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Events for November 2014 – March 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
Nov. 04, 2014 11am (Math 220)	Noncomm Geometry Raphaël Ponge (Seoul National University) X This will be report on a series of joint papers with Hang Wang (U. Adelaide, Australia). The general aim of these papers is to use tools from noncommutative geometry to study conformal geometry and some noncommutative incarnations of conformal geometry. In this talk, I will present two main results. The first main result is a reformulation of the local index formula in the setting of conformal-diffeomorphism invariant geometry. The second main result is the construction of a new family of global conformal invariants taking into account the action of the group of conformal-diffeomorphisms. These invariants are not of the same type as the conformal invariants considered by Spyros Alexakis in his celebrated solution of the Deser-Schwimmer. The arguments leading to both results heavily rely on noncommutative geometry. In particular, a crucial use is made of the conformal invariance of the Connes-Chern character. However, the main results are ultimately stated in differential-geometric fashion. Noncommutative geometry and conformal geometry: local index formula and conformal invariants.
Nov. 04, 2014 11am (MATH 350)	Number Theory Caleb Shor (Western New England University) X Given an algebraic curve C of genus g, we say a point P is a Weierstrass point if dim(L(gP))>1, where L(D) is the Riemann-Roch space associated to a divisor D. One can generalize this to define a higher-order Weierstrass point, and one can also talk about the weight of a Weierstrass point. In this talk, we will consider Weierstrass points on superelliptic curves, which are curves of the form y^n=f(x) for f(x) a polynomial of degree d. Under a mild hypothesis, it is well known that the branch points of such curves are Weierstrass points. We will investigate the weights of these branch points for given values of n and d and use them to obtain some asymptotic results. We will also look at the various weight distributions that can arise on certain families of hyperelliptic curves of low genus and prescribed automorphism groups. Ordinary and higher-order Weierstrass points on cyclic covers of the projective line
Nov. 04, 2014 3pm (Math 350)	Algebraic Geometry Pál Zsámboki (University of Washington) X Starting with linear endomorphisms of a 2-dimensional complex vector space, and moving on to line bundles on a real conic, I will discuss how the problem of finding a moduli space can involve having to extend our notion of geometrical space. After that, I will talk about how a choice of a meaningful compactification of the moduli space of vector bundles on a surface has led to the proof of irreducibility of said moduli space. Introduction to moduli spaces and compactification Sponsored by the Meyer Fund
Nov. 04, 2014 4pm (Math 350)	Algebraic Geometry Pál Zsámboki (University of Washington) X I will discuss an approach to getting a meaningful boundary to the moduli space of G-bundles on a smooth, proper scheme using derived objects. Homotopical Algebraic Geometry will arise as the natural framework to treat this problem in. No knowledge of higher stacks will be assumed. Towards the compactification of the moduli space G-torsors via perfect Lie algebra complexes Sponsored by the Meyer Fund
Nov. 05, 2014 4pm (MATH 350)	Grad Student Seminar Jonathan Lamar X Line up a deck of cards on a table and repeatedly switch pairs of them at random. One might ask how many switches are necessary and/or sufficient to produce a deck which is well-shuffled. Answering this question and others like it motivates the study of random walks on groups, which has surprising connections to representation theory. I will provide a gentle introduction to this subject. If time permits, I will also discuss the result of Diaconis and Shahshahani (1981) that for the above shuffling process on an n card deck, a rapid shift from low to high entropy occurs after about (n/2)*log(n) switches. From Card Shuffling to Representation Theory and Back
Nov. 06, 2014 3pm (Weber 201…	Algebraic Geometry Melody Chan (Harvard) X The genus of Brill-Noether curves I'll discuss an ongoing joint project with Lopez, Pflueger, and Teixidor, in which we use a surprising theorem on random standard Young tableaux to give a new proof, using degenerations and limit-linear series, of the classical formula of Eisenbud-Harris and Pirola for the genus of the Brill-Noether curve. No background will be assumed. Topology of tropical moduli spaces The moduli space of n-marked, genus g tropical curves is a cell complex that was identified in work of Abramovich-Caporaso-Payne with the boundary complex of the complex moduli space M_{g,n}. I will report on new results on the topology of tropical M_{1,n} and M_{2,n}, obtaining as corollaries new calculations of the top-weight cohomology of the complex moduli spaces M_{1,n} and M_{2,n}. Joint work, in part, with Galatius and Payne. The genus of Brill-Noether curves
Nov. 11, 2014 11am (Math 220)	Noncomm Geometry Josh Frinak X If is morphism of complex vector bundles over a smooth manifold which is an isomorphism over a subset , then determines a class in the relative -group . I intend to present Quillen's construction of the Chern character which involves the use of a -graded variant of the concept of a connection called a superconnection. Given a convenient choice of superconnection we are interested in the curvature . It will turn out that the super trace of will represent the Chern character, , in de Rham cohomology. Chern character via superconnections.
Nov. 11, 2014 11am (MATH 350)	Number Theory Leonhard Summerer (University of Vienna) X In this talk I want to introduce a method based on the geometry of numbers and in particular on the study of the successive minima functions with respect to a certain lattice and a suitable one-parameter family of convex bodies. This approach leads to the definition of some new approximation constants and in turn to some new transference inequalities. The related geometric picture in conjunction with some recent results of D. Roy motivates some best possible bounds in the case of simultaneous approximation of several real numbers. Geometry of Numbers and simultaneous Approximation
Nov. 11, 2014 12:10pm (MATH …	Kempner Anca Radulescu (SUNY New Paltz) X The behavior of orbits for iterated logistic maps has been widely studied since the dawn of discrete dynamics as a research field. Existing results refer not only to the family of real polynomials , , for , but also to the context of complex maps , parametrized as , with . However, little is is known about orbit behavior if the map changes along with the iterations. \\ \noindent We investigate how the theory changes if the dynamical scheme involves two functions, and , iterated according to a prescribed binary sequence of s and s. In particular, we observe the effects of the structure of the symbolic sequence (periodicity, complexity, etc) on the complexity of the resulting system and (visually) on the topological structure of its Julia set.\\ \noindent This direction is of potential interest to a variety of applications (including genetic and neural coding), since it investigates how an occasional or a reoccurring error in a replication or learning algorithm may affect the outcome. Real and complex dynamics for symbolic sequences of logistic maps
Nov. 11, 2014 2pm (MATH 350)	Lie Theory Derek Wise (University of Erlangen) X While groups formalize the notion of symmetry, "2-groups" include also the idea that symmetries might be related to each other by symmetries of their own. I will explain 2-groups, which are categories equipped with a group-like multiplication, and discuss their representation theory. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to vector spaces. For a simple reason that I will explain, the most interesting representations of Lie 2-groups are all infinite-dimensional. I will describe Lie 2-group representations on an appropriate kind of infinite-dimensional 2-vector space, and show that these can be nicely understood in terms of objects familiar from ordinary group representation theory and geometry. Higher Symmetry: Lie 2-Groups and their Representations Sponsored by the Meyer Fund
Nov. 13, 2014 3pm (Weber 201…	Algebraic Geometry Dmitry Zakharov (NYU) X Given an algebraic curve with marked points and integer weights on the markings, we can consider the corresponding divisor. This defines an Abel--Jacobi map from the moduli space of marked curves of compact type to the universal Jacobian variety. Pulling back relations from the universal Jacobian for various values of the weights gives a plethora of tautological relations on the space of curves of compact type. These relations include those discovered by Faber, Getzler, Belorousski and Pandharipande, and Tavakol, and this method was also used by Hain to compute the double ramification cycle. I will talk about the extension of the Abel--Jacobi map to a larger moduli space of marked curves, the corresponding extended formula for the double ramification cycle, and Pixton's conjectures about the extensions of these relations to the entire Deligne-Mumford compactification. The Abel--Jacobi map and relations in the tautological ring
Nov. 13, 2014 3pm (MATH 350)	Probability Douglas Rizzolo (U. Washington) X We will discuss the motion a tagged gas particle in a gravitational field. Our starting point will be a Markov approximation to a Lorentz gas model with variable density. We investigate how the density of the ambient gas impacts the recurrence or transience of the tagged particle and we will show that there are multiple scaling regimens leading to nontrivial diffusive limits. This talk is based on joint work with Krzysztof Burdzy. A gas particle in a gravitational field
Nov. 14, 2014 4pm (Math 350)	Functional Analysis Christopher Seaton (Rhodes College, Memphis, TN) X In many applications, one is interested in a topological groupoid whose object and arrow spaces are not smooth manifolds yet do come equipped with a natural notion of smoothness. Often, this notion of smoothness can be understood by giving each space the structure of a differentiable or differentiable stratified space. In this talk, we will discuss differentiable spaces, differentiable stratified spaces, and differentiable stratified groupoids. We will give examples of naturally occurring differentiable stratified groupoids and consider some of the consequences of these structures. Differentiable stratified groupoids (joint work with Carla Farsi and Markus Pflaum) Sponsored by the Meyer Fund
Nov. 18, 2014 11am (Math 220)	Noncomm Geometry Alex Nita Fredholm Modules
Nov. 18, 2014 11am (MATH 350)	Number Theory Bianca Viray (University of Washington) X In 1970, Manin showed that the Brauer group can obstruct the existence of rational points. Colliot-Thélène and Sansuc have conjectured that this obstruction completely explains the failure of rational points on del Pezzo surfaces. We show that on degree 4 del Pezzo surfaces, this Brauer-Manin obstruction manifests itself through linear projections. As a consequence of the proof, we obtain a simple and efficient for computing the Brauer classes of a degree 4 del Pezzo surface. This is joint work with Anthony Várilly-Alvarado. Obstructions to the Hasse principle on degree 4 del Pezzo surfaces
Nov. 18, 2014 12:10pm (MATH …	Kempner Robert Varley (University of Georgia) X Rational homotopy theory as developed by Quillen and Sullivan allows a complete description of its objects of study by differential graded algebras over the rational numbers. Thus rationally some of the basic classification problems in homotopy theory become "moduli problems" in algebraic geometry and number theory. Halperin, Schlessinger, and Stasheff showed how to describe all rational homotopy types having a given cohomology algebra. I will discuss this moduli construction and how completion of the rationals is useful for its analysis. In particular, homotopy theory over the finite adeles seems to accommodate the known fact that even for smooth projective varieties defined over a number field, the rational homotopy type of the associated topological space can depend on the embedding of the number field in the complex numbers. Rational homotopy theory and its completions
Nov. 18, 2014 3pm (Math 350)	Algebraic Geometry Robert Varley (University of Georgia) X I will recount some of the interesting features of principally polarized abelian varieties and their theta divisors, emphasizing the case of Jacobians of curves, connections with linear series, and the intensive study made by Kempf of the presentation matrix for the determinantal structure of Jacobian theta divisors. The analogous setup for Pryms is more complicated. Roy Smith and I have been trying to supplement Casalaina-Martin's multiplicity formula by using a natural geometric parametrization of the tangent cone. I will describe the skew-symmetric presentation matrix for the (local) Pfaffian structure of Prym theta divisors. It is then clear which essential higher order information is still lacking for very exceptional singular points, and I will explain how the unknown 2nd order terms seem to be governed by a Massey product structure. Towards understanding all the singular points of Prym theta divisors Sponsored by the Meyer Fund
Nov. 19, 2014 4pm (MATH 350)	Grad Student Seminar Leo Herr (CU-Boulder) In Defense of Projective Space
Nov. 20, 2014 3pm (MATH 350)	Probability Ke Wang (IMA) X We start with a simple, yet useful, concentration inequality concerning random weighted projections in high dimensional spaces. The inequality is then used to prove a general concentration inequality for random quadratic forms. In another application, we show the optimal infinity norm O(sqrt{(log n)/n}) for most unit eigenvectors of a large class of random matrices, including the adjacency matrix of random graphs. This is joint work with Van Vu. Random weighted projections, random quadratic forms and random eigenvectors
Nov. 20, 2014 3pm (Weber 201…	Algebraic Geometry Maria Angelica Cueto (Columbia University) X The aim of my two talks will be to show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in the sense of Berkovich. Our proof is constructive and it relies on the combinatorial description of the tropical Grassmannian as a space of phylogenetic trees by Speyer-Sturmfels. In the first talk, I will discuss the connection between tropical geometry and analytification of algebraic varieties, and explain the notion of faithful tropicalization. I will also describe the combinatorics of the aforementioned space of trees inside tropical projective space. The second talk will be devoted to the proof of the main theorem and its interpretation in terms of the tropicalization map. Namely, the fiber over each point is an affinoid domain with a unique Shilov boundary point. Our homeomorphism identifies points in the tropical Grassmannian with the unique Shilov boundary point on their fibers, allowing for further generalizations of our results by Gubler-Rabinoff-Werner. This is joint work with M. Haebich and A. Werner. Faithful tropicalization of the Grassmannian of planes
Dec. 01, 2014 4pm (MATH 350)	PDE/Analysis David Ebin (Stony Brook University) X We will consider the initial value problem for the motion of incompressible inviscid fluids with free boundary and surface tension on the boundary. We will show how to find solutions. We also show that If the initial boundary has constant mean curvature, then the motion converges to motion of a fluid with fixed boundary as the surface tension constant gets large. We shall also show that the absence of surface tension can lead to an ill-posed problem. The initial value problem for fluids with free boundary and surface tension Sponsored by the Meyer Fund
Dec. 02, 2014 11am (MATH 350)	Number Theory Doug Ulmer (Georgia Tech) X For every genus g>0 and most primes p, we write down a curve over Fp(t) with Jacobian of large rank and with explicit divisors to fill out a finite index subgroup of the Mordell-Weil group. The height pairings among the explicit points have a very nice group-theoretic description, and we have an analytic class number formula of the type \|sha\|=square of index of the explicit points in the full Mordell-Weil group. Explicit high ranks in higher genus
Dec. 02, 2014 11am (Math 220)	Noncomm Geometry Alex Nita Fredholm Modules (continued)
Dec. 02, 2014 2pm (MATH 350)	Number Theory Katherine Stange (University of Colorado Boulder) X This is a number theory talk of interest to those who care about matrix groups. A Schmidt arrangement is the orbit of the real line under the Bianchi group PSL(2,OK) (realised as Mobius transformations) where OK is the ring of integers in an imaginary quadratic field K. The arrangement takes the form of a dense collection of intricately nested circles in the complex plane. I'll explain how the number theory of K influences the arrangement, and I'll use these arrangements to generalise Apollonian circle packings and define a new collection of thin groups of arithmetic interest. In other words, for those who walk the third floor, I will be explaining the circles on my door. The talk will require some basic algebraic number theory (e.g. the ideal class group), and a passing familiarity with Mobius transformations. (SPECIAL TIME) The Apollonian structure of quadratic imaginary fields / cross listed with Lie Theory Seminar
Dec. 02, 2014 2pm (MATH 350)	Lie Theory Katherine Stange (University of Colorado Boulder) X This is a number theory talk of interest to those who care about matrix groups. A Schmidt arrangement is the orbit of the real line under the Bianchi group PSL(2,OK) (realised as Mobius transformations) where OK is the ring of integers in an imaginary quadratic field K. The arrangement takes the form of a dense collection of intricately nested circles in the complex plane. I'll explain how the number theory of K influences the arrangement, and I'll use these arrangements to generalise Apollonian circle packings and define a new collection of thin groups of arithmetic interest. In other words, for those who walk the third floor, I will be explaining the circles on my door. The talk will require some basic algebraic number theory (e.g. the ideal class group), and a passing familiarity with Mobius transformations. The Apollonian structure of quadratic imaginary fields
Dec. 02, 2014 3pm (Math 350)	Algebraic Geometry Martin Ulirsch (Brown University) X This talk will begin with a quick introduction to the topology of Berkovich analytic spaces and the tropical geometry of toric varieties. In particular, we are going to talk about skeletons of toric varieties and their relationship to tropicalization. Moreover, I am going to explain how a generalization of these notions to the setting of toroidal embeddings, open embeddings that are étale locally isomorphic to the torus embedding of a toric variety, can be used to study the tropical geometry of moduli spaces. Part I: Tropicalization of toric varieties and moduli spaces Sponsored by the Meyer Fund
Dec. 02, 2014 4pm (Math 350)	Algebraic Geometry Martin Ulirsch (Brown University) X Recent work by J. and N. Giansiracusa, myself, and O. Lorscheid suggests that the tropical geometry of a toric variety X, or more generally of a logarithmic scheme X, can be formalized as a "Berkovich analytification" of a scheme over the field with one element that is canonically associated to X. The goal of this talk is to introduce the theory of Artin fans, originally due to D. Abramovich and J. Wise, which can be used to lift rather unwieldy F₁-geometric objects to the more familiar realm of algebraic stacks. Artin fans are étale locally isomorphic to quotient stacks of toric varieties by their big tori and their glueing data has a completely combinatorial description in terms of Kato fans. I am going to explain how to use the ideas surrounding the notion Artin fans to study tropicalization maps associated to toric varieties and logarithmic schemes. Surprisingly these techniques allow us to give a reinterpretation of Tevelev's theory of tropical compactifications that can be generalized to compactifications of subvarieties in logarithmically smooth compactifcations of smooth varieties. For example, we can introduce definitions of tropical pairs and schoen varieties in terms of Artin fans that are equivalent to Tevelev's notions. Part II: Artin fans in tropical geometry Sponsored by the Meyer Fund
Dec. 03, 2014 4pm (MATH 350)	Grad Student Seminar Steve Weinell (CU-Boulder) X In this talk we will attempt to state and then give a rough outline of the proof of G\"{o}del's first incompleteness theorem. We will explore the notions of G\"{o}del numbers and recursive functions and relations, along with explanations of basic concepts in mathematical logic as needed. We will then state without proof some key theorems and outline the proof of G\"{o}del's incompleteness. This talk will follow notes written by Dr. Don Monk. Godel's First Incompleteness Theorem
Dec. 09, 2014 9am (MATH 350)	PDE/Analysis Boris Khesin (University of Toronto) KAM theory and the 3D Euler equation Sponsored by the Meyer Fund
Dec. 09, 2014 11am (Math 220)	Noncomm Geometry Alex Nita Fredholm Modules (continued)
Dec. 09, 2014 2pm (Math 350)	Functional Analysis Elizabeth Gillaspy (University of Colorado - Boulder) X This talk will be a merged seminar of the "Analysis and Groupoids" seminar and the "Algebraic Lie Theory" seminar. After a quick overview of the "deformations of Lie algebras" picture of quantum groups, I will introduce a few of the operator algebraic approaches of quantum groups. The idea of twisting a quantum group by a 2-cocycle has also received a fair amount of attention; I will explain this construction and discuss how it connects to the twisted groupoid C-algebras I have studied. [joint with Algebraic Lie Theory] Quantum Groups as C-algebras and Lie algebras Sponsored by the Meyer Fund
Dec. 09, 2014 2pm (Math 350)	Lie Theory Elizabeth Gillaspy X This talk will be a merged seminar of the "Analysis and Groupoids" seminar and the "Algebraic Lie Theory" seminar. After a quick overview of the "deformations of Lie algebras" picture of quantum groups, I will introduce a few of the operator algebraic approaches of quantum groups. The idea of twisting a quantum group by a 2-cocycle has also received a fair amount of attention; I will explain this construction and discuss how it connects to the twisted groupoid C-algebras I have studied. Quantum Groups as C-algebras and Lie algebras Sponsored by the Meyer Fund
Dec. 09, 2014 3pm (MATH 350)	Kempner Boris Khesin (University of Toronto) X A plane curve is called nondegenerate if it has no inflection points. How many classes of closed nondegenerate curves exist on a sphere? We are going to see how this geometric problem, solved in 1970, reappeared along with its generalizations in the context of the Korteweg-de Vries and Boussinesq equations. Its discrete version is related to the 2D pentagram map defined by R.Schwartz in 1992. We will also describe its generalizations, pentagram maps on polygons in any dimension and discuss their integrability properties. This is a joint work with Fedor Soloviev. Pentagram maps and nondegenerate curves Sponsored by the Meyer Fund
Jan. 21, 2015 11am (Math 350)	Functional Analysis Sooran Kang (University of Otago) X We review KMS states on C-algebras associated to directed graphs and higher-rank graphs. Then we discuss recent joint work with Iain Raeburn and Astrid an Huef on spatial realizations of KMS states on C-algebras associated to higher-rank graphs. KMS states on C*-algebras associated to higher-rank graphs Sponsored by the Meyer Fund
Jan. 30, 2015 4pm (MATH 350)	PDE/Analysis Jae Min Lee (CU Boulder) X We show that if the dispersion of the KdV equation is replaced by a higher order dispersion , where is an odd integer, then the critical Sobolev exponent for local well-posedness on the circle does not change. That is, the resulting equation is locally well-posed in , . In this first talk, I will introduce the solution space for the IVP for KdV, state the lemmas, and state and prove the main theorem. Keywords: KdV equation, Local well-posedness, Sobolev spaces, Fourier transform. Well-posedness of KdV with higher dispersion - Part 1
Feb. 03, 2015 1pm (MATH 220)	Grad Algebra/Logic Jason Hill (CU Boulder (doctoral candidate) and SpotXchange (data scientist)) X The aim of these talks is to discuss the algorithms and data structures used to examine finite permutation groups, and to question where parallel processing can be a benefit. In the first talk, we'll aim to answer the following questions: How are permutations stored and multiplied on a computer? How are permutation groups stored, and how do we determine invariants such as group order? How are stabilizer subgroups computed? Parallel Permutation Group Algorithms
Feb. 03, 2015 2pm (Math 220)	Functional Analysis Marty Walter (University of Colorado, Boulder) Negative Definite Functions Sponsored by the Meyer Fund
Feb. 04, 2015 9am (MATH 350)	PDE/Analysis Jae Min Lee (CU Boulder) X We show that if the dispersion of the KdV equation is replaced by a higher order dispersion , where is an odd integer, then the critical Sobolev exponent for local well-posedness on the circle does not change. That is, the resulting equation is locally well-posed in , . In this second talk, the results leading to the main Theorem (proved in Part 1) will be established. Keywords: KdV equation, Local well-posedness, Sobolev spaces, Fourier transform. Well-posedness of KdV with higher dispersion - Part 2
Feb. 04, 2015 4pm (MATH 350)	Grad Student Seminar Robert Hines (CU-Boulder) X e is transcendental [and maybe pi -PIZZA PIE- not that I'd describe Blackjack pizza as transcendent, but I digress]. You know that e is transcendental, right? Have you seen a proof? Well here's your big chance! We'll warm up by showing that e is irrational and proving the existence of transcendental numbers (a la Liouville). Hermite's (1873) proof of the transcendence of e uses nothing more than basic calculus, arithmetic, and a heaping tablespoon of genius. Time permitting (or forcing), we will discuss generalizations due to Lindemann-Weierstrass, Gelfond-Schneider, and Baker. Trancendenz
Feb. 10, 2015 11am (MATH 350)	Number Theory Rachel Pries (Colorado State University) X We study the relationship between the p-rank of a curve and the p-ranks of the Prym varieties of its cyclic covers in characteristic p >0. For all p and g, we generalize a result of Nakajima by proving that the Prym varieties of all unramified cyclic degree L covers of a generic curve X of genus g and p-rank f are ordinary. When p > 3, we prove that there exists a curve of genus g and p-rank f having an unramified degree 2 cover whose Prym is almost ordinary. Using work of Raynaud, we use these two theorems to prove results about the (non)-intersection of the L-torsion group scheme with the theta divisor of the Jacobian of a generic curve X of genus g and p-rank f. On the existence of ordinary and almost ordinary Pryms
Feb. 10, 2015 12:10pm (MATH …	Kempner Jim Carlson (University of Utah) X As Cantor showed us, there are uncountably many transcendental numbers. Despite being overwhelmingly more numerous than algebraic numbers, it is not so easy to find them in Nature: we know, for example, that pi is transcendental, but proving this is a major result. We will discuss these facts, then go on to discuss the Kontsevich-Zagier theory that most transcendentals belong to a countable class, the periods of integrals of algebraic functions with rational coefficients. Finally, we consider Yoshinaga’s number: a specific number that is not a period. Transcendental numbers and periods
Feb. 10, 2015 1pm (MATH 220)	Grad Algebra/Logic Jason Hill (CU Boulder (doctoral candidate) and SpotXchange (data scientist)) Parallel Permutation Group Algorithms
Feb. 10, 2015 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X This talk is an introduction to random walks on groups and some of the tools in representation theory used to study these walks. In particular, we will explore conditions under which a group's representation theory can still be used to study clumped variants of the group walk. At the heart of this talk is a family of unipotent groups, whose walks have a wide variety of flavors, including walks on positive integers, hypercubes, bounded partitions and more. This work is joint with Theplertboon. Superclass walks on unipotent groups
Feb. 10, 2015 3pm (Math 220)	Functional Analysis Judith Packer (University of Colorado, Boulder) X We review the quantum Heisenberg manifolds of M. Rieffel, who originally defined them as generalized fixed point algebras. These are shown to be twisted groupoid algebras as first defined by A. Kumjian. This is joint work with S. Kang and A. Kumjian. Quantum Heisenberg manifolds as twisted groupoid algebras Sponsored by the Meyer Fund
Feb. 11, 2015 4pm (MATH 350)	Grad Student Seminar Christian Klevdal X For topological spaces that are "nice" (by which we mean semi-locally simply connected), there is a close connection between covering spaces and the fundamental group. In this talk we will explain this relationship and show how it fails for more general spaces. We will then discuss some potential ways to generalize the theory. As far as background, we assume only basic knowledge of the fundamental group and covering spaces. The Hawai'ian Earring
Feb. 12, 2015 3pm (Webber 20…	Algebraic Geometry Jim Carlson (University of Utah) X As Cantor showed us, there are uncountably many transcendental numbers. Despite being overwhelmingly more numerous than algebraic numbers, it is not so easy to find them in Nature: we know, for example, that pi is transcendental, but proving this is a major result. We will discuss these facts, then go on to discuss the Kontsevich-Zagier theory that most transcendental belong to a countable class, the periods of integrals of algebraic functions with rational coefficients. Finally, we consider Yoshinaga’s number: a specific number that is not a period. Transcendental numbers and periods
Feb. 12, 2015 3pm (MATH 350)	Probability Sean O'Rourke (CU Boulder) X The famous Gauss-Lucas theorem describes a geometric connection between the roots of a polynomial and the roots of its derivative. In particular, the theorem states that the critical points of a polynomial p all lie within the convex hull of the roots of p. Building on the works of Pemantle-Rivin and Kabluchko, I will describe a probabilistic version of the Gauss-Lucas theorem, which holds for a large class of random polynomials. Critical points of random polynomials
Feb. 13, 2015 3pm (ECCR 265)	Kempner Andrea Bertozzi (UCLA) X Note the unusual time and place! Modeling of particle laden flow, especially in the case of higher particle concentrations, does not readily allow for first principles models. Rather, semi-empirical models of the bulk dynamics require careful comparision with experiments. At UCLA we have developed this theory for the geometry of viscous thin film flow with non-neutrally buoyant particles. We have found that for these slower flows, that diffusive flux models, involving a balance between shear-induced migration and hindered settling, can provide reasonably accurate predictive models. I will discuss the current state of this work including recent extensions to bidensity slurries and the relevant mathematics needed to understand the dynamics. Lubrication theory can be derived for this problem and results in a coupled system of conservation laws including regular shock dynamics and singular shocks. I will also briefly discuss relevant applications such as spiral separators. Particle laden thin films: theory and experiment
CANCELED Feb. 16, 2015	Jeff Taylor Test - please ignore
Feb. 17, 2015 12:10pm (MATH …	Kempner Piotr M. Hajac (Polish Academy of Sciences) X Assuming that both temperature and pressure are continuous functions, we can conclude that there are always two antipodal points on Earth with exactly the same pressure and temperature. This is the two-dimensional version of the celebrated Borsuk-Ulam Theorem which states that for any continuous map from the n-dimensional sphere to n-dimensional real Euclidean space there is always a pair of antipodal points on the sphere that are identified by the map. Our quest to unravel topological mysteries in the Middle Earth of quantum spaces will begin with gentle preparations in the Shire of elementary topology. Then, after riding swiftly through the Rohan of C-algebras and Gelfand-Naimark Theorems and carefully avoiding the Mordor of incomprehensible technicalities, we shall arrive in the Gondor of compact quantum groups acting on unital C-algebras. It is therein that the generalized Borsuk-Ulam-type statements dwell waiting to be proven or disproven. Time permitting, we shall pay tribute to the ancient quantum group SUq(2), and show how the proven non-trivializability of the SUq(2)-principal instanton bundle is a special case of two different noncommutative Borsuk-Ulam-type conjectures. (Based on joint work with Paul F. Baum and Ludwik Dabrowski.) There and back again: from the Borsuk-Ulam theorem to quantum spaces
Feb. 17, 2015 1pm (MATH 220)	Grad Algebra/Logic Jason Hill (CU Boulder (doctoral candidate) and SpotXchange (data scientist)) Parallel Permutation Group Algorithms
Feb. 17, 2015 2pm (MATH 350)	Lie Theory Martin Walter (CU) X Two years ago a student in Nat's class was also in mine and asked to do an independent study. One thing led to another and what I thought was going to be an easy question of functional analysis about abelian groups turned out to be mostly about number theory which perhaps everyone knows, even me, now. The answer: ``It'' is a negative definite function. A Special Negative Definite Function
Feb. 18, 2015 4pm (MATH 350)	Grad Student Seminar Aleesha Moran (CU-Boulder) X Numerical semigroups have long been studied for their interesting algebraic properties. Given a numerical semigroup and , the collection of arithmetic sequences with step size contained in naturally forms a monoid, denoted as . We refer to these monoids as Leamer monoids. These monoids were originally constructed to study special cases of the Huneke-Wiegand conjecture. We provide foundational results for the factorization theory of Leamer monoids, including a full description of their elasticities. We also give a complete description of the Leamer monoid when is generated by an arithmetic sequence with step size . We also study the collections of numerical semigroups which generate the same Leamer Monoid. Factorization Properties of Leamer Monoids
Feb. 19, 2015 12:10pm (MATH …	Kempner Alison Reddy (University of Illinois Urbana-Champaign) X Placing students into an appropriate first math course is conducive to their success in the first two years of college, while placement into the wrong course can have devastating effects. Accurate placement requires an accurate and recent assessment, but many students are placed based on standard test scores that are years out of date. Alison Reddy will discuss what makes a good placement program and give data the from the University of Illinois Placement Program which has assessed over 70,000 students over the past 8 years. Appropriate placement of students into a first math course: Placement program best practices
Feb. 19, 2015 3pm (MATH 350)	Probability Iddo Ben-Ari (U. Connecticut) X Consider Brownian motion on a bounded interval which is redistributed back into the interval whenever hitting the boundary. The redistribution from each boundary point is according to some fixed probability distribution associated with it, and, conditioned on the boundary point ``hit", the redistribution is independent of the past. It is not hard to show that this is an exponentially ergodic Markov process. The convergence rate has an analytic characterization as the spectral gap for an operator that can be viewed as the infinitesimal generator of the process. In some distinguished cases the gap has an explicit numerical expression. The main thrust of the work presented is to provide a probabilistic and intuitive interpretation for the convergence rate through construction of an efficient coupling, that is a coupling in which time for coupling has a tail that coincides with the convergence rate. The coupling is elementary but obtaining efficiency is not straightforward due to the fact that the redistribution ``reshuffles" the system. This is in sharp contrast with reflected BM on an interval for which it is well known that essentially any coupling is efficient. Time permitting, I will also discuss probabilistic treatment of spectral problems for related models. Efficient Coupling for Diffusion with Redistribution
Feb. 24, 2015 11am (MATH 350)	Number Theory Jeff Achter (Colorado State University) X An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a beautiful product formula, purely in terms of congruence considerations involving that polynomial, for the size of such an isogeny class; an equidistribution hypothesis too strong to be true apparently calculates this cardinality! I will give a new, transparent explanation, worked out with Julia Gordon, for this phenomenon. It turns out that Gekeler's formula computes an adelic orbital integral which, thanks to work of Langlands and Kottwitz, visibly calculates the desired quantity. Local heuristics and exact formulas for elliptic curves over finite fields
Feb. 24, 2015 3pm (Math 350)	Algebraic Geometry Zheng Zhang (Stony Brook) X Based on the work of Gross and Sheng-Zuo, Friedman and Laza have classified variations of real Hodge structure of Calabi-Yau type over Hermitian symmetric domains. In particular, over every irreducible Hermitian symmetric domain there exists a canonical variation of real Hodge structure of Calabi-Yau type. In this talk, we will first review Friedman and Laza’s classification. A natural question to ask is whether the canonical Hermitian variations of Hodge structure of Calabi-Yau type come from families of Calabi-Yau manifolds (geometric realization). In general, this is very difficult and is still open for small dimensional domains. We will discuss an intermediate question, namely does the canonical variations occur in algebraic geometry as sub-variations of Hodge structure of those coming from families of algebraic varieties (motivic realization). In particular, we will give motivic realizations for the canonical variations of Calabi-Yau type over irreducible tube domains of type A using abelian varieties of Weil type. On motivic realizations for variations of Hodge structure of Calabi-Yau type over Hermitian symmetric domains Sponsored by the Meyer Fund
Feb. 24, 2015 3pm (Math 220)	Functional Analysis Ian Long (University of Colorado, Boulder) X A finite measure on the real line is said to be spectral if there exists a set of exponential functions which forms an orthonormal basis for . Moreover, a bounded Lebesgue measurable set of positive measure is spectral if the renormalization of Lebesgue measure on is spectral. I will introduce examples of spectral measures and sets on the real line, describe their relation to an unproven conjecture of Fuglede, examine a related conjecture made by Gabardo and Lai, and introduce a recent result by Duktay and Jorgensen which attempts to answer the question: when do two spectral measures Spectral Measures Sponsored by the Meyer Fund
Feb. 25, 2015 5pm (MATH 350)	Grad Student Seminar Keli Parker (CU-Boulder) X In this brief talk I would like to go over some of the basics of blowing up in algebraic geometry and provide some explicit examples that arise when thinking about families of curves. For this first Slow Pitch of 2015 I would like to the keep the discussion at the introductory level, and there should be very little background knowledge needed. Note the late start (5pm). Blowing up with a view toward stable reduction
Feb. 26, 2015 3pm (MATH 350)	Probability Hyun Lee (CRM) X Interacting particle systems on the integer lattice that the coordinate Bethe Ansatz method can be applied to have connections to the random matrix theory and the symmetric function theory. We will survey the Bethe Ansatz method, and some known results and recent progresses in this direction. In particular, we will consider two initial configurations of the systems that are related to the GUE Tracy-Widom distribution and the GOE Tracy-Widom distribution. The coordinate Bethe Ansatz solvable interacting particle systems
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

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