Organizers: Selman Akbulut (Michigan State University), Anar Akhmedov (University of Minnesota), Weimin Chen (University of Massachusetts, Amherst), Cagri Karakurt (University of Texas, Austin), TianJun Li (University of Minnesota).
The main focus of this workshop will be on the topology and invariants of smooth 4manifolds. The workshop is a part of the FRG: Collaborative Research: Topology and Invariants of Smooth 4Manifolds. It is funded by NSF Focused Research Grant DMS1065955. There will be 5 days of minicourses (July 31  August 4), primarily for graduate students, followed by a conference during the second week (August 5  10).
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*** .
Abstract of mini courses :
4manifolds via their handlebodies (by Selman Akbulut)
Abstract:
Handlebody descriptions of 3 and 4manifolds will be discussed, and as time permits their various applications to 4manifold 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 AkhmedovPark exotic manifolds.
Construction of exotic 4manifolds (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 4manifold. 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 4manifolds. In this talk, using Luttinger surgery, I will present a new constructions of Lefschetz fibration over 2sphere 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 4manifolds
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 4manifold X. In this lecture we will address the geography problem of simplyconnected spin and nonspin symplectic 4manifolds with nonnegative signature or near the BogomolovMiyaokaYau line c1^2(X) = 9chi(X) (joint results with B. Doug Park).
Group actions on 4manifolds (by Weimin Chen)
Abstract:
In this minicourse we give an introduction to some basic questions and basic techniques
in the study of finite group actions on 4manifolds, with an emphasis given to
symplectic finite group actions.
Lecture 1: A general introduction to finite group actions on 4manifolds, with emphasis
on locally linear topological actions: basic properties, construction, and obstructions to smoothability.
Lecture 2: Symplectic finite group actions. Jholomorphic curves in 4orbifolds. Orbifold
SeibergWittenTaubes theory.
Lecture 3: Symplectic actions on CP^2 and K3 surfaces. Some open problems.
Elements of HM and ECH (by YiJen Lee)
Abstract:
I will go over some background of SeibergWittenFloer homology and ECH (in a generalized sense), their relations, and if time permits, some computations.
The Kodaira dimension of symplectic 4manifolds (by TianJun Li)
Abstract:
Lecture 1. Kodaira dimension for low dimensional manifolds
Lecture 2. Symplectic 4manifolds with nonpositive Kodaira dimension
The triangulation conjecture (by Ciprian Manolescu)
Abstract:
First lecture: The SeibergWitten equations in 3 dimensions
I will review the KronheimerMrowka construction of monopole Floer homology for 3manifolds, 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^1equivariant SeibergWitten Floer homology, based on finite dimensional approximation and Conley index theory. The construction yields something more, an S^1equivariant 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 SeibergWitten Floer homology for rational homology 3spheres equipped with a spin structure. The analogue of Froyshov'’s correction term in this setting is an integervalued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that the 3dimensional homology cobordism group has no elements of order 2 that have Rokhlin invariant one. By previous work of GalewskiStern and Matumoto, this implies the existence of nontriangulable highdimensional manifolds.
Lefschetz fibrations and symplectic fillings of contact 3manifolds (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 4manifolds 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 blowups of Stein fillings (a joint result with Niederkrueger), and a recent generalization of that result in joint work with Lisi and Van HornMorris. 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 ThurstonWinkelnkemper), 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 minicourses :
Wednesday 7/31  Thursday 8/1  Friday 8/2  Saturday 8/3  Sunday 8/4 
9:009:30 Registration and Coffee (math department lounge) 
9:009:30 Coffee 
9:009:30 Coffee 
9:009:30 Coffee 
9:009:30 Coffee 
9:3010:30 W. Chen I 
9:3010:30 W. Chen II 
9:3010:30 W. Chen III 
9:3010:30 C. Manolescu II 
9:3010:30 YJ. Lee I 
11:0012:00 TJ. Li I 
11:0012:00 TJ. Li II 
11:0012:00 C. Manolescu I 
11:0012:00 A. Akhmedov II 
11:0012:00 YJ. Lee II 
Lunch  Lunch  Lunch  Lunch  Lunch 
2:003:00 S. Akbulut I 
2:003:00 S. Akbulut II 
2:003:00 C. Wendl III 
2:005:00 Short talks 

3:304:30 C. Wendl I 
3:304:30 C. Wendl II 
3:304:30 A. Akhmedov I 

5:30  6:30 Soccer 
5:30 Dinner (math department lounge) 
5:30  6:30 Soccer 
Monday 8/5  Tuesday 8/6  Wednesday 8/7  Thursday 8/8  Friday 8/9  Saturday 8/10 
9:009:30 Registration and Coffee 
9:009:30 Coffee 
8:309:00 Coffee 
9:009:30 Coffee 
9:009:30 Coffee 
8:309:00 Coffee 
9:3010:30 C. Manolescu 
9:3010:30 S. Akbulut 
9:0010:00 A. Stipsicz 
9:3010:30 M. Ue 
9:3010:30 M. Furuta 
9:0010:00 HJ. Kim 
11:0012:00 R. Gompf 
11:0012:00 B. Ozbagci 
10:1511:15 T. Cochran 
11:0012:00 Y. Ni 
11:0012:00 Z. Wu 
10:1511:15 J. Wang 
Lunch  Lunch  11:3012:30 C. Karakurt 
Lunch  Lunch  
1:302:30 D. Auckly 
1:302:30 I. Baykur 
free afternoon 
1:302:30 H. Endo 
1:302:30 M. Usher  
2:453:45 W. Zhang 
2:453:45 L. Starkston 
2:453:45 M. Hamilton 
2:453:45 W. Wu  
4:155:15 F. Arikan 
4:155:15 T. Lidman 
4:155:15 J. Dorfmeister 
4:155:15 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 4dimensional 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 3manifold. 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 twodisk with Stein regular fibers, such that the monodromy of the fibration is a symplectomorphism induced by compositions of righthanded Dehn twists along embedded Lagrangian nspheres 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 LoiPiergallini and AkbulutOzbagci. This is a joint work with Selman Akbulut.
Strict Onestable Equivalence in Four Dimensions (by David Auckly)
Abstract:
A wellknown theorem of Wall establishes that two simplyconnected, 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 HeeJung Kim)
Topological complexity of symplectic 4manifolds and Stein fillings (by Inanc Baykur)
Abstract:
Following the groundbreaking works of Donaldson and Giroux,
Lefschetz pencils and open books have become central tools in the
study of symplectic 4manifolds and contact 3manifolds. 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
4manifold, 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 3manifold, 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 HornMorris
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 selflinking 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 selflinking zero has nonzero Arf invariant and nonzero 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 4manifolds, one fundamental question is under what conditions the sum is minimal, i.e. contains no embedded sphere of selfintersection 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 fibersum 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 SeibergWitten monopoles Nakamura showed a similar result as well as a
10/8type 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 3spheres are introduced in terms of
Pin(2)equivariant Floer K group to formulate 10/8 type inequality for
spin 4manfolds 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 TianJun Li.
Exotic smoothings of open 4manifolds (by Robert Gompf)
Abstract:
Smoothing theory for open 4manifolds 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 4manifolds, and these tools are often complementary to traditional techniques.
Homology classes of negative square and embedded surfaces in 4manifolds (by Mark Hamilton)
Abstract:
Problem 4.105 from the Kirby list asks if the selfintersection number of embedded spheres in any given simplyconnected closed oriented 4manifold 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 3manifolds 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 HeeJung 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 CassonGordon 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 SeibergWitten and Heegaard Floer theory (by YiJen Lee)
Abstract:
Heegaard Floer homology of Seifert fibered homology spheres and nonzero 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 nonzero degree maps. This is joint work with Cagri Karakurt.
The triangulation conjecture (by Ciprian Manolescu)
Abstract:
We define Pin(2)equivariant SeibergWitten Floer homology for rational homology 3spheres equipped with a spin structure. The analogue of Froyshov'’s correction term in this setting is an integervalued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that the 3dimensional homology cobordism group has no elements of order 2 that have Rokhlin invariant one. By previous work of GalewskiStern and Matumoto, this implies the existence of nontriangulable highdimensional manifolds.
Applications of 4dimensional techniques to Dehn surgery (by Yi Ni)
Abstract:
In recent years, techniques from 4dimensional 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 twodimensional 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 biinvariant 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 2sphere (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 blowdowns
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 4manifold 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 mubar invariants and the eta invariants for Seifert rational homology 3spheres (by Masaaki Ue)
Abstract:
We discuss the relation between the mubar invariants defined by Neumann and Siebenmann and the eta invariants for the Dirac and the signature operators for Seifert rational homology 3spheres. Our arguments are based on the finite dimensional approximation of the SeibergWitten invariants by Furuta.
Lagrangian submanifolds of CP^n (by Michael Usher)
Abstract:
Quasihomomorphisms on mapping class groups (by Jiajun Wang)
Abstract:
A quasihomomorphism on a group is a realvalued function which is a homomorphism up to a constant. Quasihomomorphisms on the mapping class groups of surfaces include signature, linking number, Rasmussen invariant, OzsvathSzabo invariants, etc. We show that the quasihomorphisms on a surface vanishing on the handlebody group is an infinitedimensional 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.
Abstract:
The problem of symplectic ballpacking 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 4manifolds.
Abstract:
In this talk, we will discuss recent progress on the 3dimensional analogue of Thomconjecturetype 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.
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 Jholomorphic subvariety when the class of the subvariety is Jnef. 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 TianJun Li.