Department of

Mathematics


Seminar Calendar
for Differential Geometry events the next 12 months of Saturday, January 1, 2005.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, January 20, 2005

2:00 pm in 345 Altgeld Hall,Thursday, January 20, 2005

Geometry of Hermitian algebraic functions

Dror Varolin   [email] (UIUC Math)

Abstract: I will consider a class of functions, called Hermitian algebraic, whose domains are the total space of a holomorphic line bundle. I will define these functions, and show that they are generalizations of very natural mathematical objects, namely bihomogeneous Hermitian polynomials. I will talk about a few elementary properties of Hermitian algebraic functions, and then proceed to state a Hermitian algebraic analogue of Hilbert's 17th problem for such functions. (Recall Hilbert's 17th problem = Artin's Theorem: Any non-negative polynomial is a sum of squares of rational functions.) The analog can be vaguely stated as follows: Is any non-negative Hermitian algebraic function a quotient of squares of holomorphic mappings? This Hermitian algebraic analogue of Hilbert's 17th problem was considered by D. Quillen in a certain important case, and later by D'Angelo and Catlin in the general setting. Both Quillen and Catlin-D'Angelo settled the problem in the affirmative under the hypothesis that the Hermitian function is strictly positive away from the zero section. D'Angelo also showed that in the Hermitian algebraic setting, there are non-negative Hermitian algebraic functions that are not quotients of squares. I will state necessary and sufficient conditions under which a non-negative Hermitian algebraic function is a quotient of squares. I will then indicate how to prove this result in the case considered by D'Angelo-Catlin. If time permits, I will indicate how one goes on to the general case.

Thursday, February 10, 2005

2:00 pm in 345 Altgeld Hall,Thursday, February 10, 2005

Isotopies of Lagrangian spheres

Richard Hind (Notre Dame)

Abstract: Lagrangian spheres arise naturally in symplectic manifolds as vanishing cycles of Lefschetz fibrations. We try to classify such spheres up to Hamiltonian diffeomorphism in some simple symplectic 4-manifolds. Sometimes, like in S^2 x S^2, there is a unique Lagrangian sphere, but other manifolds contain nonequivalent Lagrangian spheres which can be homotopic or even smoothly isotopic.

Thursday, February 24, 2005

2:00 pm in 345 Altgeld Hall,Thursday, February 24, 2005

Anti-self-dual connections from a symplectic viewpoint

Katrin Wehrheim (Institute of Advanced Study)

Abstract: The space of connections over a Riemann surface is symplectic, and the gauge action of bundle isomorphisms can be viewed as Hamiltonian group action. The corresponding symplectic vortex equation is the anti-self-duality equation for connections over the product of two surfaces. So it is natural to consider anti-self-dual connections with Lagrangian boundary conditions. I will explain this framework and show that the boundary behaviour of these connections is governed by a holomorphic part of the equation. In a joint project with Dietmar Salamon we use these connections to define a new Floer theoretic invariant for 3-manifolds with boundary. This is an intermediate object between the instanton Floer homology of a homology sphere and a symplectic Floer homology arising from a Heegard splitting. It might thus be used to prove a conjecture of Atiyah and Floer relating these.

Thursday, March 3, 2005

2:00 pm in 345 Altgeld Hall,Thursday, March 3, 2005

Nondegeneracy of CMC Surfaces

John M Sullivan   [email] (UIUC Math)

Abstract: We give a new geometric interpretation of recent work by Korevaar, Kusner and Ratzkin, which proves that many CMC surfaces are nondegenerate in the sense of having no square-integrable Jacobi fields. This sheds new light on our earlier classification of triunduloids.

Thursday, March 10, 2005

2:00 pm in 345 Altgeld Hall,Thursday, March 10, 2005

On the complexity of matrix multiplication

Joseph Landsberg   [email] (Texas A&M Math)

Abstract: I will explain how I solved a 20 year old problem in computational complexity using elementary differential and algebraic geometry. The problem in question was to determine if the best known algorithm for multiplying two by two matrices is the best possible, even after allowing for a small error. I will then explain a more general program for studying the asymptotic exponent for matrix multiplication - the number x such that asymptotically one can multiply n by n matrices using n^x multiplications. (Using naive row-column multiplication technique we know x is at most 3 and much work over the years has brought it close to 2.) This program involves studying higher secant varieties of compact Hermitian symmetric spaces, a subject of interest in its own right. Most of the talk will be elementary, acessible to anyone who knows how to multiply matrices.

Thursday, March 17, 2005

2:00 pm in 345 Altgeld Hall,Thursday, March 17, 2005

Constructing invariants for hyperbolic manifolds using geodesic currents

Martin Bridgeman (Boston College)

Abstract: Geodesic currents are a natural generalization of closed geodesics. We will use geodesic currents to describe geometric invariants for Kleinian groups. In particular we will show that Thurston's description of the Weil-Peterson metric for Teichmuller space can be extended to give a two-form on quasifuchsian space.

Thursday, April 21, 2005

2:00 pm in 345 Altgeld Hall,Thursday, April 21, 2005

Arithmetic harmonic analysis and cohomology of hyperkahler manifolds

Tamas Hausel (University of Texas, Austin)

Abstract: Motivated by mirror symmetry considerations, I will study the problem of calculating Hodge numbers of hyperkahler varieties, like moduli spaces of Higgs bundles, flat connections and character varieties of Riemann surfaces or toric hyperkahler varieties, and Nakajima's quiver varieties, such as twisted ADHM spaces and Hilbert schemes of points on the complex plane. Various forms of Fourier transform over finite fields are then shown to calculate the Betti numbers (or at least certain Hodge numbers) of these hyperkahler varieties. Finally a recent conjecture comparing the cohomology of character varieties and certain Nakajima quiver varieties is explained, which hints at a deep connection between the character tables of finite Lie groups and Lie algebras; and also features a mysterious appearance of Macdonald polynomials in the cohomology of character varieties.

Thursday, April 28, 2005

2:00 pm in 243 Altgeld Hall,Thursday, April 28, 2005

Minimal and constant mean curvature surfaces in Carnot groups

Scott Pauls (Dartmouth)

Abstract: In the past decade many authors have investigated variational problems in the setting of Carnot-Carath\'eodory spaces. I will give a brief survey of known results on two basic problems, the minimal surface problem and the isoperimetric problem, starting with the basic existence results of Garofalo and Nhieu, Danielli Garofalo and Nhieu, and Leonardi and Rigot. I will then discuss several aspects of both problems in the Heisenberg group including considerations of uniqueness and smoothness as well as a connection between these two variational problems and the geometry of the minimal/isoperimetric surfaces. Last, I will discuss some new work in which I develop a horizontal second fundamental form for hypersurfaces in general Carnot groups and provide a general link between the solutions to these problems and the geometry and curvature of hypersurfaces.

Thursday, August 25, 2005

3:00 pm in 147 Altgeld Hall,Thursday, August 25, 2005

Organizational Meeting

Thursday, September 8, 2005

3:00 pm in 241 Altgeld Hall,Thursday, September 8, 2005

Hyperbolic 3-manifolds with discrete length spectrum

Christopher Leininger   [email] (UIUC Math)

Abstract: If $N$ is a complete hyperbolic 3-manifold with finitely generated fundamental group, then $N$ has discrete length spectrum (not counting multiplicities) if and only if one of the following holds 1. $N$ is geometrically finite or 2. $N$ (or a two-fold cover) is an infinite cyclic (locally isometric) cover of a finite volume fibered hyperbolic $3$-manifold. This is joint work with R. D. Canary.

Thursday, September 15, 2005

3:00 pm in 241 Altgeld Hall,Thursday, September 15, 2005

Collapsing manifolds with boundary

Jeremy Wong (UIUC Math)

Abstract: I give existence and regularity results for Gromov-Hausdorff limits of sequences of certain classes of manifolds with boundary. In addition to metric structure of the limits, I will present a fibering theorem which asserts that the manifolds in the sequence are locally trivial fiber bundles over their limit, if the limit is weakly geodesically extendible.

Thursday, September 29, 2005

3:00 pm in 241 Altgeld Hall,Thursday, September 29, 2005

Kaehler metrics on toric varieties

Eugene Lerman   [email] (UIUC Math)

Abstract: This is joint work with D. Burns and V. Guillemin. Geometric Invariant Theory quotients of C^N by torus actions have natural Kaehler metrics. In the case when the quotient is a smooth compact manifold, a formula for such a metric was found by Guillemin about 10 years ago. We show that the same formula works for any quotient, even if it is singular and non-compact. The proof is a simple exercise with Legendre transforms.

Thursday, October 6, 2005

3:00 pm in 241 Altgeld Hall,Thursday, October 6, 2005

The Geometry of Reconfiguration

Valerie Peterson (UIUC Math )

Abstract: There are a number of settings in which a dynamic system coordinates local rules in order to effect global changes in the states of the system; that is, the system reconfigures. I will give a general definition of a reconfigurable system and several examples of how and where they arise (in robotics, biology and computer science, to name a few places). I will also define an abstract cubical complex - the state complex - that keeps track of local moves and global states, as well as the independence of moves (i.e. which ones "commute"). Finally, I plan to prove that there are only finitely many time-optimal geodesics between two points on the state complex of a reconfigurable system, due to the presence of non-positive curvature. CAT(0) geometry is the main tool used in the result, but prior knowledge will not be assumed. This is joint work with R. Ghrist.

Thursday, October 13, 2005

3:00 pm in 241 Altgeld Hall,Thursday, October 13, 2005

Lagrangian fibrations on holomorphic symplectic manifolds

Justin Sawon   [email] (SUNY Stony Brook)

Abstract: A somewhat surprising result (due to Matsushita) is that a holomorphic symplectic manifold (compact hyperkahler manifold) can only be fibred by Lagrangian complex tori, i.e. no other kind of fibration can exist. An example in two dimensions is an elliptic K3 surface. Now by carefully studying elliptic K3 surfaces, it is not too difficult to show that all K3 surfaces are deformation equivalent. What can Lagrangian fibrations tell us about holomorphic symplectic manifolds in higher dimensions? In this talk I will describe some results on Lagrangian fibrations and their moduli spaces.

Thursday, October 20, 2005

3:00 pm in 241 Altgeld Hall,Thursday, October 20, 2005

Symplectomorphism groups and embeddings of balls

Martin Pinsonnault   [email] (Fields Institute)

Abstract: Let $(X,\omega)$ be a closed symplectic $4$-manifold and let $Emb (B_c, X)$ be the space of symplectic embeddings of the standard ball $B_c$ of capacity $c$ into $X$. This space of embeddings encodes a lot of information about the diffeomorphism type of the symplectic form. In some cases, this space is homogeneous with respect to the natural action of the symplectomorphism group of $X$ and the stabilizer of an embedding can be identified to a subgroup of the symplectomorphism group of the corresponding blow-up. Therefore, one can compute the homotopy type of $Emb(B_c, X)$ provided one has a good understanding of these symplectomorphism groups. We will apply these ideas to the projective plane $CP^2$ and to all symplectic ruled surfaces.

Friday, October 28, 2005

3:00 pm in 241 Altgeld Hall,Friday, October 28, 2005

A new approach to the Weinstein conjecture in dimension three

Casim Abbas   [email] (Michigan State)

Abstract: In my lecture I will describe a general strategy for approaching the Weinstein conjecture in dimension three which is joint work with Kai Cieliebak and Helmut Hofer. The conjecture dating back to 1979 asserts that a Hamiltonian system on an arbitrary (four dimensional) symplectic manifold must have periodic trajectories on each closed energy hypersurface of contact type. We have applied this approach to prove the Weinstein conjecture for a new class of three dimensional contact manifolds (planar contact manifolds). I will discuss how the present approach reduces the general Weinstein conjecture in dimension three to a compactness problem for the solution set of a first order elliptic PDE.

Friday, November 4, 2005

3:00 pm in 141 Altgeld Hall,Friday, November 4, 2005

The K-theory of symplectic quotients

Megumi Harada   [email] (University of Toronto)

Abstract: The topology of symplectic quotients is a topic of great interest in many different contexts, including combinatorics, algebraic geometry, representation theory, and gauge theory. We first give an overview and motivation for this subject, and then discuss a surjectivity result which expresses the K-theory of a symplectic quotient M//G in terms of the equivariant K-theory of the original manifold M. This result is the natural K-theoretic analogue of the Kirwan surjectivity theorem for rational cohomology. Along the way, we prove a K-theoretic version of a key lemma of Atiyah and Bott, which states that the equivariant K-theory Euler class of a G-bundle is not a zero divisor, provided that an S^1 subgroup fixes precisely the zero seciton. This lemma is a key result in equivariant symplectic geometry, and (time permitting) we discuss some further applications of this lemma. This is joint work, and work in progress, with Gregory D. Landweber.

Thursday, November 10, 2005

3:00 pm in 241 Altgeld Hall,Thursday, November 10, 2005

A (partial) Lagrangian Piunikhin-Salamon-Schwarz morphism

Peter Albers   [email] (NYU)

Abstract: One of the major tools in symplectic topology is Floer homology, which can be assigned either to a Hamiltonian function on a symplectic manifold or a Lagrangian submanifold. Hamiltonian Floer homology is known to be isomorphic to the singular homology of the symplectic manifold. A particular isomorphism was constructed by Piunikhin, Salamon and Schwarz in 1994. Under some topological assumption Lagrangian Floer homology can be shown to be isomorphic to the singular homology of the Lagrangian submanifold, but in general this is not true; it might even vanish identically. In 1996 Oh constructed a spectral sequence relating Lagrangian Floer homology and singular homology. We show to what extent the approach of Piunikhin, Salamon and Schwarz can be adapted to Lagrangian Floer homology. Furthermore, this leads to a comparison morphism between Lagrangian and Hamiltonian Floer homology resembling the natural homomorphism in singular homology, which is induced by the inclusion. This leads to some new insights into the topological structure of Lagrangian submanifolds.

Thursday, November 17, 2005

3:00 pm in 241 Altgeld Hall,Thursday, November 17, 2005

Orbifold cohomology of abelian symplectic reductions

Tara Holm   [email] (University of Connecticut)

Abstract: I will talk about the topology of symplectic (and other) quotients. I will briefly review Kirwan's techniques for proving that the restriction map from the equivariant cohomology of the originial space to the ordinary cohomology of the symplectic reduction is a surjection. I will show how this result can be used to understand various aspects of the topology of quotients, focusing on the theme of orbifolds and computing the Chen-Ruan orbifold cohomology ring of symplectic reductions by torus actions.

Thursday, December 1, 2005

3:00 pm in 241 Altgeld Hall,Thursday, December 1, 2005

Teichmuller geodesic rays which do not have a limit in PMF

Anna Lenzhen (UIC)

Abstract: We construct a Teichmuller geodesic ray which does not have a limit on the Thurston boundary of Teichmuller space.

Friday, December 2, 2005

3:00 pm in 141 Altgeld Hall,Friday, December 2, 2005

Symplectic Anosov structures on Riemann

Anna Wienhard (IAS)

Abstract: Flat $G$-bundles on Riemannian manifold M are parametrised by representations of the fundamental group $\pi_1(M)$ into $G$. I will explain that for a special subset of the space of representations of a surface group into the symplectic group $Sp(2n,R)$ we can construct a splitting into two (non-flat) Lagrangian subbundles. Contraction and expanding properties of these subbundles imply that the holonomy representations of these symplectic vector bundles are quasiisometric embeddings, in particular they are faithful with discrete image. These special subsets of the space of representations can be viewed as generalized Teichm\"uller space. If time permits I will explain some relations with other moduli spaces.

Thursday, December 8, 2005

3:00 pm in 241 Altgeld Hall,Thursday, December 8, 2005

Knot invariants from contact homology

Lenny Ng   [email] (AIM / Stanford)

Abstract: I will describe a technique of constructing invariants of knots and other submanifolds through contact topology and holomorphic curves. The knot invariant can be described purely topologically, with no reference to contact geometry, and is fairly strong. In particular, it encodes the Alexander polynomial and is closely related to the A-polynomial.