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Events for April – August 2014

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Apr. 12, 2014 6:19am (HUMN 1…	Algebraic Geometry Western Algebraic Geometry Symposium WAGS April 12-13
Apr. 01, 2014 1pm (MATH 350)	Number Theory Amy Feaver (CU Boulder) X TBA Euclid's Algorithm in Multiquadratic Fields (thesis defense)
Apr. 02, 2014 4pm (MATH 350)	Grad Student Seminar Justin Keller Topological Quantum Field Theories and ******-Algebras
Apr. 03, 2014 1pm (MATH 220)	Number Theory Lenny Fukshansky (Claremont College) X A celebrated theorem of Cassels (1955) asserts that an integral quadratic form, which is isotropic over Q, has a non-trivial integral zero of small height; the bound is in terms of the height of the coefficient vector of the quadratic form. In the later years, analogues of Cassels’ result have been proved over other global fields: over number fields by Raghavan (1975), over rational function fields by Prestel (1987), and over algebraic function fields by Pfister (1997). Further extensions of Cassels’ theorem to small-height isotropic subspaces of a quadratic space, using the contemporary theory of height functions as developed by W. M. Schmidt, have been obtained by Schlickewei (1985), Schlickewei & Schmidt (1987), and Vaaler (1987, 1989). More recently, I have done some work on effective (with respect to height) decompositions of bilinear spaces, as well as further generalization of this theory to the situations with additional algebraic conditions and over quaternion algebras. In this talk, I will discuss several results in this area, starting from Cassels’ original theorem and up until my own recent work. Some of the results discussed are joint work with W. K. Chan and G. Henshaw. Heights and effective theory of quadratic forms over global fields (NOTE: THURS)
Apr. 04, 2014 4pm (Benson 185)	Kempner Seth Sullivant (North Carolina State) X The main problem in phylogenetics is to reconstruct evolutionary relationships between collections of species, typically represented by a phylogenetic tree. In the statistical approach to phylogenetics, a probabilistic model of mutation is used to reconstruct the tree that best explains the data (the data consisting of DNA sequences from genes appearing in all species being analyzed). In algebraic statistics, we interpret these statistical models of evolution as geometric objects in a high-dimensional probability simplex. This connection arises because the functions that parametrize these models are polynomials, and hence we can consider statistical models as algebraic varieties. The goal of the talk is to introduce this connection and explain how the algebraic perspective leads to new theoretical advances in phylogenetics, and also provides new research directions in algebraic geometry. The talk material will be kept at an introductory level, with background on both phylogenetics and algebraic geometry. Phylogenetic algebraic geometry
Apr. 07, 2014 4pm (MATH 350)	Kempner Kyle Petersen (DePaul) X The face vector, or f-vector, of a simplicial complex counts the number of faces in the complex according to dimension (number of vertices, edges, triangles, etc.). An old problem in combinatorial topology is to characterize the f-vectors that can be realized for a given topological space. For example, there is no simplicial complex with 4 vertices and 7 edges (why not?), and while you can construct a simplicial complex with 5 vertices, 8 edges, and 5 triangles, such a complex cannot be realized on a sphere (this fact is deeper). In this talk, I describe some of the history of attempting to characterize f-vectors in the context of something called the Charney-Davis conjecture, which is itself a discretized version of a conjecture of Hopf. Due to work of S. Gal, the Charney-Davis conjecture follows if one can prove that a certain transformation of the f-vector is nonnegative. This is the gamma vector of the title. Interesting combinatorics ensue. What is the gamma vector? (And what does it count?)
Apr. 07, 2014 4pm (MATH 350)	Lie Theory Kyle Petersen (DePaul University) X The face vector, or f-vector, of a simplicial complex counts the number of faces in the complex according to dimension (number of vertices, edges, triangles, etc.). An old problem in combinatorial topology is to characterize the f-vectors that can be realized for a given topological space. For example, there is no simplicial complex with 4 vertices and 7 edges (why not?), and while you can construct a simplicial complex with 5 vertices, 8 edges, and 5 triangles, such a complex cannot be realized on a sphere (this fact is deeper). In this talk, I describe some of the history of attempting to characterize f-vectors in the context of something called the Charney-Davis conjecture, which is itself a discretized version of a conjecture of Hopf. Due to work of S. Gal, the Charney-Davis conjecture follows if one can prove that a certain transformation of the f-vector is nonnegative. This is the gamma vector of the title. Interesting combinatorics ensue. What is the gamma vector? (And what does it count?) Sponsored by the Meyer Fund
Apr. 08, 2014 9am (MATH 350)	PDE/Analysis Alejandro Sarria (CU Boulder) X The Constantin-Lax-Majda equation (CLM) was proposed as a one dimensional model for the three dimensional vorticity of an incompressible fluid. We provide background on the CLM model and discuss some of its drawbacks. Several of its proposed generalizations are also considered. In particular, blowup results and criteria leading to global solutions among these models are compared. The Constantin-Lax-Majda equation and generalizations thereof
Apr. 08, 2014 1pm (MATH 220)	Grad Algebra/Logic Jeffrey Shriner (CU Boulder) X A chip-firing game on a graph G begins with a configuration (the number of 'chips' on each vertex). When a vertex 'fires', it sends a chip to each adjacent vertex, so the only rule of the game is that a vertex may 'fire' if and only if it has at least deg(v) chips. We discuss a variant of the chip-firing game, called the dollar game, in which one vertex is required to always be in debt. Using the dollar game, we form an abelian group structure K(G), called the critical group of G. We compute the order of K(G), and look at how the dollar game may be used as a tool for analyzing the structure of the group. The Dollar Game and the Critical Group of a Graph, Part 1
Apr. 08, 2014 1pm (MATH 350)	Number Theory Rachel Davis (Purdue University) X Let $E$ be an elliptic curve over $ℚ$ with surjective mod-2 representation. Let $K$ be the splitting field of the $2$ -division polynomial of $E$ . We show that for every element $ξ$ of the cohomology group $H1(GQ,E[2])$ , there exists $E^{'}$ , a mod-2 companion of $E$ and $P ∈ E(Q)$ such that $E, P)$ maps to $ξ$ under the Kummer map. I will also discuss how a finite subgroup of this cohomology group is related to $S4$ -extensions of $ℚ$ that contain $K$ and are unramified outside a finite set of primes. (joint work with Edray Goins and Jamie Weigandt) Cohomology and S4 extensions of Q
Apr. 10, 2014 1pm (MATH 350)	Number Theory Damien Roy (University of Ottawa) X In a series of recent papers, W.M. Schmidt and L. Summerer develop a remarkable theory of parametric geometry of numbers which enables them to recover many results about simultaneous rational approximation to families of Q-linearly independent real numbers, or about the dual problem of forming small linear integer combinations of such numbers. They recover classical results of Khintchine and Jarnik as well as more recent results by Bugeaud and Laurent. They also find many new results of Diophantine approximation. Their theory provides constraints on the behavior of the successive minima of a natural family of one parameter convex bodies attached to a given n-tuple of real numbers, in terms of this varying parameter. In this talk, we are interested in the converse problem of constructing n-tuples of numbers for which the corresponding successive minima obey given behavior. We will present the general theory of Schmidt and Summerer, mention some applications, and report on recent progress concerning the above problem. On Schmidt and Summerer parametric geometry of numbers (NOTE: THURS)
Apr. 10, 2014 3pm (Webber 20…	Algebraic Geometry Dusty Ross (Michigan) X Given a quasi-homogeneous polynomial of degree d, Landau-Ginzburg theory studies certain intersection numbers on the moduli space of d-spin curves (parametrizing curves with d-th roots of the canonical bundle). I will describe a generalization of these intersection numbers obtained by allowing some of the points on the curves to be weighted in the sense of Hassett. As one changes the weights, the invariants thus obtained can be related by a wall-crossing formula. I will explain how the wall-crossing formula generalizes the mirror theorem of Chiodo-Iritani-Ruan, and in particular how it gives a completely enumerative (A-model) interpretation of the mirror phenomenon. Wall-crossing in genus zero Landau-Ginzburg theory
Apr. 14, 2014 10am (Eng E206…	Algebraic Geometry Douglas Ulmer (Georgia Tech) Jacobians with many points
Apr. 15, 2014 9am (MATH 350)	PDE/Analysis Boramey Chhay (CU Boulder) X I will introduce these two types of manifolds including basic definitions, properties, and applications. Contact and Symplectic Manifolds
Apr. 15, 2014 1pm (MATH 220)	Grad Algebra/Logic Jeffrey Shriner (CU Boulder) X A chip-firing game on a graph G begins with a configuration (the number of 'chips' on each vertex). When a vertex 'fires', it sends a chip to each adjacent vertex, so the only rule of the game is that a vertex may 'fire' if and only if it has at least deg(v) chips. We discuss a variant of the chip-firing game, called the dollar game, in which one vertex is required to always be in debt. Using the dollar game, we form an abelian group structure K(G), called the critical group of G. We compute the order of K(G), and look at how the dollar game may be used as a tool for analyzing the structure of the group. The Dollar Game and the Critical Group of a Graph, Part 2
Apr. 16, 2014 4pm (MATH 350)	Grad Student Seminar Jonathan Lamar X This sales pitch is loosely based on a paper of Barcelo and Ram (http://arxiv.org/abs/math/9707221), in which they attempt to describe the desired aesthetic of a result in combinatorial representation theory. I will do my best to communicate this aesthetic to you through two examples: the finite-dimensional representations of the symmetric groups, and the polynomial representations of GL(n,k) for k an algebraically closed field of characteristic zero. As it turns out, there is a nice combinatorial description of each theory. If time permits, I also hope to describe the relationship between the two examples using the language of polynomial functors. What Makes Representation Theory Combinatorial ?
Apr. 17, 2014 1pm (MATH 350)	Number Theory Gabrielle Nebe (RWTH Aachen) X This talk presents recent constructions and classifications of extremal even unimodular lattices whose dimension is a multiple of 24. We know six such lattices, the Leech lattice in dimension 24, four lattices of dimension 48 and, since 2010, also one extremal lattice in dimension 72. The existence of the latter was a longstanding open question. I will comment on its construction before turning to the classification of possible automorphisms of extremal even unimodular lattices in dimension 48. This investigation led to the discovery of a new extremal lattice, whose automorphism group is very small. Automorphisms of extremal lattices (NOTE: THURS)
Apr. 22, 2014 9am (MATH 350)	PDE/Analysis Steve Preston (CU Boulder) X Following up on Boramey Chhay's talk last week, I will discuss some geometry of the group of diffeomorphisms preserving a symplectic form or a contact structure. In particular I will talk about the geodesic equation in these spaces with a right-invariant Riemannian metric. Symplectomorphisms and contactomorphisms
Apr. 22, 2014 1pm (MATH 220)	Grad Algebra/Logic Jeffrey Shriner (CU Boulder) X A chip-firing game on a graph G begins with a configuration (the number of 'chips' on each vertex). When a vertex 'fires', it sends a chip to each adjacent vertex, so the only rule of the game is that a vertex may 'fire' if and only if it has at least deg(v) chips. We discuss a variant of the chip-firing game, called the dollar game, in which one vertex is required to always be in debt. Using the dollar game, we form an abelian group structure K(G), called the critical group of G. We compute the order of K(G), and look at how the dollar game may be used as a tool for analyzing the structure of the group. The Dollar Game and the Critical Group of a Graph, Part 3
Apr. 23, 2014 4pm (MATH 350)	Grad Student Seminar Xiaorong Liu X Stochastic functional differential equations is a kind of SDEs including past states of the solutions. I will present sufficient conditions ensuring the pth mean and almost sure exponential stability, as well as the pth mean integrability of solutions to stochastic functional differential equations. Moment exponential stability and integrability of stochastic functional differential equations
Apr. 28, 2014 4pm (MATH 350)	Kempner Hortensia Soto-Johnson (University of Northern Colorado) X One area of research in undergraduate mathematics education, which has not received much attention, is the teaching and learning of complex variables. In this presentation, I will share results from two projects related to research participants’ use of gesture as they conveyed their geometric reasoning about tasks related to complex variables. In the first study, we compared the gestures of six experts and four pairs of novices as they described their geometric reasoning of the complex number arithmetic operations z+w, zw, and 1/z. In the second project, we explored the interplay between gesture and diagrams as undergraduates reasoned about complex-valued equations. I will discuss the importance of gesture awareness in the classroom and directions for further research. Reasoning about complex variables concepts – the role of gesture
Apr. 29, 2014 9am (MATH 350)	PDE/Analysis Alejandro Sarria (CU Boulder) X We discuss a particular class of solutions to 3D Euler which blowup in finite time from smooth initial data. This family of exact solutions has infinite energy and was derived by Peter Constantin. Some qualitative properties of blowup are also addressed. Riccati Blowup in Semibounded Solutions to 3D Euler
Apr. 29, 2014 1pm (MATH 350)	Number Theory Hershy Kisilevsky (Concordia University) X For an elliptic curve $E$ defined over the rational field, $ℚ$ , we will be interested in the variation of the Mordell-Weil groups $E(K)$ as $K$ ranges over finite extensions of $ℚ$ . In particular we will examine the ranks of $E(K)$ (both algebraic and analytic), as $K$ varies and give some of our results and conjectures on the values obtained. Acquiring Algebraic Points on Elliptic Curves
Apr. 29, 2014 1pm (MATH 220)	Grad Algebra/Logic Michael Pinsker (Universite Diderot, Paris) X A function clone on a set D is a set of finitary operations on D which is closed under composition and which contains the projections. Every function clone carries a natural algebraic structure given by the equations that hold in the clone, as well as a natural topological structure given by pointwise convergence. Let P be the function clone consisting of all projections on a two-element set. Given any function clone C on a countably infinite set, we compare the following questions: 1) Is there a homomorphism from C to P? 2) Is there a continuous homomorphism from C to P? These questions have applications in constraint satisfaction. More context will be provided in the subsequent talk in the logic seminar. Decomposing function clones on infinite sets Sponsored by the Meyer Fund
Apr. 29, 2014 2pm (MATH 350)	Lie Theory Jonathan Lamar (CU) X I will be presenting on a paper of Halasi and Palfy in which they prove that for certain pattern subgroups of UT(n,q), the number of conjugacy classes is not given by a polynomial equation in q. Moreover, they also prove that for some pattern groups, the number of conjugacy classes is not given by even a finite set of polynomials in q. This result is related to a famous open problem posed by Graham Higman, who asked if the number of conjugacy classes of UT(n,q) is given by a polynomial in q. Pattern groups whose number of conjugacy classes is not given by a polynomial
Apr. 29, 2014 2pm (MATH 220)	Michael Pinsker (Universite Diderot, Paris) X The Constraint Satisfaction Problem (CSP) of a finite or countable first-order structure S in a finite relational language is the problem of deciding whether a given conjunction of atomic formulas in that language is satisfiable in S. Many classical computational problems can be modelled this way. The study of CSPs involves an interesting combination of techniques from universal algebra, Ramsey theory, and model theory. This talk will present an overview over these techniques as well as some wild conjectures. Introduction to Constraint Satisfaction over infinite structures Sponsored by the Meyer Fund
Apr. 29, 2014 3pm (Math 220)	Algebraic Geometry Samouil Molcho (Brown) X The space of stable log maps of Abramovich-Chen and Gross-Siebert is a central object of study in singular Gromov-Witten theory. In this talk, I will give an explicit combinatorial description of the space of genus 0, 2 pointed stable maps to a toric variety: it is an appropriate enrichment of the Chow Quotient of Kapranov-Sturmfels-Zelevinsky into a toric stack. The idea for this description comes from Morse Theory. This is joint work with William Gillam. Logarithmic Stable Maps as Moduli of Broken Orbits Sponsored by the Meyer Fund
Apr. 30, 2014 4pm (MATH 350)	Grad Student Seminar Trevor Jack X Cantor Spaces are precisely those spaces that are totally disconnected, compact metric spaces with no isolated points and this discussion will determine the number of nonequivalent metrics that can be defined on such spaces. Nonequivalent Metrics on Cantor Spaces
May. 01, 2014 3pm (Webber 20…	Algebraic Geometry Nicola Tarasca (Utah) X Inside the moduli space of curves of genus 2 with 2 marked points, the loci of curves admitting a map to P1 of degree d totally ramified at the two marked points have codimension two. In this talk I will show how to compute the classes of the compactifications of such loci in the moduli space of stable curves. I will also discuss the relation with the related work of Hain, Grushevsky-Zakharov, Chen-Coskun, Cavalieri-Marcus-Wise. Double total ramifications for curves of genus 2
May. 08, 2014 3pm (Webber 20…	Algebraic Geometry Ekin Ozman (UT Austin) The p ranks of Prym varieties
Aug. 27, 2014 4:05pm (MATH 3…	Grad Student Seminar Paul Lessard (CU-Boulder) X Eilenberg and Mac Lane developed or discovered category theory in the context of algebraic topology; the very utility of� topological invariants such as homology, co-homology, and the homotopy groups is contingent on their functoriality. These� particular invariants are moreover homotopy invariants. This property can be rephrased as those invariants factoring through the homotopy category, where homotopic maps are identified. We try and reverse this observation. Can we, starting with a functor, interpret it as a homotopy invariant by finding the right notion of homotopy? What data do we need to do so, and how universal are these constructions? In part one we will motivate and define Quillen's concept of a model category, and in part two we will outline how such a structure begets a homotopy category. In lieu of pizza, it is suggested that one attend the department picnic. From Invariants, to functors, to homotopy, to model categories, and back to homotopy: part 1

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