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Events for September 2012 – January 2013

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Sep. 04, 2012 2pm (MATH 350)	Lie Theory Matt Douglass (U. North Texas) X In their proof of the Deligne-Langlands conjecture, Kazhdan and Lusztig construct an explicit isomorphism between the equivariant K-group of the Steinberg variety of a complex, reductive, algebraic group G and the extended affine Hecke algebra of the Weyl group of G. Later, Ostrik used their construction to define a "Kazhdan-Lusztig" type basis for the equivariant K-group of the nilpotent cone in the Lie algebra of G. In these talks, I'll explain known and conjectural connections between (1) Ostrik's construction, (2) the lowest two-sided cell ideal and some special Kazhdan-Lusztig basis elements in the extended, affine Hecke algebra, and (3) G-equivariant vector bundles supported on {0} in the nilpotent cone. In order to keep the talks as accessible as possible, I'll talk only about the group SL_2(C). Equivariant vector bundles supported on a point and Kazhdan-Lusztig theory I
Sep. 11, 2012 2pm (MATH 350)	Lie Theory Matt Douglass (U. North Texas) X In their proof of the Deligne-Langlands conjecture, Kazhdan and Lusztig construct an explicit isomorphism between the equivariant K-group of the Steinberg variety of a complex, reductive, algebraic group G and the extended affine Hecke algebra of the Weyl group of G. Later, Ostrik used their construction to define a "Kazhdan-Lusztig" type basis for the equivariant K-group of the nilpotent cone in the Lie algebra of G. In these talks, I'll explain known and conjectural connections between (1) Ostrik's construction, (2) the lowest two-sided cell ideal and some special Kazhdan-Lusztig basis elements in the extended, affine Hecke algebra, and (3) G-equivariant vector bundles supported on {0} in the nilpotent cone. In order to keep the talks as accessible as possible, I'll talk only about the group SL_2(C). Equivariant vector bundles supported on a point and Kazhdan-Lusztig theory II
Sep. 18, 2012 2pm (MATH 350)	Lie Theory Nicolas Proudfoot (Oregon) Toric and hypertoric varieties
Sep. 25, 2012 2pm (MATH 350)	Lie Theory Sarah Kitchen (Freiburg) X This is a report on work in progress with Achar, McGerty, and Nevins. Koszul duality is a type of derived equivalence that appears in representation theory. It is expected to relate various categories of representations for Langlands dual pairs of Lie algebras, and underlie observed examples of duality in symplectic geometry. For a given abelian category, the derived equivalence does not occur at the level of the derived category of our abelian category, but for some graded version of the category. Obtaining a graded version and verifying that we have given one suitable for Koszul duality are not usually easy processes. In earlier work with Achar, we produced a grading on perverse sheaves with a fixed stratification from a candidate subcategory of mixed Hodge modules. In this talk, I will describe a more general context in which we can check if a mixed category is a grading suitable for Koszul duality. Obtaining Koszul Gradings Sponsored by the Meyer Fund
Oct. 02, 2012 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X Recent innovations in supercharacter theory have allowed us to get our hands on the representations of the finite groups of unipotent uppertriangular matrices. Instead of a prohibitively hard problem (the usual representation theory of these groups), we instead obtain a theory whose foundation is the combinatorics of set partitions. I will give a brief introduction to supercharacters and give our favorite example in the case of unipotent groups. The remainder of the talk will explore some of the set partition combinatorics that pops out of the theory. This is joint work with D. Bragg. The combinatorics of unipotent groups
Oct. 03, 2012 4:15pm (MATH 3…	Grad Student Seminar Nathan Wakefield X Let \((g_i)_{i \geq 1}\) be a sequence of Chebyshev polynomials, each with degree at least two, and define \((f_i)_{i \geq 1}\) by the following recursion: \(f_1=g_1\), \(f_n=g_n\circ f_{n-1}\) for \(n \geq2\). Choose \(\alpha \in \mathbb{Q}\) such that \(\{ g_1^n (\alpha) : n \geq 1 \}\) is an infinite set. The main result of this talk is as follows: If \(f_n(\alpha)=\frac{A_n}{B_n}\) is written in lowest terms, then for all but finitely many \(n>0\) the numerator \(A_n\) has a primitive divisor; that is, there is a prime \(p\) which divides \(A_n\) but does not divide \(A_i\) for any \(i \lt n\). I will also talk about some further directions of my future work at the end. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials
Oct. 10, 2012 4:15pm (MATH 3…	Grad Student Seminar Cliff Blakestad X We will introduce p-adic numbers and some of their basic properties including Hensel's lemma. We will then explore consequences of these properties for polynomial factorization via Newton polygons. p-Adics, Polynomials, and Polygons
Oct. 16, 2012 2pm (MATH 350)	Lie Theory Rahbar Virk (CU) X I will explain how some very elementary abstract nonsense with monoidal categories can be leveraged to produce some highly non-trivial symmetries for categories of representations of the quantum group U_q(gl_n) (uniformly for q generic and/or a root of 1). To keep things simple I will explain how the ideas work for the Lie algebra gl_n and then indicate the modifications necessary to get the more interesting quantum version. This is essentially a summary of the main ideas of an old paper of mine (that I never popularized or gave talks on) available at http://arxiv.org/abs/1006.0999. Silly games with tensor categories
Oct. 23, 2012 2pm (MATH 350)	Lie Theory Robert Marsh (Leeds) X Joint work with Michael Barot. We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system. Reflection group presentations arising from cluster algebras
Oct. 24, 2012 4:15pm (MATH 3…	Grad Student Seminar Bryce Chriestenson X In this talk I will describe the construction of the complex of de Rham currents. This complex is dual to the complex of compactly supported differential forms. Time permitting I will explain how to generalize this construction to a smooth manifold. I will also try to describe how currents are related to non commutative geometry. de Rham currents on an open subset of ℝⁿ
Oct. 30, 2012 2pm (MATH 350)	Lie Theory Justin Keller (CU) X We define the notion of a supercharacter theory (SCT) for an arbitrary finite group and view each SCT as a coarsening or approximation of the usual character theory of the group. Generalizing the relationship between the center of the group algebra and its character theory, we describe how certain subalgebras (Schur rings) of the group algebra play an analogous role for SCTs. In this way, Schur rings can be viewed as an approximate center for the group algebra, and as a tool for finding and understanding SCTs. The goal will be to provide a foundation and motivating example (the group algebra) for the following talk, which will introduce the notion of a Schur ring in the setting of Hopf algebras. Supercharacter Theories and Schur Rings I
Oct. 31, 2012 4:15pm (MATH 3…	Grad Student Seminar Boramey Chhay X Fractals are sets which have fractional Hausdorff dimension. We will construct some of these sets and see how n-dimensional Hausdorff measure relates to Lebesgue measure. Time permitting we will discus probability. Fractal Sets in ℝⁿ
Nov. 06, 2012 2pm (MATH 350)	Lie Theory Justin Keller (CU) X We define the notion of a supercharacter theory (SCT) for an arbitrary finite group and view each SCT as a coarsening or approximation of the usual character theory of the group. Generalizing the relationship between the center of the group algebra and its character theory, we describe how certain subalgebras (Schur rings) of the group algebra play an analogous role for SCTs. In this way, Schur rings can be viewed as an approximate center for the group algebra, and as a tool for finding and understanding SCTs. The goal will be to provide a foundation and motivating example (the group algebra) for the following talk, which will introduce the notion of a Schur ring in the setting of Hopf algebras. Supercharacter Theories and Schur Rings II
Nov. 07, 2012 4:15pm (MATH 3…	Grad Student Seminar Trubee Davison X Given a complete and separable metric space (X,d), and a finite family, S, of contractions on X, one can construct a compact subset K of X that is invariant under S. This set K is called the fractal set, or attractor set, associated to S. In this talk, we will discuss the construction of the fractal set. As a somewhat loose, but fun, application of fractals, we will then look at one example of the 'fractal-type' phrasing that occurs in many of Bach's compositions. The speaker will conclude with a performance on the violin of a Bach composition. Constructing the Fractal Set with an Application to the Music of Bach
Nov. 13, 2012 2pm (MATH 350)	Lie Theory Scott Andrews (CU) X Classifying the irreducible characters of the group of unipotent upper triangular matrices over a finite field is a provably difficult problem. We can instead consider a coarsening of the usual character theory, which we will refer to as a supercharacter theory. I will construct a supercharacter theory of the group of unipotent upper triangular matrices and look at analogous constructions in other types. Supercharacter Theories of Unipotent Groups
Nov. 27, 2012 2pm (MATH 350)	Lie Theory Aaron Lauda (USC) X It is a well understood story that one can extract link invariants from quantum deformations of the universal enveloping algebra of simple Lie algebra called quantum groups. These invariants are called Reshetikhin-Turaev invariants and the famous Jones polynomial is the simplest example. Kauffman showed that the Jones polynomial could be described very simply by replacing crossings in a knot diagrams by various smoothings. In this talk we will explain Cautis-Kamnitzer-Licata's simple new approach to understanding these invariants using basic representation theory and the quantum Weyl group action based on the representation theoretic notion called skew-Howe duality. Even the graphical (or skein theory) description of these invariants can be recovered in an elementary way from this data. The advantage of this approach is that it suggests a `categorification' where knot homology theories arise in an elementary way from the structure of categorified quantum groups. Why representation theorists already know about quantum knot invariants Sponsored by the Meyer Fund
Nov. 28, 2012 4:15pm (MATH 3…	Grad Student Seminar Ryan Rosenbaum X A transcendental number is a complex number which is not the root of a polynomial in ℚ[x]. We will give an accessible introduction to the theory of real transcendental numbers. We will touch on basic properties of transcendental numbers (like they actually exist), Liouville's theorem, irrationality measure and Roth's theorem. Time permitting we will include a construction of transcendental numbers relating to the undecidability of the halting problem over Turing machines. An Introduction to Transcendental Number Theory
Dec. 11, 2012 2pm (MATH 350)	Lie Theory Vivek Dhand (MSU) X Young's partition lattice L(m,n) consists of unordered partitions having m parts where each part is at most n, ordered by inclusion of the corresponding Young diagrams. Many years ago, Richard Stanley conjectured that this poset has a decomposition into saturated rank-symmetric chains. Despite many efforts, this conjecture has only been proved when min(m,n) is less than or equal to 4. In this talk, we decompose L(m,n) into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, and we show that resulting subposets have an elementary raising and lowering algorithm. As a corollary, we obtain a symmetric chain decomposition for the subposet of L(m,n) consisting of "sufficiently generic" partitions. Tropical Decomposition of Young's Partition Lattice
Jan. 30, 2013 4:15pm (MATH 3…	Grad Student Seminar Nathan Wakefield X Heights are an important component of the number theorist's toolbox. This presentation will describe the basic elements of heights, and how they play a role in many major areas of number theory; ranging from elliptic curves to dynamical systems. The canonical height is a particularly important tool in arithmetic dynamics. In this talk, we will discuss recent developments in the area of canonical heights, and explain an extension to generalized iteration. We will conclude with an open conjecture and discuss how heights may contribute to its solution. Canonical Heights in Generalized Iterations

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