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Events for October 2013 – February 2014

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Oct. 01, 2013 12:17am (Math …	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 12:17am (MATH …	Algebraic Geometry Paul Johnson (CSU) Topology and combinatorics of Hilbert schemes of points on orbifolds
Oct. 03, 2013 12:17am (CSU: …	Algebraic Geometry Salim Ali Altug (Columbia) Beyond Endoscopy via the Trace Formula
Oct. 08, 2013 12:17am (MATH …	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 12:17am (Math …	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
Nov. 05, 2013 12:17am (Math …	Lie Theory Erica Shannon (CU) X The simple Lie algebra d_n has three minuscule representations: the natural representation and the two spin representations. I'll explain how to construct the Lie algebra d_n and how to use actions on polytopes to build these three representations. This talk will be as concrete as possible given that the smallest example is when n=4. No background knowledge of Lie algebras or representation theory will be assumed. Minuscule Representations of d_n via Polytopes
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
Nov. 05, 2013 10am (MATH 350)	Ulam Thomas Ivey X I'll give a broad introduction to the subject of integrablesystems of PDE. While there is no clear-cut definition of integrability, the most well-known integrable equations share several common features which I will review: soliton solutions, Lax pairs, Backlund transformations, and bi-Hamiltonian structures leading to infinitely many conservation laws and commuting flows. Integrable PDE, part 1
Nov. 05, 2013 1pm (MATH 220)	Grad Algebra/Logic Clifford Bridges (CU Boulder) A Method to Realize Groups as Galois groups: Rigidity in A_n, Part 3
Nov. 05, 2013 1pm (MATH 350)	Number Theory Stefan Erickson X A fundamental question in number theory is enumerating number fields and function fields with discriminant up to some bound. While the situation for quadratic fields is fairly straightforward, cubic fields pose a greater challenge. After a brief introduction to zeta functions and function fields, we will describe the techniques behind enumerating cubic function fields with a given quadratic resolvent, building on the work of Cohen-Morra for cubic number fields. This is joint work with Colin Weir (Simon Frasier) and Ravi Donepudi (Colorado College). Enumerating Cubic Function Fields
Nov. 05, 2013 3pm (Math 350)	Algebraic Geometry Anton Dzhamay (University of Northern Colorado) X The goal of this talk is to describe some interesting geometry that underlines a special class of discrete integrable systems known as discrete Painlevé equations. In the first hour we'll consider some concrete examples that we then use as a motivation for the general introduction into the Sakai's theory of classification of discrete Painlevé equations using the Okamoto surface of the equation. In the second hour we'll describe various sources of such equations and then show how the underlying geometry can help us relate different equations of the same type, up to finding very explicit change of coordinates that transforms one equation into the other. On the geometry of discrete Painlevé equations
Nov. 05, 2013 3pm (MATH 350)	PDE/Analysis Taylor Klotz (University of Colorado at Boulder) X The Ricci Flow has become a famous topic for its role in the solution of the Poincare conjecture which is a special case of the Geometrization conjecture. We will begin with the statement of these famous conjectures, briefly discuss the relationship to the Ricci Flow, and then present some special solutions. Ideas of Perelman's work on the topic may be mentioned but specific details will not be covered. Some Basics of Ricci Flow
Nov. 11, 2013 2pm (MATH 220)	H. Peter Gumm (Philipps-University, Marburg) Modal logics, coalgebraicallly
Nov. 11, 2013 3pm (Math 350)	Algebraic Geometry Anton Dzhamay (UNCO) X Abstract: The goal of this talk is to describe some interesting geometry that underlines a special class of discrete integrable systems known as discrete Painlevé equations. In the first hour we'll consider some concrete examples that we then use as a motivation for the general introduction into the Sakai's theory of classification of discrete Painlevé equations using the Okamoto surface of the equation. In the second hour we'll describe various sources of such equations and then show how the underlying geometry can help us relate different equations of the same type, up to finding very explicit change of coordinates that transforms one equation into the other. On the Geometry of Discrete Painlevé Equations
Nov. 12, 2013 10am (MATH 350)	Ulam Pearce Washabaugh X We will describe techniques to solve the forward and inverse scattering problems for the Schrödinger Operator, and then discuss how these can be used to solve the KdV equation. An Introduction to the Inverse Scattering Transform
Nov. 12, 2013 1pm (MATH 220)	Grad Algebra/Logic Heinz Peter Gumm (Philipps-University, Marburg, Germany, and Chapman University, Ora…) X Universal Co-Algebra is a general theory of state based systems, encompassing familiar structures, such as automata, Kripke Structures, dynamical systems, stochastic automata, even topological spaces considered as doubly nondeterministic transition systems. In order to capture and generalize systems with such diverse behaviours, the mathematical language to deal with universal coalgebra is Category theory. The results obtained this way are very concrete: homomorphism theorems, Birkhoff style theorems, classification theorems, and they can be very elegantly derived, using only elementary categorical notions and tools. Universal Co-Algebra is in many ways dual to Universal Algebra, and when translating co-algebraic proofs, one even obtains elegant proofs of universal algebraic results, albeit in a more general setting than that of classical universal algebra. Universal Co-Algebra
Nov. 12, 2013 1pm (MATH 350)	Number Theory Phillip Habegger X Conjectures on unlikely intersections generalize various problems in diophantine geometry such as the Manin-Mumford Conjecture. They originated in work of Bombieri-Masser-Zannier, Pink, and Zilber. Using a point counting result of Pila-Wilkie formulated in the language of o-minimal structures, Zannier devised a new strategy to prove the Manin-Mumford Conjecture. After providing some historical background on unlikely intersections I will explain what an o-minimal structures is. Then I'll describe a possible approach for dealing with higher dimensional unlikely intersections in abelian varieties defined over number fields. This is joint work in progress with Jonathan Pila. Unlikely Intersections and o-Minimal Structures
Nov. 12, 2013 2pm (MATH 350)	Lie Theory Uriya First (Hebrew University) X In short: I will explain how to use bilinear forms to prove theorems of Albert and Saltman about algebras with involution as well as several other related results. In detail: A well-known theorem of Albert states that a central simple algebra A of period 2 in the Brauer group admits an involution of the first kind (i.e. fixing the center). This was generalized by Saltman to Azumaya algebras with the difference that the conclusion is that there is some algebra in the Brauer class of A admitting an involution of the first kind. Albert and Saltman also gave necessary and sufficient conditions for existence of involutions of the second kind (i.e. not fixing the center). Another proof of Saltman's Theorem was later given by Knus, Parimala and Srinivas. I will discuss a new approach to Saltman's Theorem based on bilinear forms. This approach gives a new perspective on the proof of Knus-Parimala-Srinivas and also give rise to further results and generalizations. I shall also explain how to apply this approach to construct Azumaya algebras of period 2without involution. All notions (e.g. Brauer Group, Azumaya algebra) will be defined during the talk. Azumaya Algebras With and Without Involution
Nov. 12, 2013 3pm (MATH 220)	PDE/Analysis Jiahong Wu (Oklahoma State University) X The Boussinesq equations concerned here model geophysical flows such as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq equations serve as a lower-dimensional model of the 3D hydrodynamics equations. In fact, the 2D Boussinesq equations retain some key features of the 3D Euler and the Navier-Stokes equations such as the vortex stretching mechanism. The global regularity problem on the 2D Boussinesq equations with partial or fractional dissipation has attracted considerable attention in the last few years. This talk reviews recent results on various cases of partial dissipation and presents most recent developments on the 2D Boussinesq equations with partial and fractional dissipation. Recent developments on the 2D Boussinesq equations with partial dissipation
Nov. 12, 2013 3pm (MATH 350)	Algebraic Geometry Qile Chen (Columbia) X The theory of stable log maps are developed for studying the degeneration of Gromov-Witten invariants. In this talk, I will introduce another interesting application of stable log maps to classicalbirational geometry ---- We construct very free rational curves on Fano complete intersections in projective spaces over an algebraically closed field of arbitrary characteristics. This a joint work with Yi Zhu. Very free rational curves on Fano complete intersections
Nov. 19, 2013 10am (MATH 350)	Ulam Thomas Ivey X I'll continue a broad introduction to the subject of integrable systems of PDE. I'll discuss the construction of an infinite sequence of commuting flows and local conservation laws for the KdV equation, and if time permits I will also Backlund transformations, which can be used to construct multi-soliton solutions. Integrable PDE, part 2
Nov. 19, 2013 1pm (MATH 220)	Grad Algebra/Logic Charlie Scherer (CU Boulder) X Whitehead's problem is a question about abelian groups that was motivated by complex analysis via homology and eventually resolved by set theory. I use the word "resolved" rather than "solved" because it was shown that Whitehead's problem is undecidable using ordinary mathematics (ZFC). I want to discuss Whitehead's problem as an example of how Jensen's diamond principle from set theory can be applied in algebra. This week I will state Whitehead's problem and give the solution for countable groups. Diamonds are a Group's Best Friend, Part 1
Nov. 19, 2013 1pm (MATH 350)	Number Theory Mehmet Kiral (Brown University) X Given a modular form $f$ , the corresponding $L$ -function $L(s,f)$ satisfies a functional equation as $s ↦ 1-s$ . One can view bounds of the form $L(1/2,f)$ as improvements towards the $α =0$ case, the Lindelof Hypothesis. The functional equation together with the Phragmen Lindelof theorem give the bound for $α = 1/4$ . This is called the convexity bound, and any result with $α < 1 / 4$ is called the subconvexity result. The Lindelof Hypothesis is a consequence of the Riemann Hypothesis for $L(s,f)$ and thus are expected to be true for all L-functions with a functional equation and an Euler product. We prove a subconvexity bound for $L(1/2,f,χ)$ in the conductor aspect where f is a half integral weight modular form. In this case the L function does not posess an Euler product. Subconvexity bounds for half Integral weight modular forms
Nov. 19, 2013 3pm (MATH 350)	Algebraic Geometry Matthew Satriano (U. Michigan) X We will discuss a technique which allows one to approximate singular varieties by smooth spaces called stacks. As an application, we will address the following question, as well as some generalizations: given a linear action of a group G on complex n-space C^n, when is the quotient C^n/G a singular variety? We will also mention some applications to Hodge theory and to derived equivalences. Stacky Resolutions of Singularities Sponsored by the Meyer Fund
Nov. 19, 2013 3pm (MATH 220)	PDE/Analysis Ryan Rosenbaum X Weyl's law describes the asymptotic distribution of the eigenvalues of the Laplacian on a compact Riemann manifold M with Dirichlet boundary condition. In the case that M is a torus, this quickly reduces to the classical problem of counting lattice points in a ball. We will prove Weyl's law in this case using a trace formula argument which generalizes considerably. Time permitting, we will sketch the proof of Weyl's law in a more general setting. Weyl's law via trace formulas
Dec. 03, 2013 10am (MATH 350)	Ulam Thomas Ivey X I will review integration by the method of Darboux for second-order hyperbolic PDE in the plane, and its natural extension to hyperbolic exterior differential systems (EDS). I will then discuss the extension of this technique to decomposable EDS, as studied by Anderson, Fels and Vassiliou in their groundbreaking paper "Superposition Formulas for Exterior Differential Systems." Peter Vassiliou (University of Canberra) will be visiting the department this week and next, and this week's Ulam seminar is designed to provide background for Vassiliou's seminar talks next week. Generalizations of Darboux Integrability
Dec. 03, 2013 1pm (MATH 220)	Grad Algebra/Logic Charlie Scherer (CU Boulder) Diamonds are a Group's Best Friend, Part 2
Dec. 03, 2013 2pm (Math 350)	Lie Theory Bridget Tenner (DePaul University) X We will discuss two varieties of permutation patterns: classical patterns and the more generally (and more recently) defined mesh patterns. We will analyze completely when a mesh pattern is equivalent to a finite collection of classical patterns, in a sense declaring that the additional data of the mesh was superfluous for these patterns. We will also describe the permutations having the fewest superfluous meshes, and the permutations having the most, enumerating the superfluous meshes in each case. (Mesh) Patterns Sponsored by the Meyer Fund
Dec. 03, 2013 3pm (MATH 350)	PDE/Analysis Thomas Ivey X Austere submanifolds are a special sub-class of minimal submanifolds in Euclidean space. Harvey and Lawson defined this class, and showed that the conormal bundle of a submanifold M is special Lagrangian if and only if M is austere. Later, Bryant classified austere 3-folds in Euclidean space, as well as linear-algebraic models for austere 4-folds. More classification results in dimension four were obtained by Marianty Ionel and myself, and we have recently generalized the austere condition to submanifolds in complex projective space. An introduction to austere geometry
Dec. 09, 2013 4pm (MATH 350)	Kempner Anca Radulescu (CU Boulder) X Many natural systems, including the brain, are organized as dynamic networks, with nodes interacting in a time-dependent fashion. The way in which various parts of the brain (from the micro-scale of neurons to the macro-scale of functional regions) are wired together is one of the great scientific challenges of the 21st century, currently being addressed by large-scale research collaborations, such as the Human Connectome Project. The aim of our own research is to understand in particular the relationship between a networks hardwired circuitry and its dynamics. Key factors for optimal (critical) dynamics in a network of neural nodes are a combination of a well-balanced adjacency scheme together with well-tuned connection strengths between nodes. We investigate whether small perturbations in edge density or geometry can produce large effects in the systems phase space dynamics, or if phenomena like phase transitions (bifurcations) are robust to either of such changes. We finally illustrate, (using imaging results from 97 human subjects with a wide range of emotional responses) how such properties can be useful as biomarkers for different emotional profiles, or neuropsychiatric conditions. Graph theory, nonlinear networks and emotional responses
Dec. 10, 2013 1pm (MATH 220)	Grad Algebra/Logic Alexandr Kazda (Vanderbilt University) X There is a lot happening on the boundary between universal algebra and computer science. It is not very surprising that algebraists want to have fast algorithms. However, the connection runs in the other direction as well: It turns out that universal algebra is a very good tool for describing the complexity of the Constraint Satisfaction Problem (CSP). An instance of the CSP asks us to decide if we can assign values to variables in such a way that a list of constraints is satisfied (examples: graph coloring, logical formula satisfiability, Sudoku). If we restrict the sort of constraints to be used, the complexity of solving the CSP can vary from polynomial to NP-complete. We will talk about how to use universal algebra to classify the complexity of many such restricted CSPs. Universal Algebra Meets Algorithmic Complexity
Dec. 10, 2013 3pm (MATH 350)	PDE/Analysis Peter Vassiliou (University of Canberra, Australia) X ABSTRACT: Gaston Darboux (1842 - 1917) was much occupied with the theory of orthogonal coordinate systems throughout his productive career. Despite his many distinguished researchers as well as vast contributions by others like L. Bianchi (1856 -1928) and G. Lame (1795 - 1870) much remains to be understood and discovered. In this talk I will introduce the basic ideas and then go on to show how the theory of Darboux integrable systems can be used to construct orthogonal coordinate systems. It will be shown that there is a natural relationship between this circle of ideas and certain Einstein metrics as well as a particular completely integrable system, the so called $n$ -wave resonant interaction system. A number of open questions will be framed Einstein Meets Darboux
Dec. 14, 2013 4pm (MATH 350)	Ulam Jeff Taylor (CU Boulder) X This is my abstract This is a test Sponsored by the Meyer Fund
Jan. 01, 2014 3pm (Math 350)	Algebraic Geometry Previous talks Previous talks
Jan. 01, 2014 4pm (MATH 350)	Kempner Previous talks Previous talks
Jan. 23, 2014 2pm (MATH 350)	PDE/Analysis NA X Among other things, we will discuss topics and speakers for this semester's seminar. All are welcomed to attend. Organizational Meeting
Jan. 28, 2014 9am (MATH 350)	PDE/Analysis Alejandro Sarria (CU Boulder) X The two-component Camassa-Holm system models the propagation of unidirectional shallow water waves over a flat bed and, in some instances, admits an interpretation as a geodesic equation for a right-invariant metric on the diffeomorphism group. In this talk, a short-wave model for this system is studied with breakdown (or global existence in time) of classical solutions being discussed. On a short-wave model for the two-component Camassa-Holm system
Jan. 28, 2014 1pm (MATH 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) Infinite Groups with Finite Conjugacy Classes, Part 1
Jan. 28, 2014 1pm (MATH 350)	Number Theory Jeff Thunder (University of Northern Illinois) X In his paper Northcott's Theorem on Heights II. The Quadratic Case, W.M. Schmidt gave asymptotic estimates for the number of points in projective space that have height no greater than a given bound and that generate a quadratic extension of the rational number field. For the special case where the points lie in projective 2-space, he needed to prove a result on sums of quotients of certain L-series (coming from quadratic extensions of the rational number field). Motivated by the function field case of Schmidt's result, we will look at L-series over function fields and discuss some recent results. L-series over function fields
Feb. 04, 2014 9am (MATH 350)	PDE/Analysis Pearce Washabaugh (CU Boulder) X The Euler-Arnold equation is essentially a coordinate invariant geodesic equation that can be written down on any Lie group, even those of infinite dimension, with a specific kind of inner product on its Lie algebra. I'll review and expand upon Boramey's introduction to this topic last semester. Time permitting I'll talk about curvature calculations on the group of volume preserving diffeomorphisms of a manifold with applications to weather prediction, as well as magnetic extensions of Lie groups, which allow one to phrase the system of ideal magnetohydrodynamics equations as geodesic equations on an infinite dimensional Lie group. An Introduction to the Euler-Arnold Equation
Feb. 04, 2014 1pm (MATH 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) Infinite Groups with Finite Conjugacy Classes, Part 2
Feb. 11, 2014 9am (MATH 350)	PDE/Analysis Stephen C. Preston (CU Boulder) X I will give an overview of infinite-dimensional Riemannian geometry, especially those arising in physical applications. In particular I will discuss the sign of the curvature for the volume-preserving diffeomorphisms and the space of inextensible strings, and what it means. The talk will assume some knowledge of finite-dimensional Riemannian geometry but not much beyond that. Curvatures of infinite-dimensional manifolds -Part I
Feb. 11, 2014 1pm (MATH 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) Infinite Groups with Finite Conjugacy Classes, Part 3
Feb. 18, 2014 9am (MATH 350)	PDE/Analysis Boramey Chhay (CU Boulder) X I will introduce an example of an infinite dimensional manifold which is the manifold of Riemannian metrics. That is, given a finite dimensional, compact, smooth, orientable manifold $M$ without boundary, we can consider the Riemannian structures that can be placed on $M$ . We will then put a metric on this space and discuss its properties. Metrics on the Space of Metrics - Part I
Feb. 25, 2014 9am (MATH 350)	PDE/Analysis Boramey Chhay (CU Boulder) X I will introduce an example of an infinite dimensional manifold which is the manifold of Riemannian metrics. That is, given a finite dimensional, compact, smooth, orientable manifold $M$ without boundary, we can consider the Riemannian structures that can be placed on $M$ . We will then put a metric on this space and discuss its properties. Metrics on the Space of Metrics - Part II
Feb. 25, 2014 1pm (MATH 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) Burnside's Theorem on Transitive Permutation Groups of Prime Degree, Part 1

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Events for October 2013 – February 2014

Lie Theory

Algebraic Geometry

Algebraic Geometry

Algebraic Geometry

Lie Theory

Lie Theory

Grad Algebra/Logic

Number Theory

PDE/Analysis

Grad Algebra/Logic

Lie Theory

Kempner

Grad Algebra/Logic

Probability

Grad Algebra/Logic

Number Theory

Lie Theory

Ulam

Number Theory

Grad Algebra/Logic

Lie Theory

PDE/Analysis

Algebraic Geometry

Ulam

Grad Algebra/Logic

Number Theory

Algebraic Geometry

PDE/Analysis

Algebraic Geometry

Ulam

Grad Algebra/Logic

Number Theory

Lie Theory

PDE/Analysis

Algebraic Geometry

Ulam

Grad Algebra/Logic

Number Theory

Algebraic Geometry

PDE/Analysis

Ulam

Grad Algebra/Logic

Lie Theory

PDE/Analysis

Kempner

Grad Algebra/Logic

PDE/Analysis

Ulam

Algebraic Geometry

Kempner

PDE/Analysis

PDE/Analysis

Grad Algebra/Logic

Number Theory

PDE/Analysis

Grad Algebra/Logic

PDE/Analysis

Grad Algebra/Logic

PDE/Analysis

PDE/Analysis

Grad Algebra/Logic