Colloquium Archive

November 10, 2009

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 methods

Abstract: 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.

April 28, 2009

The dynamics of protein binding

Speaker: L.A. Peletier, Leiden University, Netherlands
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.

April 08, 2009

On van der Waerden and tall groups

Speaker: Dieter Remus, Universitat Paderborn (Germany)
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 homomorphism to a compact group is continuous. It is a classical result of van der Waerden that every compact, connected, semisimple Lie group is a vdW-group. In abstract harmonic analysis a compact group G is called tall if for each positive integer n there are only finitely many pairwise inequivalent irreducible continuous unitary representations of G of degree n. It is known that every vdW-group is tall, but there are totally disconnected tall groups which 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.

April 02, 2009

Invariants of Legendrian Knots and the Legendrian Mirror Problem

Speaker: Joshua Sabloff, Haverford College
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.

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.

March 05, 2009

The existence and moduli problems for algebraic surfaces with zero geometric genus

Speaker: Caryn Werner, Allegheny College
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.

February 19, 2009

Cotorsion pairs and pure-injectivity

Speaker: Pedro Guil Asensio, Universidad de Murcia, Spain
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.

February 06, 2009

Rigid properties for metric structures on surfaces

Speaker: Enrico LeDonne, Yale University
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.

December 04, 2008

Soap films in Electric Fields

Speaker: John Pelesko, University of Delaware
Abstract: In 1968, in the context of investigating fundamental questions in electrohydrodynamics, G.I. Taylor studied the electrostatic deflection of elastic membranes1. Utilizing soap film as the membrane material and applying a fixed high voltage potential difference between two supported circular membranes, Taylor showed experimentally that at a critical voltage the two membranes snap together and touch. That is, the equilibrium state where the membranes remained separate that existed at smaller voltages either became unstable or failed to exist. This instability is familiar to researchers in the MEMS (microelectromechanical systems) and NEMS (nanoelectromechanical systems) fields where it is known as the pull-in instability. In fact, in an interesting historical coincidence H.C. Nathanson2 and his coworkers studied this instability in the context of a primitive MEMS device at roughly the same time as Taylor was conducting his studies. Nathanson is responsible for the pull-in nomenclature and the analysis of a mass-spring model of this effect. Taylor, in conjunction with R.C. Ackerberg3 developed and numerically analyzed a more accurate membrane based model of electrostatic deflection. Recently, a rigorous analysis of this model was completed4. Surprisingly, even this simple model of electrostatic deflection contains a rich solution set exhibiting a bifurcation diagram with infinitely many folds. In this talk, we provide an overview of recent results on the interaction of soap films with electrostatic fields. We discuss a re-creation of the Taylor experiment, some new experimental results and discuss the relevance of this research to MEMS and NEMS systems. 1 G.I. Taylor, Proc. Roy. Soc. A., 306, pp. 423-434, 1968 2 H.C. Nathanson et. al., IEEE Trans. on Elec. Dev., 14, pp. 117-133, 1967 3 R.C. Ackerberg, Proc. Roy. Soc. A., 312, pp. 129-140, 1969 4 J.A. Pelesko and X.Y. Chen, Jrnl. of Elec., 57, pp. 1-12, 2003

November 21, 2008

Elliptic modular transformations on the Teichmuller and the asymptotic Teichmuller spaces

Speaker: Ege Fujikawa, Chiba University
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.

November 14, 2008

Interpolation on Rational Surfaces

Speaker: Amanda Knecht, University of Michigan
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 the sections: 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.

November 06, 2008

On Knot Homology Theories

Speaker: Elisenda Grigsby
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.

October 23, 2008

Asymmetric Rhythms and Tiling Canons

Speaker: Rachel W. Hall, Saint Joseph's University
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.

October 17, 2008

General topology of moduli spaces of topologically infinite Riemann surfaces

Speaker: Katsuhiko Matsuzaki
Abstract: The moduli space of a compact Riemann surface has been studied in many 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 defined as 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) of the 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 the Teichmueller 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.

October 10, 2008

Invariants of Binary Forms Modulo Two

Speaker: Larry Smith (AG-Invariantentheorie, Goettingen)
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

May 01, 2008

Big numbers with small prime factors

Speaker: Tonghai Yang, University of Wisconsin, Madison

April 30, 2008

Countable dense homogeneous spaces

Speaker: Jan van Mill, Vrije Universiteit, Amsterdam
Abstract: A topological space $X$ is countable dense homogeneous (abbreviated CDH) provided that for all countable dense subsets $D$ and $E$ of $X$ there is a homeomorphism $f$ of $X$ such that $f(D)=E$. By the now standard back-and-forth technique, Brouwer proved around 1910 that the real line is CDH. Most of the well-known homogeneous spaces are CDH: the Hilbert cube, the universal Menger continua, the Cantor set and manifolds without boundaries. Ungar claimed in 1978 that an open dense subset of a locally compact CDH-space is CDH. The proof of this fact is incomplete, and it is unclear whether it is true. Fitzpatrick and Zhou asked in 1988 whether there is a connected CDH-space with a dense open connected subspace that is not CDH. By using techniques from infinite-dimensional topology, we will present an example of a convex subset $S$ of Hilbert space $\ell^2$ with a dense convex open subspace $T$ such that (1) $S$ is CDH, $T$ is rigid, (3) $S\times S \approx T\times T\approx \ell^2$. Here a space is called rigid if the identity is its only homeomorphism, and $X\approx Y$ means that $X$ and $Y$ are homeomprhic topological spaces.

April 17, 2008

TBA

Speaker: Peter Shalen, University of Illinois at Chicago

April 03, 2008

On Floer homology and knots admitting lens space surgeries

Speaker: Matt Hedden, MIT
Abstract: There is a simple procedure called (Dehn) surgery which alters three dimensional manifolds using knots. The simplest three-manifolds are called lens spaces, and there is a conjecture regarding the knots on which one can perform surgery to obtain these manifolds. In this talk, I'll review these notions and discuss this conjecture, known as the Berge conjecture. I'll then discuss a strategy, developed jointly with Ken Baker and Eli Grigsby by which the knot Floer homology invariants of Ozsvath, Szabo, and Rasmussen could be used to prove this conjecture.

March 28, 2008

An introduction to exterior differential systems

Speaker: Jeanne Clelland, University of Colorado
Abstract: Exterior differential systems were first introduced by E. Cartan as a geometric approach to the study of partial differential equations. This talk will start with some definitions, followed by several examples which demonstrate this approach and illustrate some of its advantages.

This talk should be accessible to anyone with a reasonably good understanding of multivariable calculus; students are welcome!

March 07, 2008

Some extensions of holomorphic motions

Speaker: Professor S. Mitra, CUNY
Abstract: In this talk, we will discuss extensions of holomorphic motions over more general parameter spaces. It is well-known that Slodkowski's theorem cannot be generalized to higher dimensional parameter spaces. We will give a remarkably simple example of a holomorphic motion of a finite set over a simply connected space that cannot be extended to the whole sphere.

We will also discuss quasiconformal motions, an idea that was introduced by Sullivan and Thurston. We will show that a holomorphic motion of a closed setE (in the Riemann sphere) over a simply connected complex Banach manifold always extends to a quasiconformal motion of the sphere; furthermore, this extended motion has some interesting properties. The Teichm\"uller space of the closed set E will play an important role throughout our discussion.

All basic definitions will be given.

March 04, 2008

Theorems of Stone-Weierstrass Type

Speaker: Stu Sidney, (Storrs)
Abstract: The standard (real) Stone-Weierstrass theorem says that if E is a vector space of continuous functions on a compact Hausdorff space X that contains the constant functions and separates the points of X, and if E is either a lattice or an algebra under pointwise operations, then E is dense in C(X). The lattice and algebra conditions can be reformulated to say that E is closed under composition with the absolute value function and the squaring function respectively. In 1963 de Leeuw and Katznelson generalized both versions by showing that if E is closed under composition with some non-affine function, then E is dense in C(X). Of course, if it is also hypothesized that E is uniformly closed -- equivalently, that E is complete in the supremum norm -- then the conclusion becomes E = C(X). But what if E is complete in some other norm that dominates the supremum norm? This makes E into a (real) Banach function space, and from the early 1960s on there has been slow and steady progress on the following problem: If a non-affine function f operates by composition on a Banach function space E, under what additional conditions on f and/or E can we deduce that E = C(X)? We shall recount some of this history, with the emphasis being on conditions on E, and perhaps discuss some work in progress along these lines.

March 03, 2008

Hirzebruch-type invariants of complex algebraic varieties

Speaker: Laurentiu Maxim, CUNY-Lehman College
Abstract: I will report on recent progress on the study of genera and characteristic classes of algebraic varieties. I will describe Hodge-theoretic analogues of the Atiyah-Meyer signature formula, and discuss possible extensions of these results to the singular setting. This is joint work with S. Cappell, A. Libgober and J. Shaneson.

November 29, 2007

The length spectrum of a flat metric

Speaker: Moon Duchin, UC Davis/MSRI
Title: The length spectrum of a flat metric
Abstract: Trying to hear the shape of a drum via its spectrum of eigenvalues or lengths has by now been studied in many contexts. For a hyperbolic surface with a constant-curvature metric, knowing how long all curves are (and which curves correspond to which lengths) is enough to determine the metric. It turns out that 6g-5 curves suffice on the surface of genus g>1. It is natural to ask the same for a different class of metrics on hyperbolic surfaces, arising from the study of Teichmuller geometry: Metrics that are everywhere flat except at finitely many cone points, such as the genus-two surface obtained by a Euclidean octagon with opposite sides glued by translation. We show that the (marked) length spectrum does determine the flat metric, but no finite set of curves suffices. In fact, we give conditions for the set of curves to be spectrally rigid. This is joint work with Chris Leininger and Kasra Rafi.

November 26, 2007

Generalizing Pfaffian closure of an o-minimal structure

Speaker: Sergio Fratarcangeli, College of New Rochelle
Title: Generalizing Pfaffian closure of an o-minimal structure

November 08, 2007

Martin Bridgeman, Hausdorff dimension under bending deformations and the Weil-Petersson metric

Date: Thursday, November 8, 2007
Speaker: Martin Bridgeman, Boston College
Title: Hausdorff dimension under bending deformations and the Weil-Petersson metric
Time & Place: 4:15 P.M., Room 638SC

October 18, 2007

On the Enormity of the Knot Concordance Group

Speaker: Shelly Harvey, Rice University

Abstract: We say that a knot in S^3 (the boundary of B^4, the 4-dimensional ball) is slice if it is the boundary of 2-disk in B^4. Moreover, two knots K and J are concordant if their connected sum is slice. In 1966, Fox and Milnor defined the knot concordance group as the set of knots up to concordance where the addition is given by the connected sum. In 1997, Cochran, Orr, and Teichner defined a filtration of the knot concordance group, {F_n}, called the (n)-solvable filtration. This filtration is geometrically significant because it measures the successive failure of the Whitney trick for 2-disks in 4-manifolds. It was shown by Cochran and Teichner that each of the abelian groups F_n/F_n+1 has rank at least 1. We show for each n, that F_n/F_n+1 has infinite rank. This was only previously known to be true when n+0,1,2. We also resolve a long standing question as to whether certain natural families of knots considered by Casson, Gordon, Gilmer and others contain slice knots. This is joint work with Tim Cochran (Rice University) and Constance Leidy (Wesleyan University).

October 12, 2007

Virtual Knots and Assembly Graphs (joint work with Angela Angeleska and Masahico Saito)

Natasha Jonoska, University of Southern Florida
Abstract: We propose molecular models for homologous DNA recombination events that are guided by either double-stranded RNA or single-stranded RNA templates. This recombination can be seen as topological braiding of the DNA, with the template-guided alignment proceeding through DNA branch migration. We show that a virtual knot diagram can provide a physical representation of the DNA at the time of recombination. Schematically, the braiding process can be represented as a crossing in the virtual knot diagram. The homologous recombination corresponds to removal of the crossings in the knot diagram (called smoothing). We associate operations on words and graphs with such rearrangements and investigate their properties which lead to proper order of the DNA sequence.