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Events for March – July 2015

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Mar. 04, 2015 5pm (Math 350)	Functional Analysis Frederic Latremoliere (University of Denver) X The introduction of noncommutative metric information on C-algebras opens the possibility to extend metric geometry to the realm of noncommutative geometry. In this talk, we will present an analogue of the Gromov-Hausdorff distance, called the Gromov-Hausdorff propinquity. The propinquity is a complete metric on a class of noncommutative Lipschitz algebras, induces the same topology as the Gromov-Hausdorff distance on classical compact metric spaces, yet also allows to give a precise meaning to such statements as fuzzy tori approximate quantum tori. Last, we will discuss a noncommutative generalization of a form of Gromov's compactness theorem. The talk will be a survey of these recent developments, and will only assume familiarity with basic C-algebra notions. Please note Special Date,Time and Place! The Dual Gromov-Hausdorff Propinquity Sponsored by the Meyer Fund
Mar. 03, 2015 11am (MATH 350)	Number Theory Michelle Manes (University of Hawaii at Manoa) X A fundamental observation in the study of one-dimensional complex dynamics is that the orbits of the (finite set of) critical points of a function f largely determine the dynamics of f on the whole projective line. So the study of PCF functions has a long history in complex dynamics, including Thurston's topological characterization of these maps in the 1980s and continuing to the present day. More recently, number theorists have advanced the idea of PCF functions as the right dynamical analog of elliptic curves with complex multiplication, suggesting that these functions may have special arithmetic interest as well. In this talk, I'll give an overview of some recent developments in the area of the arithmetic of PCF rational functions. Some arithmetic properties of post-critically finite rational functions
Mar. 03, 2015 1pm (MATH 220)	Grad Algebra/Logic Nikki Sanderson (CU Boulder) X In this talk, I will explain the connection between languages and automata. Focusing on regular languages and finite automata, I will present Brzozowski's algorithms to find canonical finite automata of interest - namely, the minimal DFA and the átomata. I will then briefly introduce a possible extension of Brzozowski's algorithm to the setting of stationary stochastic processes. Languages and Automata
Mar. 03, 2015 2pm (MATH 350)	Lie Theory Richard Green (CU) X Pancake sorting is the problem of sorting a disordered stack of pancakes in order of size, when a spatula can be inserted at any point in the stack and used to flip all pancakes above it. A pancake number is the maximum number of flips required for a given number of pancakes. I will survey some of what is known about these numbers and explain how they are connected to the problem of generating Coxeter groups of type A using longest elements of parabolic subgroups. I will also discuss a signed version of this problem, which relates to Coxeter groups of type B, as well as applications of these ideas to parallel processing and to genomics. Coxeter groups and pancake sorting
Mar. 03, 2015 3pm (MATH 350)	Kempner Wei Ho (University of Michigan) X The rank of an elliptic curve , defined over Q, is the rank of the finitely generated abelian group E(Q) of rational points on E. This quantity is still rather mysterious in many ways. In the last five years, there has been significant progress on understanding the average rank of elliptic curves, led by work of Bhargava-Shankar. We will survey these results and the ideas behind them, as well as mention generalizations in various directions and some corollaries of these types of theorems. We will also describe recently collected data on ranks of elliptic curves (joint work with J. Balakrishnan, N. Kaplan, S. Spicer, W. Stein, and J. Weigandt). This talk will be suitable for a general mathematical audience. Distributions of ranks of elliptic curves
Mar. 03, 2015 4pm (Math 350)	Algebraic Geometry Wei Ho (University of Michigan) X There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P^3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron-S\'everi groups contain specified lattices. These constructions, inspired by arithmetic considerations, involve explicit geometry and combinatorics. For some of these families, each K3 surface has many automorphisms with positive entropy. This is joint work with Manjul Bhargava and Abhinav Kumar. Families of lattice-polarized K3 surfaces Sponsored by the Meyer Fund
Mar. 04, 2015 4pm (MATH 220)	Grad Student Seminar Sebastian Bozlee (CU-Boulder) X A recurrence matrix is a matrix whose entries are sequential members of a linear homogeneous recurrence sequence. These matrices always have low rank. In this talk, we will explain what exactly the ranks are and why, how to correctly answer a question on your linear algebra final, and why pseudo-random numbers sometimes suck. Ranks of matrices whose elements come from a recurrence relation
Mar. 05, 2015 3pm (Webber 20…	Algebraic Geometry Beth Malmskog (Villenova) X Picard curves are genus 3 curves of the form y^3=f(x) where f(x) is a polynomial of degree 4. They are the simplest non-hyperelliptic curves. This talk will discuss recent work with Chris Rasmussen, in which we found all Picard curves defined over the rationals with good reduction at all primes except p=3. This work was inspired by Nigel Smart's enumeration of genus 2 curves with good reduction at all primes except p=2. This work is relevant to the study of modular curves, and employs several powerful tools, including Baker's method and the LLL algorithm. Picard curves with good reduction away from p=3
Mar. 10, 2015 12:10pm (MATH …	Kempner David Zureick-Brown (Emory University) X Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers satisfying , Darmon and Granville proved that the individual generalized Fermat equation has only finitely many coprime integer solutions. Conjecturally something stronger is true: for there are no non-trivial solutions. I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture. Diophantine and tropical geometry
Mar. 10, 2015 2pm (MATH 350)	Lie Theory Sarah Salmon (CU) X The Temperley—Lieb Algebra of type A is a finite dimensional associative algebra that arose in the context of statistical mechanics and can be realized in terms of certain diagrams. These diagrams can be indexed by the fully commutative elements of the Coxeter group of type A. Similarly, there is also a diagrammatic representation for the Temperley—Lieb algebra of type B involving decorated diagrams. Multiplying diagrams is easy to do. However, taking a given diagram and finding the corresponding reduced factorization is not so easy. We have an efficient (and colorful) algorithm for obtaining a reduced factorization for Temperley-Lieb diagrams of types A and B. Diagram factorizations in Temperley—Lieb algebras
Mar. 10, 2015 3pm (Math 350)	Algebraic Geometry David Zureick-Brown (Emory University) X We give a generalization to stacks of the classical theorem of Petri -- i.e., we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. (The talk will be mostly geometric and will require little understanding of modular forms.) This is joint work with John Voight. The canonical ring of a stacky curve Sponsored by the Meyer Fund
Mar. 10, 2015 3pm (Math 220)	Functional Analysis Krystal Taylor (Institute for Mathematics and its Applications, Minnesota) X We study the existence of certain geometric configurations in subsets of Euclidean space. In particular, we establish that a set with ``sufficient structure" contains an arbitrarily long chain with vertices in the set and preassigned admissible gaps. A k-chain in with gaps is a sequence In order to prove this result, we establish mapping properties of the convolution operator where is a tempered distribution, and and are compactly supported measures satisfying the growth bounds and . This is joint work with A. Iosevich and M. Bennett. Finite point configurations and Fourier analysis Sponsored by the Meyer Fund
Mar. 11, 2015 4pm (MATH 350)	Grad Student Seminar Hanson Smith Elliptic Curves, Modular Curves, the Proof of Fermat's Last Theorem, and Donuts.
Mar. 12, 2015 3pm (MATH 350)	Probability Anatolii Puhalskii (CU Denver) X We consider two Ito equations that evolve on different time scales. The equations are fully coupled in the sense that all of the coefficients may depend on both the ``slow'' and the ``fast'' variables and the diffusion terms may be correlated. The diffusion term in the slow process is small. A large deviation principle is obtained for the joint distribution of the slow process and of the empirical process of the fast variable. By projecting on the slow and fast variables, we arrive at new results on large deviations in the averaging framework and on large deviations of the empirical measures of ergodic diffusions, respectively. The proof relies on the property that an exponentially tight sequence of probability measures on a metric space is large deviation relatively compact. The identification of the large deviation rate function is accomplished by analyzing the large deviation limit of an exponential martingale. On large deviations of coupled diffusions with time scale separation
Mar. 12, 2015 3pm (Webber 20…	Algebraic Geometry Jason Cantarella (University of Georgia) X Knutson-Hausmann and Howard-Manon-Millson have given descriptions of the space of polygons of fixed edge lengths in R^3 as symplectic reductions of complex Grassmannians or products of spheres. These structures turn out to be genuinely useful in analyzing the geometric probability theory of random closed walks, and we give some example theorems along these lines. We compare these to the picture of polygon space as a GIT quotient. The applications of this picture are currently more mysterious. The talk describes joint work with Clayton Shonkwiler (CSU), Tetsuo Deguchi (Ochanomizu University), Rob Kusner (U Mass), and Alexander Grosberg (NYU). Geometry of Polygon Spaces
Mar. 13, 2015 3pm (MATH 350)	Number Theory Paula Tretkoff (Texas A&M) X A well-know construction of Kuga and Satake associates to the polarized level 2 Hodge structure on a K3 surface a polarized level 1 Hodge structure giving rise to the corresponding Kuga-Satake (abelian) variety. We show how this enables one to prove, for algebraic K3 surfaces, the analogue of Th. Schneider's result that the elliptic modular function j(t) takes an algebraic value at an algebraic argument t if and only if the lattice Z+Zt admits complex multiplications. The proof hinges on results from transcendental number theory. The talk will include an introduction to the Kuga-Satake construction. A transcendence criterion for complex multiplication on algebraic K3 surfaces [NOTE DAY/TIME]
Mar. 16, 2015 12pm (Math 350)	Noncomm Geometry Maxim Braverman (Northeastern University) X In 1984 Michael Berry discovered that an isolated eigenstate of an adiabatically changing periodic Hamiltonian acquires a phase, called the Berry phase. Barry Simon gave an interpretation of this phase in terms of the holonomy of a certain Hermitian line bundle. There are several situation described in physical literature when the Berry phase is claimed to be equal to the phase of the determinant of the corresponding imaginary-time Schroedinger operator. However not only rigorous proofs but even the accurate formulations of these results are missing in the literature. In this talk we establish and prove the precise relationship between the phase of this determinant and the Berry phase under the most general assumption on the Hamiltonian. The previously known examples are the special cases of this formula. The Berry phase and the phase of the determinant Sponsored by the Meyer Fund
Mar. 17, 2015 12:10pm (MATH …	Kempner Jordan Watts (University of Illinois Urbana--Champaign) X We will begin by looking at what an orbifold is in the classical way. Then, fixing an orbifold, we forget all of the information contained in the orbifold structure except for the topology, stratification, and some integer labels on certain strata. From here, we reconstruct the entire orbifold structure. This is important, because the "minimal data" contained in the topology, stratification, and labels can all be obtained from the ring of smooth functions on an orbifold. More categorically, there is an essentially injective functor from the "category" of orbifolds (which has different definitions depending on your perspective, not all equivalent) to the category of so-called differential spaces: sets equipped with sheaves of "smooth" functions with "smooth" maps between them. This a priori is unexpected, since in general the ring of smooth functions on a space built out of quotients, such as an orbifold, forgets the local quotient structure. The differential structure of an orbifold
Mar. 17, 2015 2pm (MATH 350)	Lie Theory C. Ryan Vinroot (William and Mary) X In the mid 1970s, Bernstein-Zelevinsky and Gelfand-Kazhdan proved that for any irreducible admissible representation \pi of GL(n,F), F a p-adic field, the dual of \pi is isomorphic to \pi twisted by the inverse-transpose map. The proof depends heavily on sheaf theory applied to distributions of totally disconnected locally compact groups. Since then, similar statements have been proved for other p-adic groups, all depending on the same algebro-geometric arguments. In 2006, Tupan gave an elementary proof of the statement for GL(n,F), depending only on linear algebra and topological properties. In joint work with Alan Roche (U. Oklahoma), we are able to extend Tupan's argument to the orthogonal and symplectic p-adic groups, and their groups of similitudes, using only linear algebra statements, p-adic topology, and some lie theory. Dual representations of classical p-adic groups Sponsored by the Meyer Fund
Mar. 17, 2015 3pm (Math 220)	Functional Analysis Palle Jorgensen (University of Iowa) X The talk will have two sides, one tilted towards applications, and the other representation theory. The applications are from symbolic dynamics, fractals, and from computational harmonic analysis, e.g., discrete wavelet algorithms; both standard wavelets and wavelets on fractals. The representations appearing in the talk refer to classes of certain non-abelian algebras (one in particular, the Cuntz-algebra.) We will show that a number of questions which are natural from the point of representation theory (equivalence classes of irreducibles, restriction, induction, decompositions etc) are key to solving questions in the applied areas mentioned. Atomic and non-atomic representations: algebras in analysis and in symbolic dynamics Sponsored by the Meyer Fund
Mar. 18, 2015 4pm (MATH 350)	Grad Student Seminar Mark Coffey (Department of Physics, Colorado School of Mines) X This talk introduces the Mellin and generalized Mellin transforms, reviewing several of their important properties, and briefly discussing mathematical applications. With also an introduction of hypergeometric functions, generalized Mellin transforms are developed that have polynomial factors with zeros confined to a vertical line in the complex plane. Properties of these polynomials including recurrences and functiona lequations are described. Connections with other areas of mathematics will be mentioned .Joint work with Matthew Lettington, Cardiff University. Mellin transforms with factors with zeros only on the critical line
Mar. 19, 2015 3pm (MATH 350)	Probability Juan Carlos Pardo Millan (CIMAT) X Motivated by the work of Boeinghoff and Huznethaler on branching diffusions on random environment, we introduce continuous state branching processes in a Brownian random environment. A process in such class is defined as a strong solution of an stochastic differential equation and satisfies a quenched branching property. Special attention is given to the case when the branching mechanism has finite mean, where we can provide more explicit results on long- term behaviour and we can include an immigration term. Restricting our attention to the stable case, we show that there two different behaviour depending on the scaling index . If , then the process can explode at finite time with positive probability and three different regimes appear. For the case , we show that four regimes arise for the speed of extinction, similarly to the discrete case. The precise asymptotics for the speed of extinction allow us to introduce the so called Q-process or process condition to be never extinct. This is a joint work with Sandra Palau. Continuous state branching processes in a Brownian random environment
Mar. 20, 2015 4pm (MATH 350)	Grad Student Seminar Kevin Berg (CU-Boulder) X What exactly are we doing when we prove mathematical statements? Are we demonstrating fundamental truths about the nature of reality? Are we simply engaging in a logical game with agreed upon rules? Or is something else entirely afoot? In this unsettlingly math-light talk, we will explore the basic schools of thought that govern the philosophical interpretations of mathematics, raise questions for which no definitive answers can be provided, and generally infringe upon another discipline’s academic turf for our own amusement. While pizza will be provided, the question of whether or not the truth of its delivery was encoded into logical statements concerning it prior to its arrival will remain unanswered. Gödel Drinking Hemlock - An Introduction to the Philosophy of Mathematics
Mar. 26, 2015 3pm (Webber 20…	Algebraic Geometry David Shea Vela-Vick (Louisiana State University) X We present a refinement of the Ozsvath-Szabo contact invariant in Heegaard Floer theory. This invariant, denoted b, takes values in the positive integers union infinity, and extends the usual contact invariant in the sense that if c(Y,\xi) is nonzero, then b is infinity. We further show that if (Y,\xi) is overtwisted, then b(Y,\xi) = 1, reflecting the usual vanishing of the usual contact invariant for such contact structures. In this talk, we will focus on the construction of b and discuss some of its basic properties and applications. A refinement of the Ozsvath-Szabo contact invariant
Mar. 31, 2015 11am (MATH 350)	Number Theory John Cullinan (Bard College) X Let A be an abelian variety defined over a number field K has the following property: Fix an integer m > 1 and for each good prime p, suppose the number of (F_p)-points of A mod p is divisible by m. Does there exist a K-isogenous abelian variety A' with rational torsion divisible by m? This question was first posed by Lang and then taken up by Katz in the early 1980s where he showed the answer is `Yes' for elliptic curves and `No' for threefolds and higher. The case of abelian surfaces is more difficult and only some partial results are known. We will survey what is known along with some recent work on the problem. In particular, we will focus on the subtlety of abelian surfaces and present new results towards a complete answer to this question. Some l-adic representations attached to low dimensional abelian varieties
Mar. 31, 2015 12:10pm (MATH …	Kempner Karola Meszaros (Cornell University) X The flow polytope associated to an acyclic graph is the set of all nonnegative flows on the edges of the graph with a fixed netflow at each vertex. We will examine flow polytopes arising from permutation matrices, alternating sign matrices and Tesler matrices. Our inspiration is the Chan-Robins-Yuen polytope (a face of the polytope of doubly-stochastic matrices), whose volume is equal to the product of the first n Catalan numbers (although there is no known combinatorial proof of this fact!). The volumes of the polytopes we study all have nice product formulas. Product formulas for volumes of flow polytopes
Mar. 31, 2015 2pm (MATH 350)	Lie Theory Lionel Levine (Cornell) X I will describe the role played by an Apollonian circle packing in the scaling limit of the abelian sandpile on the square grid Z^2. The sandpile solves a certain integer optimization problem. Associated to each circle in the packing is a locally optimal solution to that problem. Each locally optimal solution can be described by an infinite periodic pattern of sand, and the patterns associated to any four mutually tangent circles obey an analogue of the Descartes Circle Theorem. Joint work with Wesley Pegden and Charles Smart. Circles in the Sand Sponsored by the Meyer Fund
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

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