Workshop and Conference on the Topology and Invariants of Smooth 4-Manifolds
July 31 to August 10, 2013
University of Minnesota, Twin Cities

Organizers: Selman Akbulut (Michigan State University), Anar Akhmedov (University of Minnesota), Weimin Chen (University of Massachusetts, Amherst), Cagri Karakurt (University of Texas, Austin), Tian-Jun Li (University of Minnesota).

The main focus of this workshop will be on the topology and invariants of smooth 4-manifolds. The workshop is a part of the FRG: Collaborative Research: Topology and Invariants of Smooth 4-Manifolds. It is funded by NSF Focused Research Grant DMS-1065955. There will be 5 days of mini-courses (July 31 - August 4), primarily for graduate students, followed by a conference during the second week (August 5 - 10).

Registration:

Thanks to the generous support of the National Science Foundation, funds are available for partial support of participant expenses, such as lodging and meals. The applicants are kindly asked to seek for travel support from their home institutions. To apply for funding, you must register by Friday, April 12, 2013 , but early applications are strongly encouraged since there are a limited number of rooms available. Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Applicants are requested to register (see the registration form below), send a CV, and have one brief reference letter sent to Cagri Karakurt at karakurt@math.utexas.edu. The reference letter is optional for people with a Ph.D.

***Please note that we are no longer accepting application*** .

Housing and Transit:

We have reserved rooms at the UMN dorms for most of the participants. The following link is the website of the Centennial Hall, which is a 5 minute walking from the Math Department: Centennial Hall . There you can find the directions, the check in information, and the list of amenities. The rooms are single, they have air conditions and linens will be provided. The front desk of the Centennial dorm is open 24 hours every day, and the staffs have your name and and travel dates.

Please check out the information below for transportation options for you to get to the campus/dorm/hotel from the airport.

You can find the transit information in the IMA website .

1. The cheapest way to get to the Math Department or your hotel is by Light Rail. The fair is $1.75 except during peak periods when it is $2.25. It can be faster than taxi, depending on the traffic. Here are directions: take Light Rail to the Metrodome station, and then take bus #16 or #50. For more detailed information, please check the following link: Bus and Light Rail - MSP Airport .

2. There is also a shuttle service ($16)

For more detailed information, please check the following link: Super Shuttle - MSP Airport .

3. By Taxi (about $40 to $45). For more detailed information, please check the following link: Taxi - MSP Airport .

Here is a map showing both the Math dept (Vincent Hall) and Mech E Dept.

Map .

Mini-course speakers:

Abstract of mini courses :

4-manifolds via their handlebodies (by Selman Akbulut)

Abstract:

Handlebody descriptions of 3 and 4-manifolds will be discussed, and as time permits their various applications to 4-manifold problems will be given, such as carving, branch coverings, knot complements, complex surfaces, Stein manifolds, Lefschetz fibrations, BLF's, corks and plugs. From these techniques various exotic manifolds will be constructed, such as going from handlebody of logarithmic transforms to Dolgachev surfaces, and from handlebody of surface bundles over surfaces to Akhmedov-Park exotic manifolds.

Construction of exotic 4-manifolds (by Anar Akhmedov)

Abstract:

Lecture 1: Construction of Lefschetz fibrations via Luttinger surgery

Luttinger surgery is a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold. The surgery was introduced by Karl Murad Luttinger in 1995, who used it to study Lagrangian tori in R^4. Luttinger's surgery has been very effective tool recently for constructing exotic smooth structures on 4-manifolds. In this talk, using Luttinger surgery, I will present a new constructions of Lefschetz fibration over 2-sphere whose total space has arbitrary finitely presented group G as the fundamental group (a joint result with Burak Ozbagci).

Lecture 2: The geography of symplectic 4-manifolds

The symplectic geography problem, originally posed by Robert Gompf, ask which ordered pairs of nonnegative integeres are realized as (chi(X), c1^2(X)) for some symplectic 4-manifold X. In this lecture we will address the geography problem of simply-connected spin and non-spin symplectic 4-manifolds with nonnegative signature or near the Bogomolov-Miyaoka-Yau line c1^2(X) = 9chi(X) (joint results with B. Doug Park).

Group actions on 4-manifolds (by Weimin Chen)

Abstract:

In this mini-course we give an introduction to some basic questions and basic techniques in the study of finite group actions on 4-manifolds, with an emphasis given to symplectic finite group actions.

Lecture 1: A general introduction to finite group actions on 4-manifolds, with emphasis on locally linear topological actions: basic properties, construction, and obstructions to smoothability.

Lecture 2: Symplectic finite group actions. J-holomorphic curves in 4-orbifolds. Orbifold Seiberg-Witten-Taubes theory.

Lecture 3: Symplectic actions on CP^2 and K3 surfaces. Some open problems.

Elements of HM and ECH (by Yi-Jen Lee)

Abstract:

I will go over some background of Seiberg-Witten-Floer homology and ECH (in a generalized sense), their relations, and if time permits, some computations.

The Kodaira dimension of symplectic 4-manifolds (by Tian-Jun Li)

Abstract:

Lecture 1. Kodaira dimension for low dimensional manifolds

Lecture 2. Symplectic 4-manifolds with non-positive Kodaira dimension

The triangulation conjecture (by Ciprian Manolescu)

Abstract:

First lecture: The Seiberg-Witten equations in 3 dimensions

I will review the Kronheimer-Mrowka construction of monopole Floer homology for 3-manifolds, with particular emphasis on the Froyshov invariant. I will also discuss the Pin(2) symmetry that appears in the presence of a spin structure, and some notions of Pin(2)-equivariant topology.

Second lecture: Finite dimensional approximation and the Floer spectrum

I will present an alternate construction of S^1-equivariant Seiberg-Witten Floer homology, based on finite dimensional approximation and Conley index theory. The construction yields something more, an S^1-equivariant Floer spectrum. When we have a spin structure, everything can be tweaked to take into account the Pin(2)-equivariance of the equations.

Third lecture (conference talk): The triangulation conjecture

We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov'’s correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that the 3-dimensional homology cobordism group has no elements of order 2 that have Rokhlin invariant one. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.

Lefschetz fibrations and symplectic fillings of contact 3-manifolds (by Chris Wendl)

Abstract:

Lefschetz fibrations have played a major role in symplectic topology since the 1990's, when they appeared in the topological characterization of closed symplectic 4-manifolds due to Donaldson and Gompf. A short time later, Giroux introduced a corresponding characterization of contact manifolds in terms of open book decompositions, which can be understood as the natural structure arising at the boundary of a Lefschetz fibration over the disk. My goal in this minicourse will be to explain the relationship between symplectic/contact structures and these types of topological decompositions, and to sketch some striking applications that make use of holomorphic curve technology, e.g. the fact that all weak fillings of planar contact manifolds are blow-ups of Stein fillings (a joint result with Niederkrueger), and a recent generalization of that result in joint work with Lisi and Van Horn-Morris. My perspective on this subject is rather more geometric than topological: instead of following the topologist's standard approach of understanding Lefschetz fibrations and Stein structures in terms of handle decompositions, I will explain a version of the "Thurston trick" that gives existence and uniqueness (up to deformation) of symplectic and Stein structures on Lefschetz fibrations over arbitrary surfaces with boundary. I will then sketch some of the holomorphic curve techniques that give results in the other direction.

Here is an approximate outline of the content of the lectures:

LECTURE 1: "Basic notions and results"

Definitions of contact structures and weak/strong/exact/Stein fillings, Lefschetz fibrations, spinal open books and contact structures, a theorem on fillings of partially planar spinal open books, examples and applications.

LECTURE 2: "Symplectic geometry from topological decompositions"

Existence/uniqueness of contact structures on spinal open books (generalizing a result of Thurston-Winkelnkemper), existence/uniqueness of symplectic and Stein structures on (allowable) Lefschetz fibrations (generalizing results of Thurston and Gompf).

LECTURE 3: "Topological decompositions from symplectic geometry"

Stable Hamiltonian structures, holomorphic curves in completed symplectic fillings, reconstructing a Lefschetz fibration from a planar open book at the boundary.

Lecture notes and slides from the mini-courses :

  • W. Chen - 07-31-13 - Group actions on 4-manifolds I
  • W. Chen - 08-01-13 - Group actions on 4-manifolds II
  • W. Chen - 08-02-13 - Group actions on 4-manifolds III
  • T. Li - 07-31-13 - Kodaira dimensions of low dimensional manifolds
  • T. Li - 08-01-13 - Symplectic 4-manifolds with non-positive Kodaira dimension
  • A. Akhmedov - 08-02-13 - Construction of Lefschetz fibrations via Luttinger surgery
  • A. Akhmedov - 08-02-13 - Exotic smooth structures on 4-manifolds
  • Schedule of the workshop:

    On the weekdays, the lectures will be in the Mech Engineering Department, Room 18. It is right by the Math Department. It is convenient to enter the building from the left most entrance (going down the stairs). On the weekends, the lectures will be in the Math dept, Room 16. The lecture room 16 is in the basement of the Math Deptartment.

    Wednesday 7/31 Thursday 8/1 Friday 8/2 Saturday 8/3 Sunday 8/4
    9:00-9:30
    Registration and Coffee (math department lounge)
    9:00-9:30
    Coffee
    9:00-9:30
    Coffee
    9:00-9:30
    Coffee
    9:00-9:30
    Coffee
    9:30-10:30
    W. Chen I
    9:30-10:30
    W. Chen II
    9:30-10:30
    W. Chen III
    9:30-10:30
    C. Manolescu II
    9:30-10:30
    Y-J. Lee I
    11:00-12:00
    T-J. Li I
    11:00-12:00
    T-J. Li II
    11:00-12:00
    C. Manolescu I
    11:00-12:00
    A. Akhmedov II
    11:00-12:00
    Y-J. Lee II
    Lunch Lunch Lunch Lunch Lunch
    2:00-3:00
    S. Akbulut I
    2:00-3:00
    S. Akbulut II
    2:00-3:00
    C. Wendl III
    2:00-5:00
    Short talks
    3:30-4:30
    C. Wendl I
    3:30-4:30
    C. Wendl II
    3:30-4:30
    A. Akhmedov I
    5:30 - 6:30
    Soccer
    5:30
    Dinner (math department lounge)
    5:30 - 6:30
    Soccer


    Schedule of the conference: On the weekdays, the lectures will be in the Mech Engineering Department, Room 18. It is right by the Math Department. It is convenient to enter the building from the left most entrance (going down the stairs). On the weekends, the lectures will be in the Math Dept, Room 16. The lecture room 16 is in the basement of the Math Department.

    Monday 8/5 Tuesday 8/6 Wednesday 8/7 Thursday 8/8 Friday 8/9 Saturday 8/10
    9:00-9:30
    Registration and Coffee
    9:00-9:30
    Coffee
    8:30-9:00
    Coffee
    9:00-9:30
    Coffee
    9:00-9:30
    Coffee
    8:30-9:00
    Coffee
    9:30-10:30
    C. Manolescu
    9:30-10:30
    S. Akbulut
    9:00-10:00
    A. Stipsicz
    9:30-10:30
    M. Ue
    9:30-10:30
    M. Furuta
    9:00-10:00
    H-J. Kim
    11:00-12:00
    R. Gompf
    11:00-12:00
    B. Ozbagci
    10:15-11:15
    T. Cochran
    11:00-12:00
    Y. Ni
    11:00-12:00
    Z. Wu
    10:15-11:15
    J. Wang
    Lunch Lunch 11:30-12:30
    C. Karakurt
    Lunch Lunch
    1:30-2:30
    D. Auckly
    1:30-2:30
    I. Baykur
    free
    afternoon
    1:30-2:30
    H. Endo
    1:30-2:30
    M. Usher
    2:45-3:45
    W. Zhang
    2:45-3:45
    L. Starkston
    2:45-3:45
    M. Hamilton
    2:45-3:45
    W. Wu
    4:15-5:15
    F. Arikan
    4:15-5:15
    T. Lidman
    4:15-5:15
    J. Dorfmeister
    4:15-5:15
    M. Pinnsonault
    5:30 - 6:30
    Soccer
    5:30
    Dinner (math department lounge)
    5:30 - 6:30
    Soccer

    Titles and abstracts of the talks :

    Exotic Stein Fillings (by Selman Akbulut)

    Abstract:

    I will construct infinitely many simply connected compact 4-dimensional Stein handlebodies with second Betti number 2, such that they are all homeomorphic but not diffeomorphic to each other, furthermore they are Stein fillings of the same contact 3-manifold. I will briefly explain their role in the world of corks and plugs. This is a joint work with K. Yasui.

    Lefschetz Fibrations on Compact Stein Manifolds (by Firat Arikan)

    Abstract:

    We prove that up to diffeomorphism every compact Stein manifold W of dimension 2n+2>4 admits a Lefschetz fibration over the two-disk with Stein regular fibers, such that the monodromy of the fibration is a symplectomorphism induced by compositions of right-handed Dehn twists along embedded Lagrangian n-spheres on the generic fiber. Moreover, the induced open book supports the induced contact structure on the boundary of W. This result generalizes the Stein surface case of n=1, previously proven by Loi-Piergallini and Akbulut-Ozbagci. This is a joint work with Selman Akbulut.

    Strict One-stable Equivalence in Four Dimensions (by David Auckly)

    Abstract:

    A well-known theorem of Wall establishes that two simply-connected, homeomorphic smooth $4$-manifolds become diffeomorphic after taking the connected sum with enough copies of $S^2 \times S^2$. We call two manifolds $n$-stable equivalent if they are diffeomorphic after taking the connected sum with $n$ copies of $S^2 \times S^2$. It is less well known, but the analog of Wall's result holds for topologically isotopic smooth $S^2$s in a $4$-manifold and for topologically isotopic diffeomorphisms. In this talk we will present smooth $S^2$s that are strictly $1$-stable equivalent, as well as diffeomorphisms that are strictly $1$-stable equivalent. (Joint with Danny Ruberman, Paul Melvin, and Hee-Jung Kim)

    Topological complexity of symplectic 4-manifolds and Stein fillings (by Inanc Baykur)

    Abstract:

    Following the ground-breaking works of Donaldson and Giroux, Lefschetz pencils and open books have become central tools in the study of symplectic 4-manifolds and contact 3-manifolds. An open question at the heart of this relationship is whether or not there exists an a priori bound on the topological complexity of a symplectic 4-manifold, coming from the genus of a compatible Lefschetz pencil on it, and a similar question inquires if there is such a bound on any Stein filling of a fixed contact 3-manifold, coming from the genus of a compatible open book. We will present our solutions to both questions, making heroic use of positive factorizations in surface mapping class groups of various flavors. This is joint work with J. Van Horn-Morris

    Counterexamples to Kauffman's Conjectures on Slice knots (by Tim Cochran)

    Abstract:

    In the 1960's Levine introduced a program to decide whether or not a knot is a slice knot by studying curves on any one of its Seifert surfaces. In support of this philosophy, in 1982 Louis Kauffman conjectured that if a knot in S^3 is a slice knot then on any Seifert surface for that knot there exists a (homologically essential) simple closed curve of self-linking zero which is itself a slice knot, or at least has Arf invariant zero. Since that time, considerable evidence has been amassed in support of this conjecture. In particular, many invariants that obstruct a knot from being a slice knot have been explicitly expressed in terms of invariants of such curves on the Seifert surface. We give counterexamples to Kauffman's conjecture, that is, we exhibit (smoothly) slice knots that admit (unique minimal genus) Seifert surfaces on which every homologically essential simple closed curve of self-linking zero has non-zero Arf invariant and non-zero signatures. We not only explain the failure of Levine's philosophy, but also how it can be repaired

    Symplectic Sums along Spheres: Minimality and Kodaira Dimension (by Josef Dorfmesiter)

    Abstract:

    Symplectic Sums have provided the symplectic world with a plethora of examples. In the case of symplectic 4-manifolds, one fundamental question is under what conditions the sum is minimal, i.e. contains no embedded sphere of self-intersection -1. This question was answered by Usher for sums along surfaces of positive genus. I will detail the result in the case of sums along spheres. Moreover, it is possible to determine the Kodaira dimension of such sums, the positive genus case is again Usher's result. Of particular interest is the question if and when such a sum can have Kodaira dimension 0 and how such examples relate to known symplectic manifolds of Kodaira dimension 0. I will describe work adressing these questions in the genus 0 case.

    Chart description for hyperelliptic Lefschetz fibrations and their stabilization (by Hisaaki Endo)

    Abstract:

    Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for hyperelliptic Lefschetz fibrations, and show that any hyperelliptic Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations. This is a joint work with Seiichi Kamada.

    Two variants of the 10/8 inequality (by Mikio Furuta)

    Abstract:

    (1) A generalization of spin(^c) structure: Using SO(3)-instantons Froyshov showed a local coeffiecient version of Donaldson's theorem about definite intersection form. Using a local coefficient version of Seiberg-Witten monopoles Nakamura showed a similar result as well as a 10/8-type inequiality. I will explain a slight improvement of Nakamura's inequality. Joint work with Yukio Kametani and Nobuhiro Nakamura. (2) Spin manifolds with boundary: Two invariants $\omega_+,\omega_-$ for spin homology 3-spheres are introduced in terms of Pin(2)-equivariant Floer K group to formulate 10/8 type inequality for spin 4-manfolds with boundary. A similar result is obtained by Manolescu independently. I will explain an estimate of $\omega_+,\omega_-$ for Seifert fibered cases. Joint work with Tian-Jun Li.

    Exotic smoothings of open 4-manifolds (by Robert Gompf)

    Abstract:

    Smoothing theory for open 4-manifolds seems to have stagnated in the past decade or two, perhaps due to the misperception that since everything probably has uncountably many smoothings, that must be the end of the story. However, most traditional approaches involve tinkering with the end of the manifold without probing the deeper structure such as minimal genera of homology classes. We show that this genus function, together with its counterpart at infinity, can be controlled surprisingly well compared to the case of closed 4-manifolds, and these tools are often complementary to traditional techniques.

    Homology classes of negative square and embedded surfaces in 4-manifolds (by Mark Hamilton)

    Abstract:

    Problem 4.105 from the Kirby list asks if the self-intersection number of embedded spheres in any given simply-connected closed oriented 4-manifold X is always greater than some negative number depending only on X. We can consider the same question for embedded surfaces of arbitrary fixed genus. We will answer a part of this question and show that there is such a lower bound in the case that the homology class represented by the surface is divisible or characteristic.

    Heegaard Floer homology and numerical semigroups (by Cagri Karakurt)

    Abstract:

    Recently, Nemethi gave a combinatorial description of Heegaard Floer homology for a class of 3-manifolds containing all Seifert fibered spaces. I will talk about a reformulation of this description in terms of some numerical semigroups generated by Seifert invariants. This is a joint work with Mahir Bilen Can.

    Slice versus ribbon for some fibered knots (by Hee-Jung Kim)

    Abstract:

    In 1983, Casson and Gordon showed that a fibered ribbon knot has monodromy that (when capped off) extends over a handlebody. Shortly afterwards, Bonahon gave an infinite collection of genus 2 fibered knots K_n, and analyzed the cobordism classification of their monodromies. As a consequence, he showed (via the Casson-Gordon result) that K_m # -K_n is ribbon if and only if m=n. We investigate whether these knots can be slice, as a test of the `slice implies ribbon' conjecture, and show that K_m # -K_0 is slice if and only if m =0. This is a joint work with Daniel Ruberman.

    Invariants from Seiberg-Witten and Heegaard Floer theory (by Yi-Jen Lee)

    Abstract:

    Heegaard Floer homology of Seifert fibered homology spheres and non-zero degree maps (by Tye Lidman)

    Abstract:

    Using a recent combinatorial description of the Heegaard Floer homology of Seifert fibered homology spheres due to Can and Karakurt, we study the behavior of these objects under non-zero degree maps. This is joint work with Cagri Karakurt.

    The triangulation conjecture (by Ciprian Manolescu)

    Abstract:

    We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov'’s correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that the 3-dimensional homology cobordism group has no elements of order 2 that have Rokhlin invariant one. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.

    Applications of 4-dimensional techniques to Dehn surgery (by Yi Ni)

    Abstract:

    In recent years, techniques from 4-dimensional topology have been used to study many classical problems in Dehn surgery. I will talk about one of such applications.

    Symplectic fillings of lens spaces as Lefschetz fibrations (by Burak Ozbagci)

    Abstract:

    We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity. (This is a joint work with Mohan Bhupal)

    Hamiltonian Groups as Lie Groups (by Martin Pinnsonault)

    Abstract:

    Let M be a compact, connected, symplectic manifold. Because the group Ham(M) of Hamiltonian diffeomorphisms of M admits a bi-invariant metric, it is tempting to look for analogies between properties of Ham(M) and properties of finite dimensional Lie groups. In this talk, I will present 3 different analogies that lead to interesting results if we assume M is toric.

    Symplectic Fillings of Seifert Fibered Spaces over the 2-sphere (Laura Starkston)

    Abstract:

    The goal of this talk is to explain how to obtain a finite list of possible diffeomorphism types of symplectic fillings of a large class of Seifert fibered spaces over S^2 with their canonical contact structures. In many cases all of these diffeomorphism types can be realized as strong symplectic fillings, thus providing some complete classifications. The main arguments in the proof generalize those used by Lisca to classify symplectic fillings of Lens spaces with their standard contact structure. The technique is constructive and can suggest new diffeomorphism types that support a convex symplectic structures. Combinatorial analysis of the Seifert invariants, produces handlebody diagrams for the possible diffeomorphism types of symplectic fillings. These alternate fillings seem to all have smaller Euler characteristic than the standard filling given by a plumbing of spheres. Such fillings can provide operations on closed symplectic manifolds that generalize rational blow-downs

    Higher dimensional contact manifolds (by Andras Stipsicz)

    Abstract:

    We discuss the existence problem of contact structures on closed manifolds with odd dimension. In particular, we show that for a 4-manifold X the product XxS^1 is contact (joint with HJ Geiges) and using surgery theoretic methods we show that if M is contact, then so is MxS^2. This is joint work with Jonathan Bowden and Diarmuid Crowley.

    The mu-bar invariants and the eta invariants for Seifert rational homology 3-spheres (by Masaaki Ue)

    Abstract:

    We discuss the relation between the mu-bar invariants defined by Neumann and Siebenmann and the eta invariants for the Dirac and the signature operators for Seifert rational homology 3-spheres. Our arguments are based on the finite dimensional approximation of the Seiberg-Witten invariants by Furuta.

    Lagrangian submanifolds of CP^n (by Michael Usher)

    Abstract:

    Quasi-homomorphisms on mapping class groups (by Jiajun Wang)

    Abstract:

    A quasi-homomorphism on a group is a real-valued function which is a homomorphism up to a constant. Quasi-homomorphisms on the mapping class groups of surfaces include signature, linking number, Rasmussen invariant, Ozsvath-Szabo invariants, etc. We show that the quasi-homorphisms on a surface vanishing on the handlebody group is an infinite-dimensional vector space. As an application, we disprove a conjecture of Reznikov, which says that the set of Heegaard splittings with a given genus has a bounded generation. This is a joint work with Jiming Ma.

    On some new aspects of symplectic packing problems (by Weiwei Wu)

    Abstract:

    The problem of symplectic ball-packing has been one of the most important problems in symplectic geometry, especially in dimension 4. In this talk we summarize some recent developments on some aspects other than studying the packing capacity. This will include some new results on symplectic mapping class groups and Lagrangian uniqueness on symplectic 4-manifolds.

    On Rational Genus of Knots (by Zhongtao Wu)

    Abstract:

    In this talk, we will discuss recent progress on the 3-dimensional analogue of Thom-conjecture-type questions. In particular, we show that a simple knot in a lens space is genus minimizing within its homology class. This is a joint work with Yi Ni.

    J-holomorphic curves in a nef class (Weiyi Zhang)

    Abstract:

    After explaining complexity of arbitrary reducible subvariety when the ambient manifold is of dimension 4, we offer an upper bound of the total genus of a J-holomorphic subvariety when the class of the subvariety is J-nef. It seems new even when J is integrable. For a spherical class, it has particularly strong consequences. It is shown that, for any tamed J, each irreducible component is a smooth rational curve. We completely classify configurations of maximal dimension. To prove these results, we treat subvarieties as weighted graphs and introduce several combinatorial moves. This is a joint work with Tian-Jun Li.

    Confirmed List of Participants:

    Soccer:
    We will meet at front door of the Math Department in most afternoons at 5.30pm to play Soccer.