2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021
Current contact: Dave Futer or Matthew Stover
The Seminar usually takes place on Wednesdays at 2:45 PM in Room 617 on the sixth floor of Wachman Hall.
Steven Simon, New York University, Two Generalizations of the Ham Sandwich Theorem
The Ham Sandwich theorem states that for any $n$ finite Borel measures on $\mathbb{R}^n$, there exists a hyperplane which bisects each of the measures. This talk will present two generalizations of this theorem. In one direction, we ask for the number of mutually orthogonal hyperplanes which bisect a collection of numbers. This number will be related to the number of linearly independent vector fields on a sphere. The talk will also provide group-theoretic generalizations of the Ham Sandwich Theorem for fundamental regions corresponding to finite subgroups of spheres of dimension $0$, $1$, and $3$.
Matteo Castronovo, Temple University, The Effect of Confinement on Molecular Mechanism Inside Bio-Nanosensors
The explosive increase of research in biology has spurred the need for new techniques and devices that can surmount experimental roadblocks. Current in-vitro techniques cannot accurately identify small differences in concentration in samples containing few molecules in a single or a few cells. Nanotechnology overcomes these limitations with the possibility of fabricating nano-sensors that measure protein amounts down to a hundred molecules.
The pairing of two complementary strands of DNA, also called DNA hybridization, allows the formation of a stable helical structure. In turn, the pairing mechanism provides DNA molecules with a self-assembly functionality. The latter offers tremendous potential in nanotechnology toward developing programmable nano-sensors. For instance, in our work we fabricate prototypical nanosensors by locally, and chemically attaching short sequences of DNA to a surface. The latter form a confined patch of monolayer (i.e. a DNA brush) at the solid-liquid interface that can be selectively, and reversibly modified by hybridizing the DNA in the brush with a DNA-linked probe-molecule, which is able to recognize a target-molecule in solution. Little is known, however, about the effect of confinement on the mechanism of recognition between molecules inside such systems. In our experimental work we have studied the mechanism by which a restriction enzyme, i.e. a protein that binds DNA and cuts it in a specific site, works inside a DNA brush. We address the effect of confinement by varying the DNA surface density. We unequivocally show that confinement has a quantifiable effect on the reaction. Namely, enzymes do not access to the DNA directly from the solution, but 2D-diffuse inside the DNA brush exclusively from the side. Moreover, if the DNA surface density is sufficiently high, the enzyme becomes completely unable to access the substrate and, therefore, to cut the DNA molecules.
Our findings demonstrate that DNA-enzyme reaction mechanisms can be significantly altered when occurring in nanoscale materials, and may have broad implications on the design of innovative nanotechnology approaches to biomolecular detection.
Mike Davis, Institute for Advanced Study, Right-angledness, flag complexes, asphericity
I will discuss three related constructions of spaces and manifolds and then give necessary and sufficient conditions for the resulting spaces to be aspherical. The first construction is the polyhedral product functor. The second construction involves applying the reflection group trick to a "corner of spaces". The third construction involves pulling back a corner of spaces via a coloring of a simplicial complex. The two main sources of examples of corners which yield aspherical results are: 1) products of aspherical manifolds with (aspherical) boundary and 2) the Borel-Serre bordification of torsion-free arithmetic groups which are nonuniform lattices.
Johanna Mangahas Kutluhan, Brown University, Geometry of right-angled Artin groups in mapping class groups
I'll describe joint work with Matt Clay and Chris Leininger. We give sufficient conditions for a finite set of mapping classes to generate a right-angled Artin group. This subgroup is quasi-isometrically embedded in the whole mapping class group, as well as, via the orbit map, in Teichmueller space with either of the standard metrics. Subsurface projection features prominently in the proofs.
John Etnyre, Georgia Tech, The Contact Sphere Theorem and Tightness in Contact Metric Manifolds
We establish an analog of the sphere theorem in the setting of contact geometry. Specifically, if a given three dimensional contact manifold admits a compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure is tight. The proof is a blend of topological and geometric techniques. A necessary technical result is a lower bound for the radius of a tight ball in a contact 3-manifold. We will also discuss geometric conditions in dimension three for a contact structure to be universally tight in the nonpositive curvature setting. This is joint work with Rafal Komendarczyk and Patrick Massot.
Robert Ghrist, University of Pennsylvania, Braid Floer Homology
The classical Arnolīd Conjecture concerns the number of 1-periodic orbits of 1-periodic Hamiltonian dynamics on a symplectic manifold.The resolution of this conjecture was the impetus for and first triumphof Floer homology. The present talk considers the problem of periodicorbits of higher periods. In the case (trivial for the Arnolīd Conjecture)of a 2-dimensional disc, these orbits are braids.
This talk describes a relative Floer homology that is a topologicalinvariant of (pairs of) braids. This can be used as a forcing theoremfor implying the existence of periodic orbits in 1-periodic Hamiltoniandynamics on a disc.
This represents joint with with J.B. van den Berg, R. Vandervorst, andW. Wojcik.
Elena Fuchs, Institute for Advanced Study, Counting in Apollonian circle packings
Apollonian circle packings are constructed by continuously inscribing circles into the curvilinear triangles formed in a Descartes configuration of mutually tangent circles. An observation of F. Soddy in 1937 is that if any four mutually tangent circles in the packing have integer curvature, then in fact all of the curvatures in the packing will be integers. In the past few years, this observation has led to several developments regarding the number theory of such integer Apollonian packings. In this talk, I will discuss a very generalizable approach to counting integers appearing as curvatures in integer Apollonian packings. I will also discuss some natural questions to consider along these lines. This is joint work with J. Bourgain.
Tim DeVries, University of Pennsylvania, An Algorithm for Bivariate Singularity Analysis
How do you count? Of primary interest to enumerative combinatorists is obtaining counting formulas for various discrete, mathematical objects. For instance, what is the $n$th Fibonacci number? What is the $n,m$th Delannoy number? A common technique is to embed the sequence as the coefficients of a formal power series, known as a generating function. When this function is locally analytic, we hope that its analytic properties may help us to extract asymptotic formulas for its coefficients. We will explore this technique, known as singularity analysis, in the case that the generating function is bivariate rational. We then sketch an algorithm that, for many such generating functions, automatically produces these asymptotic formulas. Despite its combinatorial origins, this algorithm is quite geometric in nature (touching on topics from homology theory, Morse theory, and computational algebraic geometry).
This is joint work with Robin Pemantle and Joris van der Hoeven.
Shiva Kasiviswanathan, IBM, The Price of Privately Releasing Contingency Tables and the Spectra of Random Matrices with Correlated Rows
Contingency (marginal) tables are the method of choice of government agencies for releasing statistical summaries of categorical data. However, if the contingency tables are released exactly, one can reconstruct the individual entries of the data by solving a system of equations. In this talk, we give tight bounds on how much distortion (noise) is necessary in these tables to provide privacy guarantees when the data being summarized is sensitive. Our investigation also leads to new results on the spectra of random matrices with correlated rows.
Based on joint work with Mark Rudelson, Jonathan Ullman, and Adam Smith.
PATCH seminar, joint with Bryn Mawr, Haverford, and Penn
Joel Hass, UC Davis, Width invariants and the physical motion of curves through a medium
The method of gel electrophoresis was developed in the 1970s to separate fragments of DNA as they migrate through a gel, a porous sponge-like medium. An electric current pulls smaller molecules faster than larger ones. When the molecules are closed loops of DNA, biologists believe that the motion is determined by the "average crossing number". However other knot invariants may be relevant to such motion. We define and compute some of these, and relate them to other knot invariants. This is joint work with Hyam Rubinstein and Abigail Thompson.
PATCH seminar, joint with Bryn Mawr, Haverford, and Penn
Vera Vertesi, MIT, Transverse positive braid satelites
In this talk I investigate transverse knots in the standard contact structure on $\mathbb{R}^3$. These are knots for which $y>dz/dx$. The name "transverse" comes from the fact that these knots are positively transverse to the contact planes given by the the kernel of the $1$-form $dz-y\, dx$. The classification of transverse knots has been long investigated, and several invariants were defined for their distinction, one classical invariant is the self-linking number of the transverse knot, that can be given as the linking of the knot with its push off by a vectorfield in the contact planes that has a nonzero extension over a Seifert surface. Smooth knot types whose transverse representatives are classified by this classical invariant are called transversaly simple. In this talk I will talk about how transverse simplicity is inherited for positive braid satelites of smooth knot types.
Ryan Blair, University of Pennsylvania, Bridge Number and Conway Products
A well known theorem of Schubert tells us that the bridge number of knots is additive with respect to the cut and paste operation of connected sum. The Conway product is a vast generalization of connected sum achieved by removing rational tangles and gluing along 4-punctured spheres. In this talk, we will present a lower bound for the bridge number of a Conway product in terms of the bridge number of the factor knots. Additionally, we will present examples which show this lower bound is sharp.
-Note special location and time- Anne Thomas, Unversity of Sydney, Infinite generation of non-cocompact lattices on right-angled buildings
Let Gamma be a non-cocompact lattice on a right-angled building $X$. Examples of such $X$ include products of trees, or Bourdon's building $I_{p,q}$, which has apartments hyperbolic planes tesselated by right-angled p-gons and all vertex links the complete bipartite graph $K_{q,q}$. We prove that if Gamma has a strict fundamental domain then Gamma is not finitely generated. The proof uses a topological criterion for finite generation and the separation properties of subcomplexes of $X$ called tree-walls. This is joint work with Kevin Wortman (Utah).
Colin Adams, Williams College, Surfaces in Hyperbolic Knot Complements
Given a knot in the 3-sphere with hyperbolic complement, one would like to try to understand the geometry of Seifert surfaces, essential surfaces with boundary the knot. In unusual cases, which we will discuss, such a surface can be totally geodesic (also called Fuchsian). The existence of such surfaces says a lot about the knot. However, much more common is for the surface to be quasi-Fuchsian. It turns out that many of the results know for Fuchsian surfaces can be extended to quasi-Fuchsian surfaces. Lots of pictures will be included. No familiarity with hyperbolic knots and surfaces will be assumed.
Lenhard Ng, Duke University, Transverse Homology
Knot contact homology is a combinatorial Floer-theoretic knot invariant derived from Symplectic Field Theory. I'll discuss a filtered version of this invariant, transverse homology, which turns out to be a fairly effective invariant of transverse knots.
Dylan Thurston, Barnard College, Columbia Unversity, Stress matrices and rigidity
When do the lengths of the edges of a straight-edge framework determine the positions of the vertices? The problem comes up all the time in applications ranging molecular biology to sensor networks to computer vision. But it also turns out that the problem is NP-hard in general. It becomes easier if you require the initial position to be generic; then there is a polynomial algorithm based on the \emph{stress matrix} of the graph. But even in this case, actually reconstructing the positions from the edge-length is difficult. There is a good algorithm in case the framework is \emph{universally} rigid: the edge lengths determine the framework independently of the embedding dimension. There is again a characterization of such frameworks in terms of the stress matrix.
This is joint work with Steven Gortler and Alex Healy.
Nathan Dunfield, University of Illinois at Urbana-Champaign, The least spanning area of a knot and the Optimal Bounding Chain Problem
Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3-dimensional manifold. While these two surfaces are not necessarily the same, when the knot is embedded in a general 3-manifold, the two problems were shown earlier this decade to be NP-complete and NP-hard respectively. However, there is evidence that the special case when the ambient manifold is R^3, or more generally when the second homology is trivial, should be considerably more tractable. Indeed, we show here that the least area surface can be found in polynomial time. The precise setting is that the knot is a 1-dimensional subcomplex of a triangulation of the ambient 3-manifold. The main tool we use is a linear programming formulation of the Optimal Bounding Chain Problem (OBCP), where one is required to find the smallest norm chain with a given boundary. While the OBCP is NP-complete in general, we give conditions under which it can be solved in polynomial time. We then show that the least area surface can be constructed from the optimal bounding chain using a standard desingularization argument from 3-dimensional topology. We also prove that the related Optimal Homologous Chain Problem is NP-complete for homology with integer coefficients, complementing the corresponding result of Chen and Freedman for mod 2 homology.
Julien Roger, Rutgers University, Quantum Teichmueller theory and conformal field theory
The aim of this talk is to investigate the possible connection between the quantum Teichmueller space and a certain type of conformal field theory. I will first introduce the notion of a modular functor arising from conformal field theory, and its applications to low dimensional topology. Then I will describe the construction of the quantum Teichmueller space, emphasizing the relationship with hyperbolic geometry. Finally, I will describe a possible connection between the two constructions, focusing on the notion of factorization rule. The key ingredients here are the Deligne-Mumford compactification of the moduli space and its Weil-Petersson geometry. I will introduce these notions as well.
Rob Kusner, University of Massachusetts, Amherst, Knots and Links as Ropes, Bands and Branched Coverings
What is the geometry of tightly knotted rope? How, for example, is its length related to combinatorial or algebraic knot invariants? Or what shapes are typical of tight knots and links? We'll discuss recent progress on these "ropelength criticality" issues, and also explore some simpler, potentially more computable, ideal geometric models, including one which realizes knots and links as the "fattest" annuli on a Riemann surface branched covering the sphere.
Neil Hoffman, Boston College, Hidden symmetries and cyclic commensurability for small knot complements
Two hyperbolic orbifolds are commensurable if they share a common finite sheeted cover. Commensurability forms an equivalence relation on the set of hyperbolic orbifolds. Conjecturally, there are only three knot complements in a given commensurability class. Furthermore, if two knot complements are commensurable, Boyer, Boileau, Cebanu, and Walsh show that they are either cyclically commensurable, ie cover an orbifold with multiple finite cyclic fillings or they admit hidden symmetries, ie they cover an orbifold with a rigid cusp. After providing some of the necessary background, I will show that small, cyclically commensurable knot complements do not admit hidden symmetries.
Andrew Cooper, University of Pennsylvania, Singular time of the Ricci and mean curvature flows
The mean curvature flow (MCF) and Ricci flow (RF) are quasilinear parabolic equations; hence solutions are expected to develop singularities in finite time. It is straightforward that in each case, the relevant full curvature tensor (for MCF, the second fundamental form; for RF the Riemann tensor) must blow up at such a singularity. This talk will address whether it is possible characterise the singular time of these flows by a weaker criterion. I will present an argument of Sesum to show that the Ricci tensor must blow up at a finite-time singularity of the RF, and adapt it to show that in MCF the second fundamental form must blow up, roughly speaking, in the direction of the mean curvature vector. Time permitting, I will give two independent proofs that under a mildness assumption for the singularity, the mean curvature itself must blow up.
Will Cavendish, Princeton University, Finite sheeted covers of 3-manifolds and the Cohomology of Solenoids
Given a compact manifold $M$, the inverse limit of the set of all finite sheeted covering spaces over $M$ yields compact topological space $\widehat{M}$ called a solenoid that can be thought of as a pro-finite version of the universal cover of $M$. While such an object can in general be quite complicated, I will show in this talk that if $M$ is a compact aspherical 3-manifold then $\widehat{M}$ has the Cech cohomology of a disk. I will then talk about the relevance of this result to the study of finite sheeted covering spaces and lifting problems in 3-manifold theory.
Andras Stipsicz, Hungarian Academy of Sciences and IAS, Tight contact structures on 3-manifolds.
After reviewing results about the existence of tight contact structures on closed 3-manifolds, we show how to use Heegaard Floer theory (in particular, the contact Ozsvath-Szabo invariant) to verify tightness of certain contact structureson 3-maniolds given by surgery along specific knots in S^3.
Andrew Cooper, University of Pennsylvania, A characterisation of the singular time of the mean curvature flow
The mean curvature flow (MCF) is a quasilinear parabolic equation; hence solutions are expected to develop singularities in finite time. It is straightforward that the second fundamental form must blow up at such a finite-time singularity.
This talk will address whether it is possible to characterise the singular time by a weaker criterion. I will show that in MCF the second fundamental form must blow up, roughly speaking, in the direction of the mean curvature vector. Time permitting, I will give two independent proofs that under a mildness assumption for the singularity, the mean curvature vector itself must blow up, and mention connections to some results for the Ricci flow.
Dave Futer, Temple University, Detecting fiber surfaces
For a knot diagram $D(K)$, a state surface is a certain surface with boundary along $K$, algorithmically constructed from $D(K)$ by making a binary choice at each crossing. This construction generalizes Seifert's algorithm for constructing an orientable surface with boundary $K$. I will describe this construction and discuss a simple diagrammatic criterion that characterizes when one of these state surfaces is a fiber in the knot complement $S^3 \setminus K$.
Ryan Blair, University of Pennsylvania, Bridge number and tangle product of knots
Tangle product is a very general operation in which two knots are amalgamated together to create a third. The operation of tangle product generalizes both connected sum and Conway product of knots. The bridge number of a knot is the fewest number of maxima necessary to form an embedding of the knot in 3-space. I will present results showing that, under certain hypotheses involving the distance of a minimal bridge surface in the curve complex, the bridge number of a tangle product is at least the sum of the bridge numbers of the two factor links up to a constant error.
PATCH seminar, joint with Bryn Mawr, Haverford, and Penn
-Note different room-
Jason DeBlois, University of Pittsburgh, Algebraic invariants, mutation, and commensurability of link complements
I'll describe a family of two-component links with the property that many algebraic invariants of their complements can be easily computed, and describe the commensurability relation among its members. Some mutants have commensurable complements and others do not. I'll relate this to some open questions about knot complements.
PATCH seminar, joint with Bryn Mawr, Haverford, and Penn
-Note different room-
Matt Hedden, Michigan State University, Contact structures associated to "rational" open books and their invariants
A well-worn construction of Thurston and Winkelnkemper associates an essentially unique contact structure to an open book decomposition of a 3-manifold. Such a decomposition is essentially a choice of fibered knot or link in the 3-manifold, i.e. a link whose complement is a surface bundle over the circle in a "particular way". I'll discuss how to relax this "particular way" so knots which aren't even null-homologous can still be considered fibered. The generalized open book structures that result are also related to contact geometry, and I'll discuss invariants of these contact structures coming from Heegaard Floer homology. Our invariants can be fruitfully employed to populate the contact geometric universe with examples, and to better understand how it behaves under Dehn surgery. Using this latter understanding, I'll discuss possible implications for the Berge Conjecture, a purely topological conjecture about the knots in the 3-sphere on which one can perform surgery and obtain lens spaces. This is joint work with Olga Plamenevskaya.
Chris Atkinson, Temple University, Small volume link orbifolds
We will discuss recent joint work with Dave Futer in which we study hyperbolic 3-orbifolds having singular locus a link. We have identified the smallest volume hyperbolic 3-orbifold having base space the 3-sphere and singular locus a knot. We also identify the smallest volume hyperbolic 3-orbifold with base space any homology 3-sphere and singular locus a link. With weaker homology assumptions, we obtain a lower bound on the volume of any link orbifold.
Frédéric Bourgeois, Université Libre de Bruxelles, $S^1$-equivariant symplectic homology and contact homology
In this joint work with Alexandru Oancea, we construct an $S^1$-equivariant version of symplectic homology. We then describe various algebraic structures as well as a simpler computational approach for this invariant. Finally, we sketch the proof that this invariant coincides with (linearized) contact homology. The advantage of the first invariant is that transversality results can be established for large classes of symplectic manifolds, while for contact homology, the corresponding results would rely on the recent theory of polyfolds.
Ana Lecuona, Penn State University, Montesinos knots and the slice-ribbon conjecture
The slice-ribbon conjecture states that a knot in the three sphere is the boundary of an embedded disc in the four ball if and only if it bounds a disc in the sphere which has only ribbon singularities. This conjecture was proposed by Fox in the early 70s. There doesn't seem to be any conceptual reason for it to be true, but large families of knots (i.e. pretzel knots, two bridge knots) satisfy it. In this seminar we will prove that the conjecture remains valid for a large family of Montesinos knots. The proof is based on Donaldson's diagonalization theorem for definite four manifolds.
Haggai Nuchi, University of Pennsylvania, Geometry of triple linking
Gauss produced a formula for the linking number of a 2-component link in Euclidean space. This formula involves an integral with the property that the integrand is geometrically natural, i.e. it remains unchanged under rigid motions of the link. I will describe joint work producing an analogous integral formula for the Milnor triple linking number of a three-component link in Euclidean space, with the property that the integrand is again geometrically natural.
Feng Luo, Rutgers University, Solving Thurston's equation in the real numbers
Thurston's equation defined on triangulated 3-manifolds tends to find hyperbolic structures. It is usually solved in the complex numbers. We are interested in solving Thurston's equation in the real numbers and we establish a variational principle associated to such solutions.
Kei Nakamura, Temple University, On convex and non-convex Fuchsian polyhedral realizations of hyperbolic surfaces with a single conical singularity.
For a hyperbolic surface \(S\) with genus \(g \geq 2\) and with some conical singularities of positive curvatures, its Fuchsian polyhedral realization is an incompressible isometric embedding of \(S\) in a Fuchsian cylinder \(\mathbb{H}^3/\Gamma\) for some Fuchsian group \(\Gamma\) with genus \(g\) such that the image is a piecewise totally geodesic polyhedral surface. It is known by a theorem of Fillastre that, for any such \(S\), there exists a unique convex Fuchsian polyhedral realization. We will describe the geometry of convex and non-convex Fuchsian polyhedral realizations when \(S\) has a single conical singularity, and show that the convex case indeed corresponds to the Delaunay triangulation of \(S\).
Abhijit Champanerkar, CUNY College of Staten Island, Volume bounds for generalized twisted torus links
Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. In this talk we will discuss the geometry of twisted torus links and related generalizations. We will give upper bounds on their hyperbolic volume and exhibit many families of twisted torus knots with interesting properties. This is joint work with David Futer, Ilya Kofman, Walter Neumann and Jessica Purcell.
2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021