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Events for April – August 2015

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Apr. 01, 2015 4pm (ECCR 265)	Grad Student Seminar Athena Sparks (CU-Boulder) X The crossing number of a graph is the minimum number of edge crossings among all planar drawings of the graph. We will discuss the history of crossing number problems, including the current conjectures for the crossing number of the complete graph and the complete bipartite graph. We will also derive the cylindrical crossing number of the complete bipartite graph. This is the minimum number of crossings among all drawings of the complete bipartite graph where the two disjoint sets of vertices are placed on the top and bottom rims of a cylinder and every vertex at the top is connected to every vertex at the bottom by an edge drawn along the lateral surface of the cylinder. Drawing and Counting with Athena - The Cylindrical Crossing Number of the Complete Bipartite Graph
Apr. 02, 2015 3pm (MATH 350)	Probability Yuliy Baryshnikov (U. Illinois) X Quantum random walks on an integer lattice of rank k form a class of unitary (discrete time) transformations that are a nice toy model for certain quantum-mechanical operators. The amplitudes (after a large number of steps) for such random walks form typically beautiful moire patterns, whose structure and relation to singularity theory I will explain in this talk. Quantum Random Walks on Lattices and Singularities
Apr. 03, 2015 4pm (MATH 350)	Grad Student Seminar Matt Grimes (CU-Boulder) X In Calculus I, we learn about families of functions. For instance, y=tx is a family of lines parametrized by the real line and y=x^2+ax+b is a family of parabolas parametrized by the plane. It would be natural to ask if we can parametrize "all" lines, parabolas, or whatever geometric object we are interested in. We will discuss what it means to parametrize all objects of a given type using objects called moduli spaces. We will then discuss some of the interesting questions posed by moduli spaces and how a moduli space can be used to elegantly prove the fundamental theorem of algebra. The Geometry of Families
Apr. 07, 2015 11am (MATH 350)	Number Theory Jerzy Kocik (Southern Indiana University) X Apollonian disk packings have a surprising rich content of number theory, geometry and mathematical structures relevant to theoretical physics. These will be reviewed. Key words include: Minkowski space, integer sequences, Pythagorean triples, spinors, Stern-Brocot tree, and more. The talk is intended to be intriguing to every mathematician and accessible and (mathematically) entertaining to students as well. Apollonian disk packing -- numbers, geometry and physics.
Apr. 08, 2015 9am (MATH 350)	PDE/Analysis Jae Min Lee (CU Boulder) X Semigroup theory is the abstract study of first-order ordinary differential equations with values in Banach spaces, driven by linear, but possibly unbounded operators. Roughly speaking, the semigroup approach in partial differential equations is to regard a time-dependent partial differential equation as an ordinary differential equation on a suitable function space. In this talk, we outline the basics of the theory and present as well two applications to linear PDE; heat equation and second-order parabolic PDE, which is the generalization of the heat equation. Introduction to Semigroup Theory
Apr. 08, 2015 4pm (MATH 350)	Grad Student Seminar Braden Balentine (CU-Boulder) X Bell’s theorem is a fundamental and incredibly strange result that shows quantum mechanics is incompatible with theories in which probabilities in measurement arise from ignorance of pre-existing local properties. In this talk, we provide some historical background concerning the apparent inconsistencies of quantum mechanics with the physical world and provide simple and reasonably intuitive proofs of Bell’s inequality and Bell’s theorem. Little to no physics knowledge will be required and the mathematics involves simple probability arguments and some knowledge of inner-product spaces. Bell’s Theorem and the Weirdness of Quantum Mechanics
Apr. 09, 2015 11am (Math 350)	Functional Analysis Emily Proctor (Middlebury College) X I will discuss recent work with Farsi and Seaton in which we introduce the -extension of the spectrum of the Laplacian of a Riemannian orbifold, where is a finitely generated discrete group.This extension, called the -spectrum, is the union of the Laplace spectra of the -sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the singular set of the orbifold. In our work, we compare the -spectra of known examples of isospectral pairs and families of orbifolds and demonstrate that in many cases, isospectral orbifolds need not be -isospectral. We additionally prove a version of Sunada's theorem that allows us to construct pairs of orbifolds that are -isospectral for any choice of . -extensions of the spectrum of an orbifold Sponsored by the Meyer Fund
Apr. 09, 2015 3pm (Webber 20…	Algebraic Geometry Csar Lozano Huerta (Harvard) X In this talk, I will focus on compactifying the family of smooth quadric hypersurfaces in projective space using tools from the Minimal Model Program. I will also discuss work in progress on the Hilbert scheme of curves in projective 3-space which borrows ideas and properties from the quadric hypersurface case. Compactifying the space of smooth quadrics.
Apr. 10, 2015 4pm (MATH 350)	Grad Student Seminar Taylor "T.K." Klotz (CU-Boulder) X The Thurston-Perelman geometrization theorem is possibly the most important result for geometry and topology in dimension 3. Indeed, the truthiness of the Poincare conjecture follows easily from this theorem. How does one prove this amazing result? Very carefully and with a PDE called the Ricci flow. In this talk we will introduce some Riemannian geometry necessary to understand the statement of the Ricci flow and then present some basic results to help us understand the nature of its solutions. If time permits we will discuss how Ricci flow is used to prove a special case of the geometrization theorem. (Going With the) Ricci Flow
Apr. 15, 2015 4pm (MATH 350)	Grad Student Seminar Matthew Pierson (CU-Boulder) X In this talk, we will analyze and give a classification of dihedral and generalized quaternion groups. From this classification, we will state two further results. Dihedral and Generalized Quaternion Groups
Apr. 16, 2015 12pm (Math 350)	Functional Analysis James P. Cossey (University of Akron) X Let G be a finite group and p a prime. The Brauer graph of G encodes information regarding the connection between the representations of G over the complex numbers and the representations of G over an algebraically closed field of characteristic p. It is surprising that other than the classical case of cyclic defect groups, not much is known about the structure of Brauer graphs. This talk will begin with the basics of representation theory of finite groups and blocks. We will then discuss some recent developments about Brauer graphs in solvable groups and certain “useful” blocks of symmetric groups, and discuss some conjectures for more general results. This seminar is joint with the Lie Theory/Algebraic Combinatorics Seminar. Brauer graphs of finite groups Sponsored by the Meyer Fund
Apr. 16, 2015 12pm (MATH 350)	Lie Theory JP Cossey (University of Akron) X Let G be a finite group and p a prime. The Brauer graph of G encodes information regarding the connection between the representations of G over the complex numbers and the representations of G over an algebraically closed field of characteristic p. It is surprising that other than the classical case of cyclic defect groups, not much is known about the structure of Brauer graphs. This talk will begin with the basics of representation theory of finite groups and blocks. We will then discuss some recent developments about Brauer graphs in solvable groups and certain “useful” blocks of symmetric groups, and discuss some conjectures for more general results. Brauer graphs of finite groups Sponsored by the Meyer Fund
Apr. 16, 2015 3pm (Webber 20…	Algebraic Geometry Dustin Cartwright (Tennessee) X Dustin Cartwright The dual complex of a degeneration records how the components of the special fiber intersect. It appears in tropical geometry as a parameterizing object for the tropicalization. While any graph appears as the dual complex of a degeneration of curves, the same is not true in higher dimensions. I will discuss some obstructions to realizing specified dual complexes. Excluded homeomorphism types for dual complexes of surfaces
Apr. 16, 2015 3pm (MATH 350)	Probability Mei Yin (University of Denver) X We study the asymptotics for sparse exponential random graph models where the parameters may depend on the number of vertices of the graph. We obtain exact estimates for the mean and variance of the limiting probability distribution and the limiting log partition function of the edge-(single)-star model. They are in sharp contrast to the corresponding asymptotics in dense exponential random graph models. Time permitting, similar analysis will be done for directed sparse exponential random graph models parametrized by edges and multiple outward stars. Part of this talk is based on joint work with Lingjiong Zhu. Asymptotics for sparse exponential random graph models
Apr. 16, 2015 4pm (Webber 20…	Algebraic Geometry Anna Kazanova (UGA) X Anna Kasanova Conformal block vector bundles are vector bundles on the moduli space of stable curves with marked points defined using certain Lie theoretic data. Over smooth curves, these vector bundles can be identified with generalized theta functions. In this talk we discuss extension of this identification over the stable curves. This is joint work with P. Belkale and A. Gibney Vector bundles on moduli space of stable curves with marked points.
Apr. 17, 2015 4pm (MATH 350)	Grad Student Seminar Cherry Ng (CU-Boulder) X Conformal maps have applications both pure and applied. We will discuss some of these applications for Mobius transformations, Schwarz-Christoffel maps, and Joukowski maps. Introduction to Conformal Maps
Apr. 21, 2015 10am (Math 350)	Noncomm Geometry Henri Moscovici (The Ohio State University) Noncommutative geometry by (one) example. Sponsored by the Meyer Fund
Apr. 21, 2015 12:10pm (MATH …	Kempner Michael Fried (UC Irvine) X By 1872 we knew the space of compact surfaces of genus was connected. I discuss applications extending this. Consider pairs with , nonconstant and analytic, mapping to the sphere, , both with r branch points. A locally path connected topology on such pairs uses moving the branch points of the s. We ask: When is connected to ? We find the s must have the same geometric monodromy group with the same conjugacy classes attached to the branch points. Sometimes these invariants suffice; we get one connected component. Sometimes we need more sophisticated invariants to describe components. Two problems on rational functions with coefficients, in a number field , allow me to show the beginning tools that make this work. Find all with the Schur cover property: For infinitely many residue fields of , , with running over the residue field (including ), is one-one. Find all , with indecomposable over , but is a composition of degree rational functions over the complexes. We find is one of two ``easy" (centerless) groups: The dihedral group of order ( odd) or another slightly larger group, and all conjugacy classes are the same. Then we see that modular curves include all rational functions giving solutions. Finally, the solutions come from the two fiber types over the -line in Serre's Open Image Theorem. \vskip .1 in This talk is based on \S 6.1 and \S 6.2 of ``The place of exceptional covers among all diophantine relations," J. Finite Fields 11 (2005) 367--433, arXiv:0910.3331v1 [math.NT] http://www.math.uci.edu/~mfried/paplist-ff/exceptTower0910-3331v1.pdf Connected components of sphere covers
Apr. 21, 2015 2pm (MATH 350)	Lie Theory Renzo Cavalieri (CSU) X The goal is to review and contextualize the evolution of my thoughts and interactions with tropical geometry. This talk is based on collaborative work always with Hannah Markwig, and at different times with each one of Aaron Bertram, Paul Johnson and Dhruv Ranganathan. Back in 2007, Hannah Markwig approached me after being told by Paul Johnson that her tropical covers smelled like cut and join. Deciphering Paul’s oracle was the beginning of a fruitful and ongoing collaboration, that is pulling me closer and closer to the tropical world. Over the course of the years, we have been studying Hurwitz theory and Gromov-Witten theory, first using tropical geometry as a powerful combinatorial tool, and then trying to understand what is the conceptual reason for the remarkably tight connection between the boundary geometry of moduli spaces of curves and maps and the piecewise linear objects in tropical geometry. The introduction of the analityc point of view, brought to the moduli space of curves by Abramovich, Caporaso and Payne, offered not only a much sought for conceptual perspective, but also opened up the way for further investigation. TROPICALIZING A HURWITZ THEORIST
Apr. 21, 2015 3pm (Math 350)	Algebraic Geometry Michael Fried (UC Irvine) X Talk 1: Alternating groups and Lift Invariants Riemann developed theta functions to generalize Abel's Theorem – constructing analytic functions on an elliptic curve – to an arbitrary compact Riemann surface X. Attempts to make the thetas canonical, lead to two types: even and odd. Further, we get several of each type if X has genus > 1. We show how Riemann's thetas arise in considering components of sphere cover families. Our example is of spaces of r-branch point, 3-cycle, degree n, covers. Main Theorem: These spaces have one (resp. two) component(s) if r=n-1 (resp. r ? n). By improving a Fried-Serre result, we can explicitly recognize in which component a 3-cycle cover belongs. This generalizes: With covers of a fixed branch type, we can have several components. Still, we can often use a Lift Invariant to separate them. Application 1: Fibers of the absolute spaces of 3-cycle covers with + (resp. -) lift invariant carry canonical even (resp. odd) thetas when r is even (resp. odd). Sometimes we can assure the even thetas produce non-zero theta-nulls (odd ones are always zero). These Hurwitz-Torelli functions are like automorphic functions on the family of covers. Application 2: The Alternating group, An, n > 3, embeds in, On+, the nxn determinant 1 orthogonal matrices. We produce a mysterious covering group, Spinn, by pulling it back to the universal cover of On+. Serre's originally sought regular realizations of Spinn as a Galois group. We expand this problem to see famous results on modular curves as extremely special cases of the Inverse Galois Problem. This talk is based on Alternating groups and moduli space lifting Invariants, Arxiv #0611591v4. Israel J. Math. 179 (2010) 57–125 (DOI 10.1007/s11856-010-0073-2). Talk 2: An Open Image Theorem based on the Small Heisenberg Group My Talk 1 was about alternating groups, Hurwitz spaces from 3-cycles, and how lift invariant values gave a complete description of the Hurwitz space components. That probably didn't strike anyone as a very modular curve-like topic. For one, even if you studied modular curves, you didn't hear about lift invariants. When r = 4 conjugacy classes, C, define a reduced Hurwitz space, then, we find each component is an upper half-plane quotient. Further, by changing variables we find it ramifies only over the points 0, 1 and ? of the classical j-line, P1j. Part A: Main Modular Tower Conjecture: For simpler notation I often use the case r = 4. We use K to denote a number field. I will show each of the 1st Talk spaces are level 0 of a sequence that resembles () … ? X0(pk+2) ? X0(pk+1) ? … ? X0(p) ? P1j (k ? 0 the level), a classical modular curve sequence. For p a prime, we say a set of group elements is p' if each has order prime to p. The construction produces a sequence like () for any p perfect group G, and any p' classes C whose elements generate G. For r = 4, with G = G0, () M(G, p, C): … ? H(Gk+2, p, C) ? H(Gk+1, p, C) ? … ? H(G, p, C) ? P1j. Our Main M(G, p, C) Conjecture: There are no K points on H(Gk+1, p, C) for k large. A Modular Tower on M(G, p, C) is any (nonempty) projective sequence of components on (). The lift-invariant of the 1st Talk gives a method to identify all Modular Towers on M(G, p, C). The M(G, p, C) conjecture shows us why the inverse Galois problem is so hard. It generalizes that Modular curves, outside their cusps, have no K points at high levels. When r = 4, the main conjecture holds. We can see that case by locating a level on each Modular Tower whose genus exceeds 1. Then, that case follows from Weil's theorem. We use p-Frattini properties of Modular Tower cusps. Part B: Using Part A to Generalize Serre's Open Image Theorem: I constructed all modular curves from two ''easy'' groups: D = ZxsZ/2 (for dihedral), semidirect product of Z and Z/2 (order 2 group), and 2D = (Z)2xsZ/2. What happens when we replace D and 2D with G = (Z)2xsZ/3 and 2G = (Z)4xsZ/3, p? 3? We find here a lift invariant value appears in a Heisenberg group, instead of in a Spin group. Each value gives one component. But, we have a new phenomenon: There are several Harbater-Mumford components (definition from Talk 1). Those always have trivial lift invariant. So, separating covers in these components requires a new idea. An OIT starts by generalizing to Modular Towers a 1967 Serre result for modular curves. I explained Serre's result in the colloquium talk. I expand here to the conjectured property. Our M(G, p, C) monodromy statement: Any Modular Tower has geometric monodromy (over P1j when r = 4) that is eventually p-Frattini, and for almost all p it is actually p-Frattini. We base this talk on The Small Heisenberg Group, and l-adic representations from Hurwitz Spaces. Part I: Alternating groups and lift invariants, Part II: An Open Image Theorem based on the Small Heisenberg Group Sponsored by the Meyer Fund
Apr. 21, 2015 3pm (Math 220)	Functional Analysis Michael Kreisel (University of Maryland) X When studying Gabor systems, we often assume that the collection of time-frequency shifts forms a lattice. In this talk, we will look at Gabor systems where the time-frequency shifts are supported on a quasicrystal. Since quasicrystals do not generally form groups, we cannot study them using the same techniques as lattice Gabor frames. Instead we will study these Gabor frames by using a groupoid which is naturally associated to a quasicrystal. By studying certain twisted groupoid C* -algebras, we are able to show the connections between the physics of quasicrystals and Gabor theory. In particular, we are able to prove a non-existence result, showing that it is impossible to construct a tight multiwindow Gabor frame for certain quasicrystals if the generators are assumed to be in the modulation space M^1 . Gabor frames for quasicrystals Sponsored by the Meyer Fund
Apr. 22, 2015 4pm (MATH 350)	Grad Student Seminar Leo Herr X A sheaf is a sheaf, but a stack is a stack. Find out what this means, and more! A tour from Vector Bundles to all-too basic notions of interesting concepts floating around in the atmosphere above algebraic geometry. Stacks for Everybody for Everybody
Apr. 23, 2015 3pm (MATH 350)	Probability David Renfrew (UCLA) X We will discuss several ensembles of non-Hermitian random matrices with block structures, motivated from math and physics. We give a characterization of the limiting spectral measure and compute their spectral radius. Eigenvalue densities of Structured Random Matrices
Apr. 23, 2015 3pm (CSU Webbe…	Algebraic Geometry Richard Hain (Duke University) X Suppose that k is a field and that 2g-2+n > 0. Then one has the moduli space M_{g,n}/k of n pointed smooth projective curves of genus g over k. The generic curve of type (g,n) over k is defined to be the restriction of the universal curve over M_{g,n}/k to its generic point Spec k(M_{g,n}). This curve has n tautological rational points. Results in Teichmuller theory imply that when g > 2 and char k = 0, these are the only rational points. Recent results of Watanabe imply that this also holds when char k is positive. When n = 0, results of Hain in char 0 and Watanable in positive characteristic imply that Grothendieck's Section Conjecture holds for the generic curve of genus g > 2. This talk will survey these results and introduce the tool of weighted completion that was used to prove them. Rational points of universal curves
Apr. 24, 2015 4pm (TBA)	Kempner Vladimir Sverak (University of Minnesota) X Title: On PDE aspects of incompressible fluid flows. Abstract: We will discuss topics in PDE analysis of the incompressible Navier-Stokes and Euler equations. Certain “toy model” equations will be included. A brief exposition of classical difficulties will be given. On PDE aspects of incompressible fluid flows
Apr. 28, 2015 11am (MATH 350)	Number Theory John Boxall (Université de Caen Basse-Normandie) X Let be a CM-field of degree 2g with maximal reals subfield . The CM torus associated to is a torus over whose -rational points identify with the subgroup of whose norm down to lies in . We discuss asymptotic estimates for the number of integral points on that satisfy various properties. Counting points on CM tori
Apr. 28, 2015 12:10pm (MATH …	Kempner Charles Vial (Cambridge and IAS) X While toying with the Riemann-Roch formula for surfaces, I observed that a certain winding number in the modular group SL(2,Z) is related to the signature of a certain tridiagonal matrix M. I will of course give a precise statement, and I hope that the audience will help me understand why such a statement is true! A strange linear algebra statement
Apr. 28, 2015 2pm (Math 350)	Functional Analysis Danny Crytser (Kansas State University) X An abelian core for a C-algebra is a certain MASA which is useful in the study of representation theory. Not all C-algebras admit abelian cores, but those that do often have particularly nice structure. Two examples are the abelian core of a graph algebra, and the abelian core C(X) inside C(X) \rtimes G, where G is a countable discrete group acting topologically freely on X. We study the trace space of a graph C*-algebra as it relates to the abelian core. This is ongoing work with Gabriel Nagy. Tracial states and abelian cores Sponsored by the Meyer Fund
Apr. 28, 2015 3pm (Math 350)	Algebraic Geometry Charles Vial (Cambridge and IAS) X In the first part of the talk, I will introduce Chow groups and their relations to usual cohomology groups. The Chow groups of varieties are still largely mysterious and encompass geometric and arithmetic properties of varieties. I will review some of their important established properties and the web of conjectures that surround them. More specifically, given a smooth projective variety X, I will explain why the diagonal, when seen as an element of the Chow group of X \times X, controls the "motive" of X. In the second part of the talk, I will explain why the small diagonal {(x,x,x), x \in X} \subset X\times X\times X controls the intersection theory on X. I will then review joint work with Mingmin Shen where we give evidence that the Chow ring of hyperKaehler varieties has a similar structure as the Chow ring of abelian varieties. Examples of hyperKaehler varieties are given by K3 surfaces, and Hilbert schemes of length-n subschemes on K3 surfaces and their deformations. On diagonals and small diagonals in intersection theory Sponsored by the Meyer Fund
Apr. 29, 2015 4pm (MATH 350)	Grad Student Seminar Tyler Schrock (CU-Boulder) X Generally speaking, Morse Theory studies smooth structures on manifolds and investigates what information we can learn from them regarding the underlying topological properties. We will introduce Morse functions, prove the Morse Lemma, and state some cool theorems with examples. A Brief Introduction to Morse Theory
Apr. 30, 2015 3pm (Weber 201…	Algebraic Geometry Colin Weir (Simon Fraser) X On heuristics for a-numbers for hyperelliptic curves On heuristics for a-numbers for hyperelliptic curves
May. 15, 2015 2pm (MATH 350)	Number Theory Taku Ishii (Seikei University (Tokyo, Japan)) X I will talk about special functions related to Fourier expansions of Siegel modular forms of degree two, and application to computation of archimedean zeta integrals for some automorphic L-functions on GSp(2). Spherical functions on Sp(2,R) and archimedean zeta integrals
Jul. 14, 2015 11am (Math 220)	Noncomm Geometry Paul Bressler (Universidad de los Andes) Introduction to DG manifolds. Sponsored by the Meyer Fund
Aug. 26, 2015 4pm (MATH 350)	Grad Student Seminar Sebastian Bozlee (CU-Boulder) X An introduction to the categorical conceptualization of mathematics. Categories pt. 1
Aug. 28, 2015 4pm (MATH 350)	Grad Student Seminar Leo Herr X A further welcome to the categorical weltanschauung. Categories pt. 2

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