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Events for June – October 2013

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Sep. 03, 2013 3pm (CSU: Webe…	Algebraic Geometry Steffen Marcus (Utah) First hour: NAGlab Intersection Multiplicities via. regular chains Second hour: FRAGMENT Current results in logarithmic Gromov-Witten theory.
Sep. 12, 2013 3pm (CSU: Webe…	Algebraic Geometry Felipe Voloch (UT Austin) Local-Global principles for the moduli space of abelian varieties and Galois representations
Sep. 19, 2013 3pm (CSU: Webe…	Algebraic Geometry Agnes Beaudry (U. Chicago) Formal Group Laws, Elliptic Curves and Modular Forms in Stable Homotopy Theory
Sep. 24, 2013 3pm (Math 350)	Lie Theory Scott Andrews (CU) X Mackey's "method of little groups" describes the irreducible characters of a semidirect product G = HN when the normal subgroup N is abelian. I will introduce the method and give examples of interesting semidirect products of this form. Mackey's method can be used to construct supercharacter theories of G from supercharacter theories of H; I will give a few examples of know supercharacter theories that can be constructed using this method. Mackey's Method of Little Groups and Applications to Supercharacter Theories
Sep. 26, 2013 3pm (CSU: Webe…	Algebraic Geometry Virgil Pierce (UTPA) Integrable Systems for Map Enumeration
Oct. 01, 2013 3pm (Math 350)	Lie Theory Justin Keller (CU) X From the point of view of Frobenius algebras, we discuss how group actions can be used to coarsen the character theory of a finite dimensional semisimple Hopf algebra, when the representation ring is commutative. In particular, we show that a supercharacter table with integer entries can be constructed uniquely in such a way that it partitions the irreducible characters more finely than any other theory with integer values. Group actions and supercharacter theories
Oct. 01, 2013 3pm (MATH 350)	Algebraic Geometry Paul Johnson (CSU) Topology and combinatorics of Hilbert schemes of points on orbifolds
Oct. 03, 2013 3pm (CSU: Webe…	Algebraic Geometry Salim Ali Altug (Columbia) Beyond Endoscopy via the Trace Formula
Oct. 08, 2013 3pm (MATH 350)	Algebraic Geometry David Wayne (Boulder) X Recent progress in the field of algebraic K-theory has made it possible to compute the K-theory of certain rings from the cyclic homology and certain versions of their cdh-cohomology. The cdh-cohomology groups are constructed specifically so that they behave well with the methods of resolution of singularities. The idea then will be to write out our various computations on singular schemes, such as K-theory computations, in terms of computations on smooth schemes. Along with some historical context, we will present an overview of the cdh-topology and the construction of the cdh-cohomology groups along with their important properties. In particular, we will use the various versions of cdh-cohomology to develop a framework that makes it possible to compute some K-theory and cyclic homology groups of singular schemes. Afterwards, we will present some recent applications of this computational framework as well as some new results. K-theory and cdh-cohomology: a computational framework
Oct. 15, 2013 3pm (Math 350)	Lie Theory Ryan Rosenbaum (CU) X The Robinson-Schensted-Knuth (RSK) correspondence is a bijection between finitely supported countably infinite matrices and pairs of semi-standard Young tableaux with many applications to combinatorics and symmetric functions. Berenstein and Kirillov obtained explicit tropical formulas for the RSK bijection, and also considered geometric lifts of these formulas. Recently, O'Connell and other showed that under the geometric RSK (gRSK) correspondence the Schur functions are replaced by analytic objects called Whittaker functions which are ubiquitous in analytic number theory and physics. They also used gRSK to "lift" the Cauchy-Littlewood identity, which gave an alternative proof of a formula conjectured by Bump and proven by Stade. We will discuss RSK, its connection to the classical Cauchy-Littlewood identity, and the formulas of Berenstein and Kirillov. We will then explain how to obtain the analog of the Cauchy-Littlewood identity from gRSK. Time permitting, we will also discuss how the classical Cauchy-Littlewood shows up in number theory. The geometric Robinson-Schensted-Knuth correspondence and Whittaker functions I
Sep. 03, 2013 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X The Bruhat decomposition -- which factors group elements into unipotent, torus, and Weyl group components -- is a fundamental decomposition for groups of Lie type (such as the general linear group). Traditionally, it has been a model for classifying the representations of these groups. However, these methods have also been incomplete, extremely technical, or not open to explicit constructions. This talk reviews some of these methods, discusses some of their limitations, and poses several questions suggested by our recent better understanding of the representation theory of unipotent groups. The Bruhat decomposition of representations: Is there a reduced row echelon form?
Sep. 03, 2013 2pm (Math 350)	Lie Theory Nat Thiem (CU) X The Bruhat decomposition -- which factors group elements into unipotent, torus, and Weyl group components -- is a fundamental decomposition for groups of Lie type (such as the general linear group). Traditionally, it has been a model for classifying the representations of these groups. However, these methods have also been incomplete, extremely technical, or not open to explicit constructions. This talk reviews some of these methods, discusses some of their limitations, and poses several questions suggested by our recent better understanding of the representation theory of unipotent groups. The Bruhat decomposition of representations: Is there a reduced row echelon form?
Sep. 10, 2013 1pm (Math 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) Infinite structures with large automorphism groups
Sep. 10, 2013 2pm (MATH 220)	Keith Kearnes, CU X I will talk about the conjecture that a variety of algebras with a finite residual bound is finitely axiomatizable. Park's Conjecture
Sep. 10, 2013 2pm (MATH 350)	Lie Theory Richard Green (CU) X This talk will look at the Kazhdan--Lusztig basis of the Hecke algebra of type A from two points of view. From one point of view the basis is an example of an IC basis, or "canonical basis". It often happens that IC bases have remarkable properties, and it may be easy to prove that IC bases exist, but very hard to construct them explicitly. The other point of view regards the Kazhdan--Lusztig basis as an example of a cellular basis, which is very useful from the point of view of representation theory. IC Bases and Cellular Algebras I
Sep. 10, 2013 3pm (MATH 350)	PDE/Analysis Alejandro Sarria A generalized sign-changing Liouville equation
Sep. 17, 2013 1pm (MATH 220)	Grad Algebra/Logic Julie Linman (CU Boulder) Reducts of Ramsey Structures, Part 1
Sep. 17, 2013 2pm (Math 350)	Lie Theory Richard Green (CU) X This talk will look at the Kazhdan--Lusztig basis of the Hecke algebra of type A from two points of view. From one point of view the basis is an example of an IC basis, or "canonical basis". It often happens that IC bases have remarkable properties, and it may be easy to prove that IC bases exist, but very hard to construct them explicitly. The other point of view regards the Kazhdan--Lusztig basis as an example of a cellular basis, which is very useful from the point of view of representation theory. IC Bases and Cellular Algebras II
Sep. 17, 2013 3pm (MATH 350)	PDE/Analysis Stephen Preston The basics of shape space (part 1)
Sep. 24, 2013 1pm (MATH 220)	Grad Algebra/Logic Julie Linman (CU Boulder) Reducts of Ramsey Structures, Part 2
Sep. 24, 2013 2pm (MATH 220)	Keith Kearnes, CU X Alden Pixley asked in 1984 whether a congruence distributive variety in a finite language can have infinitely many subdirectly irreducible algebras, all finite. I will talk about the negative answer to this question. The Pixley problem
Sep. 24, 2013 3pm (MATH 350)	PDE/Analysis Stephen Preston The basics of shape space (part 2)
Sep. 30, 2013 4pm (Math 350)	Kempner Daniel Stroock (MIT) Applications of Cramer’s formula to probability theory
Oct. 01, 2013 1pm (MATH 220)	Grad Algebra/Logic Julie Linman (CU Boulder) Reducts of Ramsey Structures, Part 3
Oct. 01, 2013 1pm (MATH 350)	Number Theory Nathan Wakefield X Consider a function $f(n):N->N$ which measures the "growth" of the natural numbers in the following sense: for the sequence {1,2,3,....}, we define $f(n)$ by $f(n):=max{k in N: (n!)^2 \geq(n+k)!}$. This growth function has interesting asymptotic properties; we prove that when $f$ is defined on the sequence of positive integers, {1,2,3,...}, we have $\lim\limits_{n-> \infty}\frac{f(n)}{n}=1$. This idea can be generalized to any increasing sequence of integers. Specifically, we study this function defined on the sequence of prime numbers by considering the factorial like product of the sequence, and study the value of the associated limit. Comparing the growth of the prime numbers to that of the natural numbers
Oct. 01, 2013 2pm (MATH 220)	Natasha Dobrinen (DU) X The Tukey order on ultrafilters is a weakening of the well-studied Rudin-Keisler order, and the exact relationship between them is a question of interest. In a second vein, Isbell showed that there is a maximum Tukey type among ultrafilters and asked whether there are others. These two questions are the main guiding forces of the current research. In this talk, we present highlights of recent work of Blass, Dobrinen, Mijares, Milovich, Raghavan, Todorcevic, and Trujillo (in various combinations for various papers). Further information about results mentioned in this talk can be found in a recent survey article by the speaker. Survey on the structure of the Tukey theory of ultrafilters
Oct. 01, 2013 3pm (MATH 350)	PDE/Analysis Jeanne Clelland X The method of equivalence was introduced by Elie Cartan in 1908 as a procedure for finding invariants of geometric structures under the actions of pseudogroups, which most commonly appear as subgroups of the diffeomorphism group of a manifold preserving some underlying geometric structure. The method was further developed and systematized over the course of the 20th century, and it has been applied in a great variety of contexts. After briefly demonstrating Cartan’s method in the familiar context of Riemannian geometry, we will describe applications of the method of equivalence to some geometric structures related to control theory: 1. Sub-Riemannian geometry (Hughen, Moseley) and sub-Finsler geometry (C., Moseley, and Wilkens), both of which involve metrics which are defined only on a distribution on a manifold (i.e., on a subbundle of the tangent bundle of the manifold). Such metrics appear naturally in the context of optimal control of "driftless" kinematic systems. 2. Affine distributions (C., Moseley, and Wilkens), which appear naturally in the contexts of kinematic control systems with drift and dynamic control systems. 3. Dynamic equivalence of control systems (Stackpole). Dynamic equivalence generalizes the more thoroughly understood notion of static equivalence for control systems, which is addressed by the methods mentioned above. In his Ph.D. thesis, Stackpole shows how the method of equivalence may be used to address this more general notion of equivalence –- which is defined in terms of submersions rather than diffeomorphisms –- by extending to infinite prolongations. Equivalence of geometric structures in control theory via moving frames (part 1)
Oct. 08, 2013 1pm (MATH 220)	Grad Algebra/Logic Scott Andrews (CU Boulder) X Supercharacters and superclasses are coarser versions of the irreducible characters and conjugacy classes of finite groups. I will discuss a supercharacter theory of the group of unipotent upper triangular matrices over a finite field. Classifying the conjugacy classes of this group is a provably 'wild' problem. By considering the orbits of a group action, conjugacy classes and characters can be clumped in a compatible manner such that the resulting superclasses and supercharacters are easily indexed. I will present this construction, as well as an analogous one for the unipotent orthogonal, symplectic and unitary groups. Supercharacter Theories of Unipotent Groups Constructed Via Group Actions, Part 1
Oct. 08, 2013 2pm (MATH 220)	Agnes Szendrei (CU) X Let A be a finite algebra in a residually small variety with a cube term. I will discuss sufficient conditions which ensure that A is dualizable. Dualizable algebras
Oct. 08, 2013 2pm (MATH 350)	Lie Theory Justin Keller (CU) X We view the group algebra of a finite group as a semisimple Hopf algebra and look at three different types of subalgebra, each of which has interesting duality properties. We focus on Schur rings, and their relationship to the representation theory of the larger algebra. Hopf algebras, Frobenius algebras, and Schur rings
Oct. 14, 2013 4pm (Math 350)	Kempner Raymond Wells (Jacobs University and University of Colorado) The origins of complex geometry in the 19th century
Oct. 15, 2013 1pm (MATH 220)	Grad Algebra/Logic Scott Andrews (CU Boulder) Supercharacter Theories of Unipotent Groups Constructed Via Group Actions, Part 2
Oct. 17, 2013 3pm (MATH 350)	Probability Sonin (UNC Charlotte) X Mathematical models in the population dynamics are dealing with the special class of the infinite dimensional Markov processes. Their phase space is a set of all locally finite configurations of the points (particles) in. Dynamics of such configurations includes the birth and death of particles and their random motion (diffusion, jumps etc.). The goal of the theory is to give the conditions of the ergodicity of such systems and explain several well-known empirial facts, for instance, the high level of mom-uniformity in the spatial distribution of the particles ("patches"). The talk will contain a review of the recent results in this area and the open problems. Two Results in Probability Theory
Oct. 22, 2013 1pm (MATH 220)	Grad Algebra/Logic Clifford Bridges X We say a finite group G is realized as a Galois group over a field K if there is a finite Galois extension L/K such that G is isomorphic to Gal(L/K). We will develop a method of checking whether a given group is realized over a given field. In particular, we will show that A_n is realized over Q. A Method to Realize Groups as Galois groups: Rigidity in A_n, Part 2
Oct. 22, 2013 1pm (MATH 350)	Number Theory Ryan Rosenbaum X Tate included in his thesis an analog of Poisson summation on the adeles which he referred to as "Riemann-Roch" for number fields. Our goal is to explain Tate's justification for using this name. We will briefly review necessary properties of local and global fields along with the Fourier transform. We will then introduce the notion of the divisor group for a function field and show that Tate's "Riemann-Roch" implies a more familiar statement of theorem in this context. Riemann-Roch for function fields via Fourier analysis
Oct. 22, 2013 2pm (MATH 350)	Lie Theory Ryan Rosenbaum (CU) X The Robinson-Schensted-Knuth (RSK) correspondence is a bijection between finitely supported countably infinite matrices and pairs of semi-standard Young tableaux with many applications to combinatorics and symmetric functions. Berenstein and Kirillov obtained explicit tropical formulas for the RSK bijection, and also considered geometric lifts of these formulas. Recently, O'Connell and other showed that under the geometric RSK (gRSK) correspondence the Schur functions are replaced by analytic objects called Whittaker functions which are ubiquitous in analytic number theory and physics. They also used gRSK to "lift" the Cauchy-Littlewood identity, which gave an alternative proof of a formula conjectured by Bump and proven by Stade. We will discuss RSK, its connection to the classical Cauchy-Littlewood identity, and the formulas of Berenstein and Kirillov. We will then explain how to obtain the analog of the Cauchy-Littlewood identity from gRSK. Time permitting, we will also discuss how the classical Cauchy-Littlewood shows up in number theory. The geometric Robinson-Schensted-Knuth correspondence and Whittaker functions II
Oct. 28, 2013 10am (MATH 350)	Ulam Thomas Ivey X At our next meeting we'll take some time to discuss what the focus of the seminar should be for the balance of the semester. For folks who were attending the seminar earlier but stopped, or for folks who have been meaning to start, this would be a good opportunity to get involved and have some input. If time permits and there is interest, I'll speak about Darboux integrability; no EDS background will be assumed. Re-organizational Meeting
Oct. 29, 2013 1pm (MATH 350)	Number Theory Beth Malmskog X For m in the natural numbers, let $S_m$ be the Suzuki curve defined over the finite field of size $2^{2m+1}$. Because of the large number of rational points relative to their genus, the Suzuki curves provide good examples of Goppa codes. The automorphism group of $S_m$ is the Suzuki group $Sz(2m+1)$. However, the structure of the Jacobian of $S_m$ has not been determined. The a-number is a finer invariant than the p-rank of the isomorphism class of the 2-torsion group scheme. The a-number also places constraints on the decomposition of the Jacobian into indecomposable varieties. In this talk, I will discuss joint work with Holley Friedlander, Derek Garton, Rachel Pries, and Colin Weir in which which we computed a closed formula for the a-number of $S_m$ using the action of the Cartier operator on $H_0$, and continuing work with Pries and Weir to determine the full Eckdahl-Oort type of the curves. The a-numbers and Eckdahl-Oort types of Jacobians of Suzuki Curves
Oct. 29, 2013 1pm (MATH 220)	Grad Algebra/Logic Clifford Bridges (CU Boulder) A Method to Realize Groups as Galois Groups: Rigidity in A_n, Part 2
Oct. 29, 2013 2pm (MATH 350)	Lie Theory Richard Green (CU) X The numbers game , due to Mozes, is something close to a type of chip-firing game on a graph. It arises naturally in the context of Coxeter groups. I will discuss some of the combinatorial features of this game, as well as its points of contact with heaps and minuscule representations. Some Combinatorial Aspects of the Numbers Game
Oct. 29, 2013 3pm (MATH 350)	PDE/Analysis Boramey Chhay X We will discuss certain geometric properties such as curvature (sectional, scalar, Ricci), which lead to the characterization of unimodular 3 dimensional Lie groups. Then we will move on to the group of diffeomorphisms on the circle and hopefully will be able to do some computations and talk about the Virasoro-Bott group. Riemannian Geometry of a Lie Group
Oct. 31, 2013 3pm (CSU: Webe…	Algebraic Geometry David Swinarski (Fordham University) X Ribbons are double structures on $P1$ that arise as limits of canonical curves. They were first studied under this name in the 1990s by Bayer, Eisenbud, and Fong in connection with Green's Conjecture (now Voisin's Theorem). They also appear in Alper, Fedorchuk, and Smyth's recent proof of finite Hilbert stability for canonical curves. I will discuss some Groebner basis techniques that may be used to study degenerations of ribbons and discuss applications to computation of state polytopes. Groebner techniques for ribbons

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