ , it is well-known that the $ℂ$ -vector space $Cc(G,A)$ of compactly-supported continuous functions from $G$ to $A$ can be equipped with a convolution operation $⋆α$ and an involution operation $^{_{α}}$ . In order to form the crossed-product $C^{}$ -algebra $C(G,A,α)$ associated to $(G,A,α)$ , one then completes $Cc(G,A)$ with respect to the universal norm coming from all contractive $$ -homomorphisms from $(Cc(G,A),⋆α,α)$ to a $C^{}$ -algebra, where contractivity is with respect to the $L1$ -norm on $Cc(G,A)$ , for a given Haar measure on $G$ . One can then ask: Is contractivity obtainable for free for any $$ -homomorphism from $(Cc(G,A),⋆α,α)$ to a $C^{}$ -algebra? No counterexamples are currently known, and hardly anything in the literature has been written related to this question. In this talk, I will start the ball rolling by proving for any discrete twisted $C^{}$ -dynamical system $(G,A,α,ω)$ that a $$ -homomorphism from $(Cc(G,A),⋆α,ω,α,ω)$ to a $C^{}$ -algebra is automatically contractive with respect to the $L1$ -norm on $Cc(G,A)$ — for any positive-scalar multiple of the counting measure on $G$ . Automatic Continuity of -Representations for Discrete Twisted $C^{}$ -Dynamical Systems
Mar. 13, 2018 11am (MATH 350)	Number Theory Number Theory Lab X Carly will tell us about the Arizona Winter School. Number Theory Lab Meeting
Mar. 13, 2018 12:10pm (Math …	Kempner Dmitry Kaledin (Steklov Mathematics Institute and Independent University of Moscow) X Witt vectors were first introduced eighty years ago, but they still come up in different questions of commutative and homological algebra, algebraic geometry, and even algebraic topology. I will try to give a general introduction to this remarkable subject, and show both its classical parts and some recent discoveries. Witt vectors, commutative and non-commutative
Mar. 13, 2018 1pm (MATH 220)	Nik Ruskuc (St Andrews) X It is known that subdirect products of groups can be pretty wild even if the factors are reasonably tame. For instance, Baumslag and Roseblade showed that there are uncountably many non-isomorphic subdirect products of two copies of the free group F_2. However, this wild behaviour cannot occur for subdirect products of finitely generated abelian groups as a consequence of the fundamental theorem for finite abelian groups: they are all finitely generated, finitely presented, and there are only finitely many up to isomorphism. Finitely generated commutative semigroups often behave similarly to the abelian groups: e.g. they are finitely presented, have a solvable word problem, the subsemigroups of N (the natural numbers, the free monogenic semigroup) are essentially arithmetic progressions, etc. So we ask: How many subsemigroups (up to isomorphisms) does the direct product NxN (two copies of the semigroup of natural numbers) contain? This is joint work with my student Ashley Clayton. Subdirect products involving the semigroup of natural numbers Sponsored by the Meyer Fund
Mar. 13, 2018 2pm (MATH 350)	Lie Theory Nantel Bergeron (York University (Canada)) X I will define hypergraphs and explain a Hopf structure on them. Using orientations on hypergraphs we provide an explicit bijection between acyclic orientations on them and faces of hypergraphic polytopes. This allows us to obtain a geometric interpretation of the coefficients of the antipode map in this Hopf algebra. This interpretation differs from similar ones for a different Hopf structure on hypergraphs provided recently by Aguiar and Ardila. (Joint with C. Benedetti and J. Machacek). Hypergraphical polytope: Orientations and antipode Sponsored by the Meyer Fund
Mar. 13, 2018 3pm (Math 350)	Algebraic Geometry Dori Bejleri (Brown University) X A degree one del Pezzo surface is the blowup of P^2 at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we can construct a stable pair compactification of the moduli space of anti-canonically polarized degree one del Pezzo surfaces. The KSBA theory of stable pairs (X,D) is the natural extension to dimension 2 of the Deligne-Mumford-Knudsen theory of stable curves. I will discuss the construction of the space of interest as a limit of a space of weighted stable elliptic surface pairs and explain how it relates to some previous compactifications of the space of degree one del Pezzo surfaces. This is joint work with Kenny Ascher. Stable pair compactifications of the moduli space of degree one del Pezzo surfaces via elliptic fibrations Sponsored by the Meyer Fund
Mar. 13, 2018 4pm (MATH 350)	Algebraic Geometry Morgan Brown (University of Miami) X The essential skeleton is an invariant of a degeneration that appears in both Berkovich geometry and minimal model theory. I will show that for degenerations with a semistable model, the essential skeleton of a product of degenerations is the product of their skeleta. As an application, we are able to describe the homeomorphism type of some degenerations of hyperkähler varieties, both for a Hilbert scheme of degenerating K3 surfaces and for a Kummer variety associated to a degeneration of an abelian surface. This is joint work with Enrica Mazzon. The Skeleton of a Product of Degenerations
Mar. 14, 2018 4:03pm (MATH 3…	Grad Student Seminar Hanson Smith X In honor of pi day we will discuss where pi lives. Namely, we shall show that pi is neither rational nor algebraic. Along the way we will consider the history of transcendental numbers. Our journey will be motivated by problems, some solved and some not. The requisite background is a partial understanding of one’s own lack of understanding of the sets of numbers we generally like to play with. Transcending the irrationality of pi: Hilbert’s Seventh Problem
Mar. 14, 2018 5pm (Math 350)	MathClub Robin Deeley (CU Boulder) X Dynamical systems were introduce by Poincaré in his study of the behavior of the solar system. I will discuss a number of dynamical systems coming from linear algebra and calculus before moving to an introduction to shift spaces and their dynamics. The concept of sensitivity to initial conditions will be emphasized. Shift Dynamics
Mar. 15, 2018 2pm (MATH 350)	Functional Analysis Allan Yashinski (University of Hawaii at Manoa) X Wieler has shown that a certain natural class of hyperbolic dynamical systems, namely the class of irreducible Smale spaces whose local stable sets are totally disconnected, arises from a solenoid construction. We consider $C^{}$ -algebras associated to these Smale spaces, and use the stationary inverse limit structure of such a space to produce a stationary inductive limit structure for the $C^{}$ -algebra. The $C^{}$ -algebra in the inductive sequence is a Fell algebra with compact spectrum and trivial Dixmier-Douady invariant. We illustrate how our inductive limit can be used to compute the $K$ -theory of the Smale space $C^{}$ -algebra. This is a joint work with Robin Deeley. The Stable Algebra of a Wieler Solenoid: Inductive Limits and $K$ -Theory Sponsored by the Meyer Fund
Mar. 15, 2018 3pm (MATH 350)	Geometry/Analysis Jeanne Clelland (University of Colorado, Boulder) X This is a continuation of my talk from February 15. NOTE THE UNUSUAL TIME! Isometric embedding via strongly symmetric positive systems, Part 2 Sponsored by the Meyer Fund
Mar. 20, 2018 11am (MATH 350)	Number Theory Number Theory Lab X David will speak today. Number Theory Lab Meeting
Mar. 20, 2018 12:10pm (MATH …	Kempner Michael Kinyon (University of Denver) X The fact that the inner automorphism group of a group is isomorphic to the quotient of the group by its center leads to the following observation: the inner automorphism group is abelian if and only if the group itself is nilpotent of class at most 2. Loops are ``nonassociative groups’’, and as with most results in group theory, this observation does not extend to loops exactly as stated. The question as to how far the result does extend is of interest in loop theory because it reveals the interplay between loops and the permutation groups that act upon them. Work on this by various people over several years has led to what is now known as the AIM Conjecture (AIM = Abelian Inner Mappings), which I will state precisely in the talk. The conjecture remains open, but is known to hold in many important classes of loops. What makes all this of interest outside the seemingly esoteric world of nonassociative mathematics has been the approach to the problem. The conjecture can be formulated purely equationally and so it can be quite naturally attacked using automated deduction tools such as Prover9. This is where all of the recent success in establishing the conjecture in special cases has come from. In this talk, after a quick introduction to loops, I will talk about how automated deduction has been used in loop theory over the last couple of decades, and current work on the AIM Conjecture. No background in either loops or automated deduction is needed. Loops, Automated Deduction and the AIM Conjecture
Mar. 20, 2018 2pm (Math 350)	Lie Theory Lucas Gagnon (CU) X The finite subgroups of the special unitary group $SU(2)$ can be classified into two infinite families and three exceptional subgroups. The McKay correspondence uses representation theory to put these subgroups in bijection with simply laced affine Dynkin diagrams. For a finite subgroup $G$ of $SU(2)$ , the centralizer algebras of $G$ are the $ℂ G$ -module endomorphisms of the $k$ -fold tensor power of the defining representation, and the dimension of the $k$ th centralizer algebra of $G$ can be found by counting closed walks on the Dynkin diagram associated to $G$ . I’ll discuss how the combinatorics of each Dynkin diagram controls the structure of the corresponding family of centralizer algebras; in particular I will describe a basis for each algebra which can be derived uniformly from closed walks, and give a presentation for each algebra. Of interest may be that Temperley-Lieb algebras $TL(2)$ are the centralizers of $SU(2)$ , and subalgebras of each other centralizer. This is joint work with Tom Halverson. McKay Centralizer Algebras
Mar. 21, 2018 4:03pm (MATH 3…	Grad Student Seminar Athena Sparks X The first square is never a mine... Explosion!!! The Minesweeper Consistency Problem (MCP) poses the following question: given a partially completed Minesweeper board, is there a way to place mines that satisfies the board? We will delve into the MCP and discuss what it means to be NP-complete along with a standard approach for showing a problem is NP-complete. Some familiarity with the game Minesweeper would be useful (you can play it for free online). No other prerequisite knowledge needed; an overview of the relevant topics in computational complexity will be given. Minesweeper is NP-Complete
Mar. 22, 2018 11am (MATH 350)	Geometry/Analysis Mark Fels (Utah State University) X See abstract at http://euclid.colorado.edu/~jnc/Fels-abstract.pdf Backlund Transformations and Darboux Integrability Sponsored by the Meyer Fund

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