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Events happening on November 28, 2017

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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 350)	Lie Theory Jon Lamar (CU) X TBD Two Hopf subalgebras from restriction supercharacter theories
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 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. 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)	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

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