Friday, October 10, 2008
04:15 pm
Invariants of Binary Forms Modulo Two
Speaker: Larry Smith (AG-Invariantentheorie, Goettingen)<br>Abstract: We examine the invariant theory of binary bilinear forms over the field of two elements that arises in the classification of (standardly graded) Poincare duality algebras with two algebra generators over the field of two elements. We compute the corresponding ring of invariants and find separating invariants for the orbit space
Exley Science Center - ESC 638
Friday, October 17, 2008
04:15 pm
General topology of moduli spaces of topologically infinite Riemann surfaces
Speaker: Katsuhiko Matsuzaki<br>Abstract: The moduli space of a compact Riemann surface has been studied inmany fields of mathematics, but once we move to a topologically infinite Riemann surface, its moduli space has been hardly treated, though it can be simply definedas the quotient space of an infinite dimensional Teichmueller space by quasiconformal mapping class group action. In this talk, by looking at the dynamics of this action,we consider general topological properties (separability, countability and so on) ofthe moduli space. This is not such a nice space that a geometric structure can be induced from the Teichmueller space. Then we introduce the region of stability in theTeichmueller space and, by restricting the action of the mapping class group to it, we find a subregion of the moduli space where the metric structure can be endowed and, as the metric completion of this region, we obtain a new moduli space.Several properties of this space will be discussed.
Exley Science Center - ESC 638
Thursday, October 23, 2008
04:15 pm
Asymmetric Rhythms and Tiling Canons
Speaker: Rachel W. Hall, Saint Joseph's University<br>Abstract: This paper is concerned with classifying and counting rhythms that are maximally syncopated in the sense that, even when shifted, they cannot be synchronized with the division of a measure into two parts. These rhythms relate to asymmetric rhythms described in Aroms study of Central African music (1986). Our results have a surprising application to rhythmic canons. A canon is a musical figure produced when two or more voices play the same melody, with each voice starting at a different time; in a rhythmic canon, rhythms, and not necessarily melodies, are duplicated by each voice. A rhythmic canon tiles if there is exactly one note onset in some voice on each beat. Upon mapping beats to integers, a rhythm forms a tiling canon if and only if its rhythmic motif and sequence of voice entries correspond to sets A and B forming a tiling of the integersthat is, a finite set A of integers (the tile) together with an infinite set of integer translates B such that every integer may be written in a unique way as an element of A plus an element of B. Although many have studied this problem, the complete classification of such tilings is an open question.RACHEL W. HALL received a BA in Ancient Greek from Haverford College and a PhD in mathematics from the Pennsylvania State University. Her research interests are mathematical music theory and ethnomathematics. She is on the editorial boards of Music Theory Spectrum, Journal of Mathematics and Music, and Journal of Mathematics and the Arts. As a member of the folk trio Simple Gifts since 1995, she has toured throughout the Mid-Atlantic and released three albums. She plays the English concertina, piano, and(occasionally) tabla.
Exley Science Center - ESC 121
Thursday, November 06, 2008
04:15 pm
On Knot Homology Theories
Speaker: Elisenda Grigsby<br>Abstract:Low dimensional topologists like to study knots (smoothly imbedded circles in 3-manifolds considered up to ambient isotopy) in large part because of a theorem of Lickorish-Wallace: Every closed, connected, oriented (c.c.o.) 3-manifold can be obtained from the three-sphere (the simplest c.c.o. 3-maniofld) by doing surgery on a finite collection of knots.Yet knots are remarkably tricky to study directly; it is difficult to tell, just by staring at pictures of two knots, whether they are the same or different. We confront this problem through the use of knot invariants, algebraic objects associated to knots that do not depend upon how the knots are drawn. I will discuss a couple of these: Khovanov homology and Heegaard Floer homology, both inspired by ideas in physics. In the less than ten years since their introduction, they have generated a flurry of activity and a stunning array of applications. There are also intriguing connections between the two theories that have yet to be fully understood.
Exley Science Center - ESC 638
Friday, November 14, 2008
04:15 pm
Interpolation on Rational Surfaces
Speaker: Amanda Knecht, University of Michigan<br>Abstract: Tsen's theorem is a classical result which states that over the function field of a complex projective curve, a homogeneous polynomial has a nontrivial solution provided the degree of the polynomial is less than than the number of variables. In 2001 Graber, Harris, and Starr generalized this result by proving that every rationally connected variety over the function field of a curve has a rational point. A proper variety over an algebraically closed field is rationally connected if any two points can be connected by a rational curve. The GHS result is a generalization of Tsen because a smooth hypersurface in projective n-space is rationally connected if its degree is not greater than n.We can restate the theorems of Tsen and Graber, Harris, Starr in terms of the existence of sections of fibrations. Once we know that a section of our fibration exists, we can ask interpolation questions about thesections: Can we find a section through a prescribed number of points?Can we prescribe a Taylor series for the section at a finite number of points? I will give some examples of varieties for which we know the answers to such questions. We will discuss in more detail the case where the general fiber is a degree-two del Pezzo surface.
Exley Science Center - ESC 638
Friday, November 21, 2008
04:15 pm
Elliptic modular transformations on the Teichmuller and the asymptotic Teichmuller spaces
Speaker: Ege Fujikawa, Chiba University<br>Abstract: We consider the Teichmuller space and the quasiconformal mapping class group of a Riemann surface. Every quasiconformal mapping class acts on the Teichmuller space biholomorphically and isometrically as an Teichmuller modular transformation. A Teichmuller modular transformation is said to be elliptic if it has a fixed point on the Teichmuller space. In the case where a Riemann surface is of analytically finite, a Teichmuller modular transformation is elliptic if and only if it is of finite order. In the case where a Riemann surface is analytically infinite, an elliptic element can be of infinite order. However an elliptic element induced by a conformal automorphism fixing a simple closed geodesic on a Riemann surface is of finite order.In this talk, we consider a corresponding result to the asymptotic Teichmuller space which is a certain quotient space of the Teichmuller space. Every quasiconformal mapping class also induces an asymptotic Teichmuller modular transformation. An asymptotic Teichmuller modular transformation is said to be elliptic if it has a fixed point on the asymptotic Teichmuller space. We give a condition for an elliptic asymptotic Teichmuller modular transformation to be of finite order.
Exley Science Center - ESC 638
Thursday, December 04, 2008
04:15 pm
Soap films in Electric Fields
Speaker: John Pelesko, University of Delaware<br>Abstract: In 1968, in the context of investigating fundamental questions inelectrohydrodynamics, G.I. Taylor studied the electrostatic deflection of elasticmembranes1. Utilizing soap film as the membrane material and applying a fixed highvoltage potential difference between two supported circular membranes, Taylor showedexperimentally that at a critical voltage the two membranes snap together and touch. Thatis, the equilibrium state where the membranes remained separate that existed at smallervoltages either became unstable or failed to exist. This instability is familiar toresearchers in the MEMS (microelectromechanical systems) and NEMS(nanoelectromechanical systems) fields where it is known as the pull-in instability. Infact, in an interesting historical coincidence H.C. Nathanson2 and his coworkers studiedthis instability in the context of a primitive MEMS device at roughly the same time asTaylor was conducting his studies. Nathanson is responsible for the pull-innomenclature and the analysis of a mass-spring model of this effect. Taylor, inconjunction with R.C. Ackerberg3 developed and numerically analyzed a more accuratemembrane based model of electrostatic deflection. Recently, a rigorous analysis of thismodel was completed4. Surprisingly, even this simple model of electrostatic deflectioncontains a rich solution set exhibiting a bifurcation diagram with infinitely many folds. Inthis talk, we provide an overview of recent results on the interaction of soap films withelectrostatic fields. We discuss a re-creation of the Taylor experiment, some newexperimental results and discuss the relevance of this research to MEMS and NEMSsystems.1 G.I. Taylor, Proc. Roy. Soc. A., 306, pp. 423-434, 19682 H.C. Nathanson et. al., IEEE Trans. on Elec. Dev., 14, pp. 117-133, 19673 R.C. Ackerberg, Proc. Roy. Soc. A., 312, pp. 129-140, 19694 J.A. Pelesko and X.Y. Chen, Jrnl. of Elec., 57, pp. 1-12, 2003
Exley Science Center - ESC 638
Friday, February 06, 2009
04:15 pm
Rigid properties for metric structures on surfaces
Speaker: Enrico LeDonne, Yale University<br>Abstract: In this talk, we focus on the rigidity of certain non-smooth metric structures on surfaces.A theorem of Berestovskii states that a 2-dimensional geodesic metric space with transitive isometry group is isometric to a Finsler manifold. We present the problem of describing biLipschitz homogeneous geodesic surfaces, i.e., path metric spaces which are homeomorphic to 2-manifolds and have a transitive group of biLipschitz homeomorphisms.We will exhibit the fact that such bi-Lipschitz homogeneous geodesic surfaces are locally doubling. If there is time, I would discuss the fact that there exists a special doubling measure that behaves like the Haar measure for locally compact groups. By the fact that such properties hold, one can start apply the general theory of Analysis on Metric Spaces to further study such objects.
Exley Science Center - ESC 638
Thursday, February 19, 2009
04:15 pm
Cotorsion pairs and pure-injectivity
Speaker: Pedro Guil Asensio, Universidad de Murcia, Spain<br>ABSTRACT: A pair (A,B) of classes of objects in a Grothendieck category is called cotorsion when each class consists of all Ext^1-orthogonal objects to the other. In the first part of the talk, we will center our attention on the cotorsion pair (F,C), where F is the class of all flat modules over a ring R. We will show that many interesting properties of pure-injective modules can be extended to this new setting. We will prove a structure theorem for indecomposable flat cotorsion modules which suggests the existence of a Ziegler-like spectrum for the category of flat modules.In the second part of the talk we will sketch the connections between cotorsion pairs and model structures stablished by Hovey and Gillespie. In particular, we will use this aproach in order to construct some model structures associated to the so-called Finististic Dimension Conjecture or to certain generalizations of vector bundles recently proposed by Drinfeld.
Exley Science Center - ESC 638
Thursday, March 05, 2009
04:15 pm
The existence and moduli problems for algebraic surfaces with zero geometric genus
Speaker: Caryn Werner, Allegheny College<br>Abstract: The classification of surfaces of general type with zero geometric genus is a classical problem in algebraic geometry. There are numerous examples of such surfaces but a complete classification is still unknown. In this talk we will survey old and new results on constructing examples and computing their moduli.
Exley Science Center - ESC 638
Thursday, April 02, 2009
04:15 pm
Invariants of Legendrian Knots and the Legendrian Mirror Problem
Speaker: Joshua Sabloff, Haverford College<br>Abstract: I will introduce a special type of 2-plane field on R^3 called the standard contact structure. A Legendrian knot is a closed, embedded curve that is everywhere tangent to the contact structure. Similarly to topological knot theory, a fundamental problem in Legendrian knot theory is to determine when it is possible -- or impossible -- to deform one Legendrian knot into another through Legendrian knots.<br><br> One motivating question in the field asks whether a Legendrian knot can be deformed to its "Legendrian mirror." To find a new family of examples of Legendrian knots distinct from their Legendrian mirrors, I will introduce a "non-classical" invariant of Legendrian knots called Legendrian Contact Homology. The construction of the invariant involves some neat combinatorics and results in a fertile but complex algebraic object. I will end by indicating how the non-commutativity of the algebra is related to the Legendrian mirror question. This is joint work with G. Civan, J. Etnyre, P.
Exley Science Center - ESC 638
Wednesday, April 08, 2009
04:15 pm
On van der Waerden and tall groups
Speaker: Dieter Remus, Universitat Paderborn (Germany)<br>Abstract:In 1983 W.W. Comfort introduced the following notion: A van der Waer-den group (vdW-group) is a compact group on which every homomorphismto a compact group is continuous. It is a classical result of van der Waerdenthat every compact, connected, semisimple Lie group is a vdW-group.In abstract harmonic analysis a compact group G is called tall if foreach positive integer n there are only finitely many pairwise inequivalentirreducible continuous unitary representations of G of degree n. It is knownthat every vdW-group is tall, but there are totally disconnected tall groupswhich are not vdW-groups.In the talk different properties of vdW-groups and tall groups are pre-sented. In particular, the connected tall groups are classified and the follow-ing conjecture is discussed: Every connected tall group is a vdW-group.
Exley Science Center - ESC 638
Tuesday, April 28, 2009
04:15 pm
The dynamics of protein binding
Speaker: L.A. Peletier, Leiden University, Netherlands<br>Abstract: When a drug enters the blood stream, on its way to a pharmaceutical target, it finds many proteins on its way which are eager to bind it and thus prevent it from reaching its destination. Whilst this may first adversely affect the beneficial effect of the drug, the drug bound to the proteins is not lost. It forms a buffer, which eventually may be released back into the blood plasma and thus still reach its target.In this lecture we discuss a model proposed to study the dynamics of this process and determine the amount of drug that reaches its target over a given time span, say 24 hours. Mathematically, this results into the analysis of a sequence of singular perturbation problems involving systems of nonlinear ordinary differential equations.
Exley Science Center - ESC 638
Tuesday, November 10, 2009
04:15 pm
- 06:00 pm
Colloquium and DNA Seminar
Ronnie Pavlov (Vancouver) (4:15 PM, NOT 12 N) Estimating the entropy of some Z^2 shifts of finite type with probabilistic methodsAbstract: In symbolic dynamics, a Z^d shift of finite type (or SFT) is the set of all ways to assign elements from a finite alphabet A to all sites of Z^d, subject to local rules about which elements of A are allowed to appear next to each other.The (topological) entropy of any Z SFT is easily computable (it is the log of an algebraic number). However, for d > 1, the situation becomes more complex. There are in fact only a few nontrivial examples of Z^2 SFTs whose entropies have explicit closed forms.For a class of Z^2 SFTs (including the Z^2 golden mean shift), we use some probabilistic methods, including percolation theory and stochastic dominance, to describe a sequence of approximations to the entropy which converge at an exponential rate. As a corollary, we can then show that the entropy of any Z^2 SFT in this class is computable in polynomial time. This seminar will be held at 4:15 PM, NOT AT NOON, and will be a departmental colloquium as well.
ESC 638
Thursday, March 25, 2010
04:15 pm
- 06:00 pm
Math Colloquium- Chi-Wang Shu
High Order Methods for Convection Dominated PDEs--An OverviewIn this talk we will give an overview of algorithm development, analysis and application, with an emphasis on motivation, historical perspective and recent progress, on high order numerical methods for convection dominated partial differential equations. These partial differential equations arise in diverse application areas, including computational fluid dynamics, traffic flow models, computational electromagnetism, and astrophysics.After a discussion of motivation and overview on several standard methods, we will discuss a few modern numerical methods, including the finite difference weighted essentially non-oscillatory (WENO) schemes, finite volume WENO schemes, and discontinuous Galerkin (DG) finite element methods. A comparison of their relevant advantages and disadvantages will be given.
ESC 638
Thursday, September 09, 2010
04:15 pm
- 06:00 pm
The Hopf Algebra of Complex iterated Integrals
Speaker: Sheldon JoynerAbstract: The space of Chen iterated integrals of holomorphic differentials on a given Riemann surface is known to form a Hopf algebra with the shuffle product. In this talk, I will show that one can interpolate the number of times which any given form is iterated in a Chen iterated integral, to certain complex parameters. The resulting space of homotopy functionals can then be endowed with the structure of Hopf algebra. Some applications, such as a direct (computational) proof of the monodromy of polylogarithms, will be mentioned.
ESC 638
Thursday, October 07, 2010
04:15 pm
- 06:00 pm
Kac-Wakimoto Characters and Mock Theta Functions
Speaker: Amanda Folsom, Yale UniversityAbstract: In this talk I will discuss the role of certain "strange" functionscalled "mock theta functions" as a liaison between two different areas ofmathematics: modular forms in number theory, and the representation theory of alarge class of Lie superalgebras. Despite their "strange" appearance, the mocktheta functions in their most classical guises date back to the first part ofthe 20th century, however their roles in mathematics were not well understood.Only within the last 8 years have we finally begun to understand and develop agreater theory around the mock theta functions in mathematics - relatingmodular forms and representation theory is just one of their many interestingfacets. This talk is intended to be an introduction to this theory.
ESC 638
Thursday, October 28, 2010
04:15 pm
- 06:00 pm
A Survey of Modular Curves, and Applications To Torsion Points on Elliptic Curves.
Speaker: Alvaro Lozano-Robledo, UCONNAbstract: A modular curve X(G) is a Riemann surface, or the correspondingalgebraic curve, constructed as a compactification of the quotient ofthe complex upper half-plane by the action of a congruence subgroup Gof the modular group SL(2,Z). This talk will be an introduction tomodular curves, emphasizing the moduli interpretation that relatespoints on these curves with elliptic curves that satisfy certainproperties. We will discuss the complete classification of(non-cuspidal) rational points on the modular curves X_0(N), and applythis classification to the study of torsion subgroups of ellipticcurves.
ESC 638
Thursday, November 18, 2010
04:15 pm
- 06:00 pm
TBA
Speaker: Josh Lansky, American UniversityAbstract: TBA
ESC 638
Thursday, December 02, 2010
04:15 pm
- 06:00 pm
Cosmetic Surgeries and Heegaard Floer Homology
Speaker: Stanislav Jabuka, University of Nevada, RenoAbstract: For about 5 decades, it has been known that every 3-dimensional manifold can be obtained by Dehn surgery on a "framed link". Dehn surgery is an operation for building 3-dimensional manifolds that starts with a collection of n disjoint circles (called a "link" if n>1, a "knot" ifn=1) in the 3-sphere, "drills" out tubular neighborhoods of the circles, and re-inserts them in a novel way as specified by the framing. The framing itself is the choice of a rational number, one for each of the components of the link/knot. After explaining these notions in detail, and presenting several examples, we will focus on those 3-dimensional manifolds obtained by surgery on a knot. In this context one can ask both about the uniqueness of the knot as well as the uniqueness of the framing that yields the same 3-dimensional manifold. The "Cosmetic Surgery Conjecture" asserts that no two distinct Dehn surgeries on the same nontrivial knot can ever yield the same 3-dimensional manifold.While this 30 year old conjecture (whose dynamic name is due to SteveBleiler) is still largely open, there has been some recent progress driven by Heegaard Floer theory. The second half of the talk will focus on describing the Heegaard Floer techniques used and detailing the results thus obtained.Most of the talk will be accessible to a general mathematical audience.
ESC 638
Thursday, February 10, 2011
04:15 pm
- 06:00 pm
Counting Curves Using (Orbifold) Quantum Cohomology
Speaker: Linda Chen, SwarthmoreAbstract: Mathematicians have long been interested in counting the number of geometric objects which satisfy various conditions. A breakthrough occurred in the 1990's, inspired by ideas in physics, leading to surprising and beautiful recursions, for example for the number of degree d rational plane curves passing through 3d-1 general points. I will give a basic introduction to these tools of Gromov-Witten theory and quantum cohomology, discuss developments such as orbifold versions of the theories, and describe further applications to classical problems in enumerative geometry. The talk will be accessible to graduate students and a general mathematical audience.
ESC 638
Thursday, March 03, 2011
04:15 pm
- 06:00 pm
Secant Varieties of Smooth Toric Varieties
Speaker: Jessica Sidman, Mount Holyoke CollegeAbstract: Algebraic geometers have long been interested in secant varieties. The results that I will describe were motivated by work of Sturmfels and others in algebraic statistics. I will discuss joint work with David Cox which gives the dimension and degree of the secant variety of a smooth toric variety in terms of the corresponding polytope under suitably nice conditions.
ESC 638
Thursday, March 24, 2011
04:15 pm
- 06:00 pm
Liftings of Representations of Finite Groups of Lie Type
Speaker: Joshua Lansky, American UniversityAbstract: We give an introduction to the theory of reductive groups over finite fields and their representations. The aim of the talk is to present a construction for a lifting of classes of representations in a fairly general setting.
ESC 638
Thursday, April 07, 2011
04:15 pm
- 06:00 pm
The Arithmeitc of Quadratic Forms
Speaker: John Hanke, University of GeorgiaAbstract: Quadratic forms with integer coefficients are among the oldest objects in mathematics and lie at the crossroads between geometry and arithmetic. They have been studied by mathematicians for thousands of years, but starting with the work of Gauss and Legendre in the 1700s, new perspectives and techniques have been rapidly increasing our understanding of the answers to classical questions like "What numbers can be written as a sum of n squares?". We now know that there are deep connections between these questions and many other interesting objects in mathematics (e.g. modular forms, elliptic curves, abelian varieties, certain zeros of twists of the Riemann zeta function). This talk will describe some of these ideas and techniques, explain what questions they can be used to solve, and list some interesting problems that are still unresolved.
ESC 638
Thursday, April 14, 2011
04:15 pm
- 06:00 pm
Characterizing analyticity by means of Nonlinear Fourier transforms and Applications
Speaker: Shif Barhanu, Temple UniversityAbstract: In this talk we will discuss a nonlinear Fourier transform which has been usedextensively over the past thirty years to determine the real analyticity of functionsthat are solutions of partial differential equations. We will also present some newclasses of nonlinear Fourier transforms obtained in recent joint work with JorgeHounie.
ESC 638
Thursday, April 21, 2011
04:15 pm
- 06:00 pm
Percolation, partitions, and probability
Speaker: Karl MahlburgAbstract: I will discuss the surprising connections between finite-size scaling in families of bootstrap percolation models, the combinatorics of integer partitions, and limiting entropy rates for(Markov-type) stochastic processes. A combinatorial characterization of the percolation processes relates their metastability threshold exponents to the cuspidal asymptotics of generating functions for partitions with restricted sequence conditions. These generating functions are hypergeometric q-series that are of number-theoretic interest, as in many cases they are equal to the product of modular forms and Ramanujan's famous mock theta functions. In other cases, both the percolation processes and partition functions are best understood through entropy rate bounds for probabilistic sequences with gap conditions. These sequences can be understood as stochastic processes with varying transition probabilities, and techniques from the theory of linear operators are used to bound the dominant eigenvalues.
ESC 638
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