Yuguang Zhang, Tsinghua University and UC at San Diego
"Collapsing of Kahler metrics and analytic space"
Time: 3:30 PM
Location: Hill 425
Abstract: In this talk, we study the relationship of Kahler metrics along a degeneration of projective manifolds and the solutions of the non-archimedean Monge-Ampere equation. For a degeneration of projective manifolds, the solutions of the non-archimedean Monge-Ampere equation were obtained on the corresponding Berkovich analytic space by Bouckson, Favre and Jonsson. In this talk, we show that the solution can be approximated by a family of Kahler metrics on the original family of projective manifolds in the potential function sense. This result generalizes a theorem of Haase and Zharkov for Calabi-Yau hypersurfaces in toric varieties.
Tuesday, April 30th
Hongbin Sun, Princeton University
"A Transcendental Invariant of Pseudo-Anosov Maps"
Time: 3:30 PM
Location: Hill 425
Abstract: For each pseudo-Anosov map phi on surface S, we will associate it with a Q-submodule of R, denoted by A(S,phi). A(S,phi) is defined by interaction between Thurston norm and dilatation of pseudo-Anosov map. We will develop a few nice properties of A(S,phi) and give a few examples to show that A(S,phi) is a nontrivial invariant. These nontrivial examples give an answer to a question asked by McMullen: the minimal point of the restriction of the dilatation function on a fibered face need not be a rational point.
Friday, April 26th - CANCELLED
Special Geometry/Topology Seminar
Fuquan Fang, Notre Dame/Capital Normal University (NOTE: SPECIAL DATE AND TIME)
"Reflection groups, non-negative curvature and Tits geometry"
Time: 2:30 PM
Location: Hill 425
Abstract: A reflection in a euclidean space (sphere) is one of the fundamental notions of symmetry of geometric figures. It plays a central role in Killing and Cartan's work on Lie algebra in 19th century. Reflections groups on a hyperbolic space is important in hyperbolic geometry, and the first example goes back to F. Klein and Poincare. In this talk I will present
(i) A complete classification of reflection groups and the equivariant structures of complete non negatively curved manifolds.
(ii) A complete classification of positively curved polar manifolds of cohomogeneity at least 2, which is achieved partially based on Tits geometry.
(joint works with Karsten Grove and G. Thorbergsson)
Tuesday, April 23rd
Babak Modami, Yale
"Prescribing the behavior of Weil-Petersson geodesics"
Time: 3:30 PM
Location: Hill 425
Abstract: The Weil-Petersson (WP) metric is an incomplete Riemannian metric on the moduli space of Riemann surfaces with negative sectional curvatures which are not bounded away from 0. Brock, Masur and Minsky introduced a notion of "ending lamination" for WP geodesic rays which is an analogue of the vertical foliations of Teichm"{u}ller geodesics. In this talk we show that these laminations and the associated subsurface coefficients can be used to determine the itinerary of a class of WP geodesics in the moduli space. As a result we give examples of closed WP geodesics staying in the thin part of of the moduli space, geodesic rays recurrent to the thick part of the moduli space and diverging geodesic rays. These results can be considered as a kind of symbolic coding for WP geodesics.
Tuesday, April 16th
Marian Mrozek, Uniwersytet Jagiellonski, Krakow, Poland
"A topological algorithm for computing the Conley index of Poincare maps in time-periodic differential equations"
Time: 3:30 PM
Location: Hill 425
Abstract: Conley index is a topological invariant of dynamical systems used to prove the existence of invariant sets. The use of the Conley index in the study of recurrent dynamics of a flow is not straightforward. The Conley index for flows is not very useful and usually the Conley index of a Poincare map is studied. Unfortunately, the computation of the index is not easy, because in general no explicit formula for the Poincare map is available. An alternative is to compute the index numerically. The standard algorithmic technique requires computing outer enclosures of the Poincare map. This requires long time integration along the trajectories of the flow which is computationally expensive and often prohibits successful computations. We will discuss some earlier strategies to overcome the problem as well as a recent topological approach which does not require long time integration.
This is research in progress, joint work with R. Srzednicki and F. Weilandt.
Tuesday, April 9th
Frank Lutz, Technische Universitat Berlin
"Combinatorial Roundness of Grains in Cellular Microstructures and Random Discrete Morse Theory for Cell Complexes"
Time: 3:30 PM
Location: Hill 425
Abstract: Polycrystalline materials, such as metals, are composed of crystal grains of varying size and shape. Some of the occurring grain types are substantially more frequent than others. We will observe that the frequent types are ``combinatorially round'' -- which gives us a new starting point for the microstructure analysis of steel.
Computational homology packages such as CHomP or RedHom allow to obtain homological information for large complexes from material sciences or other. These packages extensively use (NP-hard) discrete Morse theory as a (fast) preprocessing step to avoid (slow, polynomial time) Smith Normal Form computations. In fact, it is surprisingly hard to construct ``complicated'' examples on which homology calculations perform poorly. We propose a new library of complicated triangulations and we introduce Random Discrete Morse Theory as a computational scheme to measure the complicatedness of a triangulation. An interesting infinite series of complicated triangulations is based on the Akbulut-Kirby spheres related to the Andrews-Curtis conjecture.
Thursday, April 4th
Special Geometry/Topology Seminar
Ben McReynolds, Purdue University
" Multiplicity, conjugacy, and algorithmic complexity"
Time: 11:00 AM
Location: Hill 124
Abstract: There are many algorithms for solving the conjugacy problem on
a finite rank free group. I will discuss a few different ways at the
conjugacy problem on a free group and introduce some notions of
complexity aimed at measuring efficiency. I will relate this to geodesic
geometry on hyperbolic n-manifolds. I will end with some curious
conjectures that whether true or false, have nice implications.
Tuesday, April 2nd
Allison Gilmore, UCLA
"Knot homologies and ideal quotients"
Time: 3:30 PM
Location: Hill 425
Abstract: A knot homology theory assigns a (bi- or triply-)graded chain complex to a knot such that the complex's chain homotopy type is an invariant of the knot and the complex's graded Euler characteristic is a classical knot polynomial. This talk will describe work in progress that aims to relate Khovanov and Rozansky's HOMFLY-PT homology (which lifts the HOMFLY-PT polynomial) to Ozsv'ath-Szab'o and Rasmussen's knot Floer homology (which lifts the Alexander polynomial). We will first define the two theories in apparently similar, purely algebraic terms. The main result is a relationship between the two algebraic constructions expressed in terms of ideal quotients. We will sketch a proof for this result, then discuss its consequences in the context of a spectral sequence that conjecturally relates the two homology theories.
Tuesday, March 26th
Guillaume Dreyer, Notre Dame
"Anosov representations along a geodesic lamination"
Time: 3:30 PM
Location: Hill 425
Abstract: Let λ ⊂ S be a geodesic lamination, where S is a connected, closed, oriented surface of genus g ≥ 2. We consider homomorphisms ρ: π1(S)→ PSLn(R) that satisfy a certain Anosov property along the leaves of λ. We discuss various geometric properties of these so-called λ-Anosov representations; in particular, we define invariants for these representations that generalize Thurston's length function, and give a characterization of the λ-Anosov property. We then introduce cataclysm deformations, and obtain, under some additional genericity condition, a parametrization of the subset of λ-Anosov representations via transverse n-twisted co-cycles for the geodesic lamination λ.
Tuesday, March 12th
Patricia Cahn, University of Pennslyvania
"Algebras Counting Intersections and Self-Intersections of Loops on a Surface"
Time: 3:30 PM
Location: Hill 425
Abstract: Goldman and Turaev defined a Lie bialgebra structure on the vector space generated by nontrivial free homotopy classes of loops on an oriented surface. The Goldman bracket and Turaev cobracket give lower bounds on the minimal intersection and self-intersection numbers of loops in given free homotopy classes, respectively. Chas showed that these bounds are not equalities in general. Andersen, Mattes, and Reshetikhin defined a Lie bracket that generalizes Goldman's. We show that their bracket gives a formula for the minimal intersection number. We also define an operation that generalizes Turaev's cobracket in the same way that the Andersen-Mattes-Reshetikhin bracket generalizes Goldman's bracket, and show this operation gives a formula for the minimal self-intersection number. Some of this work is joint with Vladimir Chernov.
Tuesday, March 5th
Wei Li, Capitol Normal and Princeton Universities
"Non-compactness of stationary harmonic maps in weak sense"
Time: 3:30 PM
Location: Hill 425
Abstract: Let M and N be two compact Riemannian manifolds. Given a sequence of stationary harmonic maps from M to N with bounded energies, we may assume that it converges weakly to a weakly harmonic map in W^{1,2}(M, N). One can ask whether the
limit map could be also stationary or not. In this talk, some basic definitions and results about harmonic maps
will be reviewed firstly. Then we will construct a concrete example to show that the limit map may not be stationary.
Tuesday, February 26th
Steve Ferry, Rutgers University
"Taming wild homotopies"
Time: 3:30 PM
Location: Hill 425
Abstract: We prove that if G is a group with finite asymptotic dimension, then the integral Novikov conjecture is true for G. In the course of the proof, we prove that if G is such a group, then the Higson compactification of EG is mod p acyclic for all p.
This is joint work with Dranishnikov and Weinberger.
Tuesday, February 19th
Tian Yang, Rutgers University
"Skein Algebras and the Decorated Teichmuller Space"
Time: 3:30 PM
Location: Hill 425
Abstract: The Kauffman bracket skein module K(M) of a 3-manifold M is defined by Przytycki and Turaev as an invariant for framed links in M satisfying the Kauffman skein relation. For a compact oriented surface S, it is shown by Bullock-Frohman-Kania-Bartoszynska and Przytycki-Sikora that K(Sx[0,1]) is a quantization of the SL_2 C -characters of the fundamental group of S with respect to the Goldman-Weil-Petersson Poisson bracket.
In the joint work with Julien Roger that I will be talking about, we define a skein algebra of a punctured surface as an invariant for not only framed links but also framed arcs in Sx[0,1] satisfying the skein relations of crossings both in the surface and at punctures. This algebra quantizes a Poisson algebra of loops and arcs on S in the sense of deformation of Poisson structures. This construction provides a tool to quantize the decorated Teichmuller space; and the key ingredient in this construction is a collection of geodesic lengths identities in hyperbolic geometry which generalizes/is inspired by Penner's Ptolemy relation, the trace identity and Wolpert's cosine formula.
Tuesday, February 12th
Tian Yang, Rutgers University
"Hyperbolic cone metrics on 3-manifolds with boundary"
Time: 3:30 PM
Location: Hill 425
Abstract: We prove that a hyperbolic cone metric on an ideally triangulated compact $3$-manifold with boundary consisting of surfaces of negative Euler characteristic is determined by its combinatorial curvature. The proof uses a convex extension of the Legendre transformation of the volume function. Several related results on maximum volume angle structures are obtained.
This is a joint work with Feng Luo.
Tuesday, February 5th
Priyam Patel, Rutgers University
"Quantifying Residual Finiteness and LERF-ness in Terms of Geometric Data"
Time: 3:30 PM
Location: Hill 425
Abstract: This talk will begin by defining residual finiteness (RF) and locally extended residual finiteness (LERF) for groups, followed by a brief history of the results that study the connection between these algebraic properties and the fundamental groups of surfaces and 3-manifolds. We will then describe what it means to quantify these group properties and present the results that quantify RF-ness and LERF-ness of hyperbolic surface groups in terms of geometric data. If time permits, we will conclude with an overview of similar techniques used to quantify residual finiteness for particular hyperbolic 3-manifold groups.
Tuesday, January 29th
Sergio Fenley, Florida State and Princeton University
"Structure and rigidity of totally periodic pseudo-Anosov flows in graph manifolds"
Time: 3:30 PM
Location: Hill 425
Abstract: This is joint work with Thierry Barbot.
A graph manifold is an irreducible manifold so that all pieces of the torus decomposition are Seifert fibered. We consider pseudo-Anosov flows in graph manifolds so that all pieces are periodic. We consider pseudo-Anosov flows in graph manifolds so that each Seifert fibered piece is periodic. This means that a regular fiber is freely homotopic to a closed orbit of the flow. We show that these flow are rigid, that is, they are completely determined by the dynamics and topological structure of a dynamical spine associated to the flow.
Tuesday, January 22nd
Christian Zickert, University of Maryland
"Thurston's gluing equations for PGL(n,C)"
Time: 3:30 PM
Location: Hill 425
Abstract: Thurston's gluing equations are polynomial equations invented by Thurston to explicitly compute hyperbolic structures or, more generally, representations in PGL(2,C). This is done via so called shape coordinates. We generalize the shape coordinates to obtain a parametrization of representations in PGL(n,C). We give applications to quantum topology, and discuss an intriguing duality between the shape coordinates and the Ptolemy coordinates of Garoufalidis-Thurston-Zickert. The shape coordinates and Ptolemy coordinates can be viewed as 3-dimensional analogues of the X- and A-coordinates on higher Teichmuller spaces due to Fock and Goncharov.
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