# Yamabe Memorial: Past symposia

## Speakers, abstracts, and materials

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2022

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2022 Yamabe Memorial Symposium

Friday, Sept. 30 to Sunday, Oct. 2, 2022

##### Pengfei Guan, McGill University — Curvature flows and monotonicity of associated geometric functionals

We discuss relationships of specific geometric functionals and curvature flows. In one direction, geometric functionals play essential roles for the convergence of curvature flows. Of special interest is Gauss curvature flow where entropy functional is crucial for non-collapsing estimates and convergence. In recent works, the arguments have been further implemented to deal with convergence of non-homogeneous Gauss curvature flow and Gauss curvature flow in space forms. The key is monotonicity or almost monotonicity of the entropy along the flows. In the reverse direction, curvature flows can be designed to solve isoperimetric type problems for geometric functionals. They serve as paths to optimal solutions for isoperimetric problem with proper monotonicity. This approach has been carried out successfully for classical quermassintegrals of non-convex domains and their variations in space forms. We will discuss some recent progresses and related open problems.

##### Sumio Yamada, Gakushuin University — Harmonic maps in general relativity

H. Weyl in 1916 described the Schwarzschild metric; the first exact solution to the Einstein equation by a single harmonic function with a pole. Since then, the Einstein equation has been regarded as an elliptic variational problem, and I will report on the recent progress in this direction, in particular a series of collaborative work with Gilbert Weinstein and Marcus Khuri. We will introduce spactimes of dimension four and five, and discuss the difference in 4 and 5, and the geometric consequences.

##### Song Sun, Berkeley — Complete Ricci-flat Kahler metrics asymptotic to cones

Complete Ricci-flat metrics with Euclidean volume growth provide singularity models for non-collapsed limits of Ricci-flat metrics. It is known that such a metric is always asymptotic to cones in a weak sense; when the metric had quadratic curvature decay, Colding-Minicozzi proved it is always asymptotic to a unique cone at a logarithmic rate. I will explain that when the metric is Kahler, the asymptotic rate is always polynomial. This result hinges on the global completeness of the metric, and is in sharp contrast to the case of local singularities where the logarithmic rate is optimal. This talk is based on joint work with Junsheng Zhang.

##### Ailana Fraser, University of British Columbia — When are stable minimal surfaces holomorphic?

The question of whether stable minimal surfaces are holomorphic under suitable geometric conditions has been much studied, beginning with work of Simons in complex projective space, and the proof of the Frankel conjecture by Siu-Yau. A theorem of Micallef shows that a stable minimal immersion of a complete oriented parabolic surface into Euclidean 4-space is holomorphic with respect to an orthogonal complex structure, and the same result in all dimensions if additionally the minimal surface has finite total curvature and genus zero. However, Arezzo, Micallef and Pirola gave a counterexample in general. It is therefore necessary to strengthen the stability condition in the general question. We will discuss some recent progress on this question. This is joint work with R. Schoen.

##### Antoine Song, University of California Berkeley — Groups homology classes, and Plateau's problem

Consider a countable group G, and an integer homology class h of G. The class h can be naturally represented by a geometric cycle in a Hilbert spherical quotient manifold via the regular representation of G. I will talk about the corresponding volume minimization problem and some of its properties.

##### Organizers

- Anar Akhmedov
- Ionut Ciocan-Fontanine
- Tian-Jun Li
- Al Marden
- Jiaping Wang

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2018

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The 9th Symposium: Moduli in Algebraic Geometry

Friday, September 28 to Sunday, September 30, 2018

##### Resolving singularities in families — Dan Abramovich (Brown University)

Semistable reduction is often the first step in constructing compactified moduli spaces, and can be used to discover their properties. I will describe work-in-progress with Michael Temkin and Jaroslaw Wlodarczyk in which we prove functorial semistable reduction for families of varieties in characteristic 0, refining work with Karu from 2000. Techniques developed for moduli spaces enter in unexpected ways.

##### "Smoothings of singular Calabi-Yau threefolds" — Robert Friedman (Columbia University)

Calabi-Yau threefolds can degenerate in many interesting ways. In this talk, which is a report on joint work with Radu Laza, we survey some results on Calabi-Yau threefolds acquiring canonical singularities and describe, under suitable hypotheses, how the geometry of smoothings is related to the topology of an appropriate resolution.

##### Mixed Spin P field theory and Gromov-Witten invariants of quintic Calabi-Yau threefold — Jun Li (Stanford)

Mixed Spin P field is a mathematical realization of Witten’s vision to study Gromov Witten invariants of quintic Calabi-Yau threefolds. In this talk, I will report the current progress toward solving all genus generating functions of the Gromov-Witten invariants of quintic Calabi-Yau threefolds.

##### Segre series for Hilbert schemes of points on surfaces — Alina Marian (Northeastern University)

The calculation of intersection-theoretic invariants for Hilbert schemes of points on a smooth surface is a basic problem in enumerative geometry. An interesting series of invariants is given by the top Segre classes of tautological vector bundles over the Hilbert scheme. In the case of tautological bundles arising from line bundles over the surface, conjectural formulas for the Segre series were proposed by M. Lehn in 1999, with a proof emerging recently in work of C. Voisin and a joint project with D. Oprea and R. Pandharipande. I will explain this circle of ideas, and will describe progress in the higher-rank case. The talk is based on ongoing work with Dragos Oprea and Rahul Pandharipande.

##### Umemura 3-folds as moduli of abelian surfaces, and the Horrocks-Mumford bundle — Shigeru Mukai (RIMS Kyoto)

The (compactified) moduli of elliptic curves of full d-level structure is the projective line P1 with a polyhedral group action for d=2,3,4,5. For these value of d, the moduli of (1,d)-polarized abelian surfaces with suitable level structure is P3 (d=2,3,4) or has the P3 blown-up at 60 points as its small resolution (d=5). After a quick review of these result together with principally polarized case (d=1), I discuss the reconstruction of abelian surfaces via the Horrocks-Mumford vector bundle on P4 in the case d=5.

##### Gromov-Witten theory of elliptic fibrations — Aaron Pixton (MIT)

Let X be a variety. Gromov-Witten invariants of X are defined by integrals on the moduli space of stable maps from a genus g curve to X. I will outline a conjecture interpreting these invariants as coefficients of certain automorphic forms if X is the total space of an elliptic fibration X -> B. I will also discuss a relationship between this conjecture and Givental's R-matrix action on cohomological field theories. This is joint work with Georg Oberdieck.

##### Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds — Yukinobu Toda (Kavli IPMU and University of Tokyo)

In this talk, I will discuss birational geometry for Joyce's d-critical loci, by introducing notions such as 'd-critical flips', 'd-critical flops', etc. I will show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry, e.g. certain wall-crossing diagrams of Pandharipande-Thomas stable pair moduli spaces form a d-critical minimal model program. I will also show the existence of semi-orthogonal decompositions of the derived categories under simple d-critical flips satisfying some conditions. This is motivated by a d-critical analogue of Bondal-Orlov, Kawamata's D/K equivalence conjecture, and also gives a categorification of wall-crossing formula of Donaldson-Thomas invariants.

##### On the construction of moduli of Fano varieties with K-stability — Chenyiang Xu (BICMR Beijing)

Abstract: Higher dimensional geometry naturally leads to the fundamental problem of parametrizing Fano varieties. However, it seems in general a nice parametrization doesn't exist, unless we post some stability condition. The notion of K-stability, which was first formulated by differential geometers like Tian and Donaldson, has a deep connection with Kahler-Einstein metric problem and suggests a good candidate. But to carry out it in algebraic geometry is a challenging problem, because of the 'infinite' nature involved in the original definition. In the recent years, there is a lot of progress in the algebraic geometry theory of K-stability, by the contribution of many people. In this talk, I will survey the progress, with a concentration on new techniques from birational geometry, e.g. the minimal model program.

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2016

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The 8th Symposium: Symplectic Geometry and Complex Geometry

Friday, September 30 to Sunday, October 2, 2016

##### A Cobordism Outlook on Lagrangian Topology — Paul Biran (ETH)

- Lagrangian Cobordism I
- Lagrangian Cobordism and Fukaya Categories
- Lagrangian Cobordism in Lefschetz Fibrations

Lagrangian cobordism is a natural notion introduced by Arnold in the early days of symplectic topology. In recent years it has been revived and extended to a theory which is relevant in the study of Lagrangian topology and Fukaya categories. In this talk we will survey these developments covering both topological as well as geometric aspects of the story. The talk is based on a series of joint works with Octav Cornea.

##### The Cohomological McKay Correspondence and Symplectic Cohomology — Mark Mclean (Stony Brook University)

Suppose that we have a finite quotient singularity C^n/G admitting a crepant resolution Y (i.e. a resolution with c_1 = 0). The cohomological McKay correspondence says that the cohomology of Y has a basis given by irreducible representations of G (or conjugacy classes of G). Such a result was proven by Batyrev when the coefficient field F of the cohomology group is Q. We give an alternative proof of the cohomological McKay correspondence in some cases by computing symplectic cohomology+ of Y in two different ways. This proof also extends the result to all fields F whose characteristic does not divide |G| and it gives us the corresponding basis of conjugacy classes in H^*(Y). We conjecture that there is an extension to certain non-crepant resolutions. This is joint work with Alex Ritter.

##### Legendrian fronts for affine varieties — Emmy Murphy (MIT)

Given a smooth complex affine variety, we discuss a method for describing its symplectic topology, by representing it as a Weinstein handle body, which in turn is given as an explicitly constructed Legendrian link. After describing this recipe (which is mostly algorithmic), we discuss a number of applications as time allows. Possibilities are: flexibility of Milnor fibers of some non-isolated singularities, mirror symmetry for the Legendrian trefoil, constructions of some new exact Lagrangians, or a proof that the Koras-Russell cubic is symplectomorphic to C^3. This is joint work with Roger Casals.

##### Lagrangian Floer theory - Generating criterion for Fukaya category and related topics — Kaoru Ono (Kyoto University)

I will explain some constructions in Lagrangian Floer theory and present "generating criterion for Fukaya category" and related topics. The talk is based on a joint work with M. Abouzaid, K. Fukaya, Y.-G. Oh and H Ohta and other joint works with K. Fukaya, Y.-G. Oh and H. Ohta.

##### Around Nakamaye's Theorem — Valentino Tosatti (Northwestern)

Nakamaye's Theorem from around 2000 says that the augmented base locus of a nef and big line bundle on a smooth projective variety equals its null locus (the locus of subvarieties where the restriction of the line bundle is not big). I will discuss a transcendental generalization of this theorem to all compact complex manifolds (with line bundles replaced by (1,1) cohomology classes) and some of its applications. I will also discuss a conjectural generalization to big (1,1) classes which are not nef. Joint work with T. Collins.

##### Hyper-Kaehler manifolds — Claire Voisin (Directrice de recherche, CNRS Institut de Mathmatiques de Jussieu)

Hyper-Kaehler manifolds (also called quaternionic or holomorphic symplectic manifolds) are higher dimensional generalizations of K3 surfaces. They are in particular Kaehler but projective ones are dense in the moduli space. The topology of these varieties is rather restricted, although we do not know what are all possible topological types. Their deformation theory is "easy". By Torelli theorem, it is governed by the period map. All known examples have been constructed by algebraic geometry. I will discuss these general facts and describe a few constructions.

##### Transversality and super-rigidity for multiply covered holomorphic curves — Chris Wendl (UCL/Humboldt University Berlin)

##### Hyperbolicity of moduli spaces of polarized manifolds — Sai Kee Yeung (Purdue)

In this talk, we would explain some joint work with Wing-Keung To on hyperbolicity of moduli spaces of polarized algebraic manifolds, including families or moduli spaces of Kaehler-Einstein manifolds of negative scalar curvature or trivial scalar curvature, and their quasi- projective versions. Classically, results on moduli space of Riemann surfaces of genus at least two have been obtained by Ahlfors, Royden and Wolpert with Weil-Petersson metric. Study of moduli of higher dimensional manifolds in terms of Weil-Petersson metrics began with a work of Siu thirty years ago. Eventually, we construct a negatively curved Finsler metric on such moduli from which results in hyperbol- icity follow naturally. We also apply the related techniques to study questions related to a conjecture of Viehweg.

##### Organizers

- Scot Adams
- Anar Akhmedov
- Bob Gulliver
- Tian-Jun Li
- Al Marden
- Yu-Jong Tzeng
- Jiaping Wang

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2014

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The 7th Symposium: Current Topics in Three-Manifolds

Friday, October 17 to Sunday, October 19, 2014

##### Open Problems in 3-manifold topology, Ian Agol (University of California at Berkeley)

We will survey some problems in low-dimensional topology, many relating to finite covers of 3-manifolds and properties of their fundamental groups.

##### On the asymptotic dimension of a curve complex, Mladen Bestvina (University of Utah)

It has been known that the curve complex of a compact surface has finite asymptotic dimension (Bell-Fujiwara) bounded by an exponential function in the complexity of the surface (Webb). In this joint work with K. Bromberg we show that there is a linear bound. The method is to estimate the capacity (or Assouad-Nagata) dimension of the visual boundary by directly constructing required coverings via train tracks on the surface. This uses the work of Buyalo, Gabai and Hamenstadt.

##### The Thurston program for hyperbolic 3-manifolds: an overview, Dick Canary (University of Michigan)

In the 1970s, Bill Thurston developed a program for the study of hyperbolic 3-manifolds and formulated a number of far-reaching conjectures about their structure and classification. Many of his conjectures have been resolved in the last decade. We will give an overview of the recent progress in preparation for this weekend's Yamabe symposium. We will try and avoid technical discussions and make the talk appropriate for a general mathematical audience.

##### Typical properties of hyperbolic 3-manifolds, Ursula Hamenstaedt (University of Bonn)

By recent results of Agol et al, closed hyperbolic three-manifolds have finite covers which are mapping tori of pseudo-Anosov diffeomorphisms of a closed surface. Pseudo-Anosov mapping classes can be counted according to their dilatation. We discuss some properties of hyperbolic three-manifolds which hold for most of those mapping tori (in a specific sense) and relate this to radom 3-manifolds in the sense of Dunfield-Thurston and Lubotzky-Maher-Wu.

##### Surface Subgroups, Cube Complexes, and the Virtual Haken Theorem, Jeremy Khan (CUNY Graduate Center)

In a largely expository talk, I will summarize the results leading up to the Virtual Haken and Virtual Fibered Theorem for three manifolds, including:

- The Geometrization Theorem of Thurston and Perleman
- The Surface Subgroup Theorem of the speaker and Vladimir Markovic
- The cube complex theory of Daniel Wise and others
- Ian Agol’s criterion for virtual fibering and his proof of the Virtual Haken Theorem

The emphasis will be on the flow of ideas rather than on detailed statements and proofs.

##### The triangulation conjecture — Ciprian Manolescu (University of California, Los Angeles)

The triangulation conjecture stated that any n-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology. At the end I will also mention other applications of this technique, and discuss a related version of Heegaard Floer homology.

##### Harmonic map between hyperbolic spaces —Vlad Markovic (University of Cambridge)

The Dirichlet problem we study is as follows: Given a quasiconformal map f of the (n-1)-sphere, find a (hyperbolic) harmonic map that extends f to the n-dimensional hyperbolic space. We show that for n>2, the Dirichlet problem has a unique solution, thus answering the well known conjecture of Rick Schoen for n>2. The problems remains open for n=2.

##### Convergence and divergence of Cannon-Thurston maps — Mahan Mj (Ramakrishna Mission Vivekananda University)

Several years ago Jim Cannon and Bill Thurston made an unexpected discovery. Let M be a closed hyperbolic 3-manifold fibering over the circle with fiber a closed surface S. The inclusion of S into M lifts to a map between universal covers S* (say the unit disk) and M*. Cannon and Thurston showed that this inclusion extends to a map from the disk boundary onto a space-filling (Peano) curve on the 2-sphere (regarded as the boundary of M*), equivariant under the fundamental group of S. This led Thurston to the following questions:

- Is this a general phenomenon for discrete faithful representations of a surface group into the isometry group of hyperbolic 3-space?
- How does this map behave with respect to sequences of representations?

I shall give a review of my work (partly with C. Series) leading to a complete resolution of these questions.

##### Some remarks on (virtual) properties of Kaehler groups, Stefano Vidussi (University of California at Riverside)

Abstract: It is known that there is no infinite, finitely presented group that is simultaneously the fundamental group of a Kaehler manifold and of a closed 3-manifold. I will ignore this fact, and I'll try to convince you (and myself) that it is anyhow interesting to look at what properties could Kaehler groups share with 3-manifold groups.

##### Organizers

- Scot Adams
- Anar Akhmedov
- Bob Gulliver
- Tian-Jun Li
- Al Marden
- Yu-Jong Tzeng
- Jiaping Wang

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2012

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The 6th Symposium: Geometric Analysis

Friday, October 5 to Sunday, October 7, 2012

##### Geometry of Gradient Ricci Solitons — Huai-dong Cao (Lehigh University)

Abstract: The concept of Ricci solitons was introduced by Richard Hamilton in mid 1980's to study singularity formations in the Ricci flow. Ricci solitons are natural extensions of Einstein manifolds. They are also self-similar solutions to the Ricic flow and often arise as singularity models. In this talk, I shall describe some recent progress on the geometry and classifications of Ricci solitons, especially my joint work with Qiang Chen on Bach-flat shrinking Ricci solitons.

##### Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture — Jean-Pierre Demailly (University of Grenoble)

Abstract: The goal of the talk will be to investigate the existence and properties of entire curves drawn in a complex projective variety. More specifically, we show that all such curves must satisfy certain global differential equations as soon as the variety is of general type. This is strongly related to the Green-Griffiths-Lang conjecture which asserts the algebraic degeneracy of entire curves, and also to some fundamental questions of arithmetic geometry (higher dimensional generalaizations of the Mordell-Faltings theorem). Our strategy relies on a use of holomorphic Morse inequalities and on a probabilistic estimate of the cohomology of jet bundles.

##### Minimal volume metrics, entropy, and degree — Benson Farb (University of Chicago)

Abstract: In this talk I will explain some major open problems, and some progress, on the following topics:

- Finding (suitably normalized) metrics that minimize volume and/or entropy on a fixed manifold.
- What is the relationship between closed manifolds M,N admitting a degree d>0 map f:M--->N, and the volumes Vol(M), Vol(N)?

##### Some Homology and Cohomology Theories for a Metric Space —Robert Hardt (Rice University)

Abstract: Various classes of chains and cochains may reveal geometric as well as topological properties of metric spaces. In 1957, Whitney introduced a geometric "flat norm" on polyhedral chains in Euclidean space, completed to get flat chains, and defined flat cochains as the dual space. Federer and Fleming also considered these in the sixties and seventies, for homology and cohomology of Euclidean Lipschitz neighborhood retracts. These include smooth manifolds and polyhedra, but not algebraic varieties or subspaces of some Banach spaces. In work with Thierry De Pauw and Washek Pfeffer, we find generalizations and alternate topologies for general metric spaces. With these, we homologically characterize Lipschitz path connectedness and obtain several facts about spaces that satisfy local linear isoperimetric inequalities.

##### Polyhedral complexes and topology of complex-projective varieties — Misha Kapovic (University of California, Davis)

Abstract: I will explain how to use polyhedral complexes of constant curvature and hyperbolic orbifolds in order to construct some interesting complex-projective varieties. Among other things, I will show how to construct projective varieties with normal crossing singularities and with prescribed finitely-presented fundamental groups.

One of the key tools of the construction is the geometry of 3-dimensional hyperbolic orbifolds related to the Jorgensen-Marden conjecture on cut-locus of such orbifolds.

##### Instantons in G2 geometry — Conan Leung (Chinese University of Hong Kong)

Abstract: M-theory on G2 manifolds is an analog of string theory on symplectic manifolds. The role of holomorphic curves with Lagrangian boundary conditions is replaced by associative submanifolds with coassociative boundary conditions. The work of Fukaya-Oh related holomorphic disks in cotangent bundles with Morse flow lines in Lagrangian submanifolds. Wang, Zhu and I generalized this to the G2 setting, namely thin associative submanifolds can be constructed from regular holomorphic curves in coassiciative submanifolds. This can be used to construct new examples of associative submanifolds.

##### Ancient solutions and singular solutions to the Yamabe flow — Natasa Sesum (Rutgers University)

Abstract: In the first part of the talk I will discuss a way of constructing new solutions to the Yamabe flow that look like towers of bubbles, that is, spheres joined by short necks. This is a joint work with P. Daskalopoulos. In the second part of the talk I will mention the results on the Ricci flow (same as the Yamabe flow in two dimensions) on conic surfaces. In the joint work with Mazzeo and Rubenstein we construct a solution to the Ricci flow o conic surfaces that preserves the conic angles. Long time behavior of the flow is related to the Uniformization theorem.

##### Parabolic flows in complex geometry — Ben Weinkove (UCSD/Northwestern)

Abstract: Parabolic flows are powerful tools in the study of geometric structures on manifolds. In this talk I will discuss some work (joint with Jian Song) on the behavior of the Ricci flow on Kahler manifolds. In particular, we analyze the singularities that form in complex dimension two and show how the flow can be continued through the singularities. I will also talk about some joint work with Valentino Tosatti on another parabolic flow, called the Chern-Ricci flow. This flow was first introduced by Matt Gill, and is a natural flow to consider on more general complex manifolds. I will discuss the behavior of this flow on complex surfaces, and give a number of examples.

##### Speakers

- Huaidong Cao
- Ben Weinkove
- Benson Farb
- Jean-Pierre Demailly
- Robert Hardt
- Conan Leung
- Misha Kapovich
- Natasa Sesum

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2010

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The 5th Symposium: Geometry and Low-Dimensional Topology

Friday to Sunday, October 8–10, 2010

##### Nodal Set of Eigenfunctions — Toby Colding (Massachusetts Institute of Technology)

Abstract: We discuss lower bounds for the Hausdorff measure of nodal sets of eigenfunctions. This is joint work with Bill Minicozzi.

##### Mirror symmetry between Toric A model and Landau-Gizburg B model — Kenji Fukaya (Kyoto University)

Abstract: In a series of papers (with Oh-Ohta-Ono) we studied Lagrangian Floer theory in the case of Toric manifolds and its Lagrangian fiber of the moment map. I would like to explain how it implies a kind of Mirror symmetry. I also explain how we can use it to obtain information about the Lagrangian submanifold L of Toric manifolds in the case L is not necessarily a Lagrangian fiber. (This part is a joint work with Abousaid, Oh, Ohta, Ono.) The relation of it to cyclic and Hochshild homology and to K. Saito theory over Novikov rings is also discussed.

##### Volumes of Hyperbolic 3-Manifolds — David Gabai (Princeton University)

Abstract: As part of his revolutionary work on hyperbolic geometry in the 1970's, Thurston generalizing work of Jorgensen and Gromov, showed that that the set of volumes of complete finite volume hyperbolic 3-manifolds is closed and well ordered. Recently, Robert Meyerhoff and Peter Milley and the speaker showed that the Weeks manifold is the unique lowest volume closed orientable one, culminating a 30+ year effort by many mathematicians using a wide variety of techniques. In particular, we make use of work of Agol - Dunfield which relies on Perelman's work on Ricci flow. This lecture will survey these developments and discuss various open problems.

##### Smooth Group Actions on 4-Manifolds and Equivariant Gauge Theory — Ian Hambleton (McMaster University)

Abstract: Algebraic automorphisms of algebraic surfaces provide natural examples of smooth finite group actions on 4-manifolds. This talk will describe an equivariant version of the Yang-Mills moduli spaces, and some applications to "rigidity" results for smooth finite group actions on 4-manifolds.

##### On Four-Dimensional Einstein Manifolds — Claude LeBrun (State University of New York at Stony Brook)

Abstract: A Riemannian metric is said to be Einstein if it has constant Ricci curvature. A central problem in differential geometry is to determine which smooth compact manifolds admit an Einstein metric, and to completely understand the moduli space of all such metrics when they exist . The 4-dimensional case of this problem appears to be highly atypical. This lecture will survey some recent results regarding the special case of 4-manifolds which admit either a complex structure or symplectic structure.

##### Einstein Manifolds and Extremal Kahler Metrics — Claude LeBrun (State University of New York at Stony Brook)

Abstract: This talk will present some new existence results for extremal Kahler metrics on toric surfaces, with applications to the construction of Einstein metrics. In particular, I will explain how an Einstein metric on CP2#2(-CP2) can be constructed by a bubbling-off procedure. The same results also shed new light on the corresponding uniqueness problem.

##### The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties — Melissa Liu (Columbia University)

Abstract: I will discuss (i) SYZ transformation relating equivariant coherent sheaves on a toric variety to Lagrangians in the cotangent of R^n, (ii) microlocalization functor relating the Fukaya category of the cotangent to constructible sheaves on the base (due to Nadler-Zaslow, Nadler), and (iii) a categorification of Morelli's theorem relating equivariant coherent sheaves on a toric variety to constructible sheaves on R^n. This talk is based on joint work with Bohan Fang, David Treumann and Eric Zaslow.

##### Recent Results on Cosmetic Surgeries — Yi Ni (California Institute of Technology)

Abstract: Dehn surgery is a fundamental construction in 3-dimensional topology. Two Dehn surgeries on the same knot are cosmetic if they yield homeomorphic manifolds. Such surgerie s are very rare. I will talk about some recent results on the classification of cosmetic surgeries. The proofs use Heegaard Floer homology and sutured manifold theory. Some of this talk is based on joint work with Zhongtao Wu.

##### Group Actions on Surfaces of General Type — Ron Stern (University of California at Irvine)

Abstract: The automorphism group of an algebraic surfaces of general type is known to be finite. Since a diffeomorphism of such a surface, when viewed as a smooth 4-manifold, must preserve the Seiberg-Witten and Donaldson basic classes, one would suspect that the diffeomorphism group of a surface of general type demonstrate some finiteness properties. For example: Are there only finitely many smoothly distinct but topologically equivalent smooth actions of a fixed cyclic group? We will report on joint work with Ron Fintushel and Nathan Sunukjian that many surfaces of general type have, in fact, infinitely many distinct smooth but topologically equivalent actions of a fixed cyclic group.

##### Speakers

- Toby Colding
- Kenji Fukaya
- David Gabai
- Ian Hambleton
- Claude LeBrun
- Melissa Liu
- Yi Ni
- Ron Stern

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2008

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The 4th Symposium: Geometry and Analysis

Friday to Sunday, September 26–28, 2008

##### Speakers

- Simon Brendle — Stanford University
- Alice Chang — Princeton University
- Gerhard Huisken — Albert Einstein Institute, Potsdam
- Ngaiming Mok — Hong Kong University
- Leon Simon — Stanford University
- Yum-Tong Siu — Harvard University
- Neil Trudinger — Australia National University
- Burkhard Wilking — University of Muenster

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2006

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The 3rd Symposium: Geometry & Symplectic Topology

Friday to Sunday, September 15–17, 2006

##### Speakers

- Denis Auroux — MIT
- Yasha Eliashberg — Stanford University
- Ron Fintushe — Michigan State University
- Mikio Furuta — University of Tokyo
- Helmut Hofer — Courant Institute
- Dusa McDuff — SUNY Stony Brook
- Peter Ozsvath — Columbia University
- Tongbin Ruan — University of Michigan

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2004

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The 2nd Symposium: Geometry & Physics

Friday to Sunday, September 17–19, 2004

##### Speakers

- Robert Bryant — Duke University
- Sheldon Katz — University of Illinois
- Kefeng Liu — UCLA
- Duong Phong — Columbia University
- Paul Seidel — University of Chicago
- Isadore M. Singer — MIT
- Karen Uhlenbeck — University of Texas
- Shing-Tung Yau — Harvard University

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2002

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The 1st Symposium: Geometry & Analysis

Friday to Sunday, September 20–22, 2002

##### Speakers

- Hubert Bray — MIT
- Ben Chow — University of California at San Diego
- Richard Hamilton — Columbia University
- Peter Li — University of California at Irvine
- Fang-Hua Lin — Courant Institute, NYU
- Richard Schoen — Stanford University
- Gang Tian — MIT
- Brian White — Stanford University