Abstract:
Let $M$ be a compact orientable 3-manifold, and $F$ be an incompressible surface which cuts $M$ into two components $M_{1}$ and $M_{2}$. In this case, $M$ is called the surface sum of $M_{1}$ and $M_{2}$ along $F$. In this talk, recent improvements on the Heegaard splittings of surface sums will be introduced.
Tuesday, November 24th
Shicheng Wang, Peking University, China
"Graph manifolds have virtually positive Seifert volume"
Time: 3:30 PM
Location: Hill 124
Abstract: There are two invariants of 3-manifolds which respect maps: simplicial volume and Seifert volume. For prime 3-manifolds, simplicial volume detects exactly their hyperbolic parts and 3-manifolds with zero simplicial volume are exactly graph manifolds. It is proved recently that each closed non-trivial graph manifold has a finite cover of positive Seifert volume. As a consequence for each closed orientable prime 3-manifold N, the set of mapping degrees D(M,N) is finite for any 3-manifold M unless N is finitely covered by either a torus bundle, or a trivial circle bundle, or the 3-sphere. This is joint work with Pierre Derbez.
Tuesday, November 17th
Richard Falk, Rutgers University
"Finite Element Exterior Calculus"
Time: 3:30 PM
Location: Hill 124
Abstract: We first provide an introduction to finite element methods, discussing the concepts of consistency, stability, and convergence, and introduce the notion of mixed finite element methods. We next establish a connection to exterior calculus, and in particular to the de Rham complex and to the Hodge Laplacian. We then discuss abstract approximation of the de Rham complex and of boundary value problems for the Hodge Laplacian. The key ideas leading to stable approximation schemes for the Hodge Laplacian are that the approximating spaces form a subcomplex of the de Rham complex and there is a bounded cochain projection from the de Rham complex to the approximating subcomplex. Finally, we outline the construction of two families of finite element spaces of differential forms that satisfy the abstract conditions, and hence are useful in the approximation of a number of important boundary value problems for partial differential equations arising in applications.
Tuesday, November 10th
Chris Atkinson, Temple University
" Volume estimates for right-angled hyperbolic Coxeter polyhedra"
Time: 3:30 PM
Location: Hill 124
Abstract:
We will give two-sided combinatorial volume estimates for right-angled hyperbolic polyhedra which is linear in the number of vertices. As an application, we will sketch how to use the lower bound estimate along with ideas from the proof of the Orbifold Theorem to give a lower volume bound for any hyperbolic Coxeter polyhedron.
Tuesday, October 20th
Zhang Weiyi, University of Minnesota
"The Kodaira dimension and Lefschetz fibration"
Time: 3:30 PM
Location: Hill 124
Abstract:
The Kodaira dimension, for complex manifolds was introduced by Kodaira more than 50 years ago, and has played an important role in the classification theory, especially for complex surfaces. In the symplectic world, this notion has been defined only for manifolds with dimension two and four. Some preliminary classification has also been achieved. We can also naturally introduce a Kodaira dimension for a 4-dimensional Lefschetz fibration when the base has genus at least one. We will discuss the equivalence of these different Kodaira dimensions. Finally, we will present a new interpretation of symplectic Kodaira dimension in dimension 4. We use this to generalize our results to some Lefschetz fibration when the base is a sphere.
Tuesday, October 13th
Zhenglei Zhang, Princeton University/Capital Normal University, China
" Degeneration of closed shrinking Ricci solitons"
Time: 3:30 PM
Location: Hill 124
Abstract: In this talk, I am going to show that a Gromov-Hausdorff limit of the shrinking Ricci solitons with upper bounded diameter and lower bounded volume has closed singular set of codimension at least two. The proof follows essentially from Cheeger-Colding's results. title: Degeneration of closed shrinking Ricci solitons
Tuesday, October 6th
Joseph Maher, CUNY
"Asymptotics for pseudo-Anosovs in the Teichmuller lattice"
Time: 3:30 PM
Location: Hill 124
Abstract: Given a point in Teichmuller. space, we call the orbit of the point under the mapping class group a Teichmuller lattice. We show that the asymptotic growth rate of the number of pseudo-Anosov lattice points in a ball of radius r is the same as the asymptotic growth rate of the total number of lattice points in the ball of radius r. This uses recent work of Athreya, Bufetov, Eskin and Mirzakhani.
Tuesday, September 29th
Kent Orr, Indiana University, Bloomington
"L^2 methods, knot concordance, localization and amenable groups"
Time: 3:30 PM
Location: Hill 124
Abstract: L^2 signatures play a central role in the study of knot concordance, a classical relation on knots closely allied with deep
considerations in singularity theory and the classification of 4-manifolds. Using a new approach which subsumes past
results, we extend the above techniques to the related problem of classifying manifolds up to homology cobordism, and
significantly extend key results concerning invariance of L^2 signatures and betti numbers.
We exhibit new examples of homology equivalent manifolds in low and high dimensions which are not homology
cobordant. Many of these results involve groups with torsion, unassailable via prior tools.
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