GEOMETRY / TOPOLOGY CONFERENCE
May 29 - June 3 (2017)
List of invited speakers/participants (tentative)
| T. Ekholm ||
|| R. Cavalieri ||
|| P. Massot |
| B. Tosun ||
|| A. Daemi ||
|| S.C. Lau |
| S. Courte ||
|| B. Fang ||
|| W. Chen ||
| M. Tange ||
|| R. Golovko ||
|| R. Zentner ||
| F. Arikan ||
|| G. Dimitroglou ||
|| S. Finashin ||
| M. Bhupal ||
|| S. Galkin ||
|| N. Kurnosov ||
| C. Ferahlar ||
|| H. Arguz ||
Scientific Committee : D. Auroux, Y. Eliashberg, I. Itenberg, G. Mikhalkin, S. Akbulut, A. Oancea
Organizing Commitee : T. Önder, S. Koçak, A. Degtyarev, Ö. Kişisel, Y. Ozan, T. Etgü, S. Salur
Supporting Organizations: NSF (National Science Foundation),
TMD (Turkish Mathematical Society)
ERC (European Research Council).
Application period has ended on February 17, 2017.
Preliminary List of Talks
||Mini Course (Lectures 1 and 2): Floer disks on simple Lagrangian spheres and applications|
We study the moduli space of Floer holomorphic disks with boundary on a Lagrangian homtopy sphere immersion with a single double point. This moduli space gives a parallelizable null-cobordism of the homotopy sphere restricting its smooth structure. We discuss applications of the construction to cotangent bundles of homotopy spheres and to homotopy classes of Lagrangian spheres. The talks report on joint work with Smith and Kragh.
||Mini Course (Lectures 3 and 4): Legendrian surgery and partially wrapped Floer cohomology|
We discuss an extension of the Legendrian surgery results of Bourgeois-Ekholm-Eliashberg to the setting of partially wrapped Floer cohomology. The extension involves in particular a Legendrian differential graded algebra with loop space coefficients and under simple connectivity assumptions gives rise to dualities between ordinary and wrapped Floer cohomology. We furthermore discuss various applications of the results to mirror symmetry and to knot contact homology. The talks report on joint work with Lekili and Ng and Shende.
||Holonomy perturbations and irreducible representations of integer homology 3-spheres in SL(2,C)|
We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation. Using a result of Boileau, Rubinstein and Wang (which builds on the geometrization theorem of 3-manifolds), it follows that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C).
||Obstructing pseudo-convex embeddings of Brieskorn spheres into complex 2-space|
A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. An important question that has been circulating among contact and symplectic topologist for some time asks: whether every contractible smooth 4-manifold admits a Stein structure? In this talk we will provide examples that answer this question negatively. Moreover, along the way we will provide new evidence to a closely related conjecture of Gompf, which asserts that a nontrivial Brieskorn homology sphere, with either orientation, cannot be embedded in complex 2-space as the boundary of a Stein submanifold. This is a joint work with Tom Mark.
||Towards the strong Arnold conjecture|
In the 1960’s, V.I. Arnold announced several fruitful conjectures in symplectic topology concerning the number of fixed point of a Hamiltonian diffeomorphism in both the absolute case (concerning periodic Hamiltonian orbits) and the relative case (concerning Hamiltonian chords on a Lagrangian submanifold).
The strongest form of Arnold conjecture for a closed symplectic manifold (sometimes called the strong Arnold conjecture) says that the number of fixed points of a generic Hamiltonian diffeomorphism of a closed symplectic manifold X is greater or equal than the number of critical points of a Morse function on X.
We will discuss the stable version of Arnold conjecture, which is closely related to the strong Arnold conjecture. This is joint work with Georgios Dimitroglou Rizell.
||Graph formulas for tautological cycles|
The tautological ring of the moduli space of curves is a subring of the Chow ring that, on the one hand, contains many of the classes represented by "geometrically defined" cycles (i.e. loci of curves that satisfy certain geometric properties), on the other has a reasonably manageable structure. By this I mean that we can explicitly describe a set of additive generators, which are indexed by suitably decorated graphs. The study of the tautological ring was initiated by Mumford in the '80s and has been intensely studied by several groups of people. Just a couple years ago, Pandharipande reiterated that we are making progress in a much needed development of a "calculus on the tautological ring", i.e. a way to effectively compute and compare expressions in the tautological ring. An example of such a "calculus" consists in describing formulas for geometrically described classes (e.g. the hyperelliptic locus) via meaningful formulas in terms of the combinatorial generators of the tautological ring. In this talk I will explain in what sense "graph formulas" give a good example of what the adjective "meaningful" meant in the previous sentence, and present a few examples of graph formulas. The original work presented is in collaboration with Nicola Tarasca and Vance Blankers.
||Kuga-Satake construction and cohomology of hyperkahler manifolds|
Let M be a simple hyperkahler manifold. Kuga-Satake construction gives an embedding of H2(M,C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H*(M,C) → H*+l(T,C) for some l, which is compatible with the Hodge structures and the Poincare pairing. This work is joint with A.Soldatenkov and M.Verbitsky.
||Contact splittings of symplectic rational homology CP2.|
Let (X,ω) be a symplectic 4-manifold with rational homology of CP2. We consider splittings of X by a hypersurface of contact type M, where we require M to be a rational homology 3-sphere. We shall go over a few basic properties of such splittings, and then explain how this consideration provides a unified approach to a number of interesting questions in low-dimensional topology, symplectic geometry, as well as in algebraic geometry.
|Siu Cheong Lau||
||Mirror construction for a pair-of-pants decomposition|
Seidel and Sheridan proved homological mirror symmetry for a pair-of-pants. In my previous joint work with Cho and Hong, we gave a systematic construction of localized mirror functor from a given immersed Lagrangian, which in particular produces the Landau-Ginzburg mirror for a pair-of-pants. In this talk we consider a pair-of-pants decomposition and explain how to glue the local Landau-Ginzburg mirrors together from Floer-theoretical considerations.
||The wrapped Fukaya category of a Weinstein manifold is generated by the Lagrangian cocore discs|
In a joint work with B. Chantraine, P. Ghiggini, and R. Golovko we decompose any object in the wrapped Fukaya category as a twisted complex built from the cocores of the critical (i.e. half-dimensional) handles in a Weinstein handle decomposition. The main tools used are the Floer homology theories of exact Lagrangian immersions, of exact Lagrangian cobordisms in the SFT sense (i.e. between Legendrians), as well as relations between these theories. Note that exact Lagrangians admit Legendrian lifts, and that appropriate Lagrange surgeries can be seen to give rise to an exact Lagrangian cobordism of the aforementioned type.
||BKMP remodeling conjecture and its application|
The Bouchard-Klemm-Marino-Pasquetti remodeling conjecture expresses all genus Gromov-Witten open-closed invariants in terms of the Eynard-Orantin topological recursion on its mirror curve. I will explain this conjecture and discuss some of its application, such as the modularity of Gromov-Witten invariants. This talk is based on the joint work with Chiu-Chu Melissa Liu and Zhengyu Zong.
||Instantons and the homology cobordism group|
The set of 3-manifolds with the same homology as the 3-dimensional sphere, modulo an equivalence relation called homology cobordance, forms a group. The additive structure of this group is given by taking connected sum. This group is called the homology cobordism group and plays a special role in low dimensional topology and knot theory. In this talk, I will explain how instantons can be used to construct an infinite family of invariants of the homology cobordism group. The relationship between these invariants and Froyshov’s celebrated invariant will be discussed. I will also talk about some topological applications.
||Invariant norms on contact transformation groups|
Inspired by the Hofer and Viterbo distances on groups of Hamiltonian diffeomorphisms, several recent works define and study invariant distances on groups of contact transformations. In some cases, contact and symplectic rigidity results allow to prove that such a distance is unbounded, hence sees a significant part of the group. But understanding when this happens is not obvious. For instance this happens for the standard contact structures on projective spaces but not for their universal covers, the standard spheres.
In joint work in progress with Sylvain Courte, we shed new lights on this puzzle by uncovering links with Giroux's theory of open book decompositions and Murphy’s theory of loose Legendrian embeddings.
||Cork twisting and exotic 4-manifolds|
It is classically well-known that for any two closed, simply-connected exotic 4-manifolds there exists a contractible (Stein) 4-manifold C such that the two exotic 4-manifolds are related to each other by removing and regluing C. A pair of such C and the boundary diffeomorphism (gluing map) t is called cork. In this talk we give a review of recent developments of corks by some 4-dimensional topologists: finite order corks, Z-corks, Zn-corks, and constraints for infinite cork twists.
||Generating functions and sheaves for Legendrian links in R3|
To a (generic) one-parameter family of functions (fx) on a manifold M we associate the graph of all critical values: this is the front projection of a Legendrian link L in R3 and (fx) is called a generating function for L. Which Legendrian links admit a generating function? How many up to equivalence? To deal with such questions it is natural to associate to a generating function a sheaf on R2 microsupported on the Legendrian. We will discuss to what extent this is a bijective correspondence. This is joint work (in progress) with Vivek Shende.
||Tight contact structures on some closed hyperbolic three-manifolds|
We show the existence of tight contact structures on infinitely many hyperbolic three-manifolds obtained via Dehn surgeries along sections of hyperbolic surface bundles over circle. This is a joint work with M. Secgin.
||Vanishing theorems on a compact complex manifold, applications to Hopf conjecture and complex structures on 6-sphere|
Vanishing theorems are important tools in algebraic geometry and differential geometry as they give information about the structure of varieties and manifolds. I generalize a Weitzenbock formula of Wu on Kahler manifolds for complex manifolds and use this to get vanishing theorems on compact complex manifolds under certain conditions, then obtain information on the plurigenera, such as geometric genus, arithmetic genus, irregularity. Hopf conjecture is one of the well-known questions in differential geometry that relates the local geometry of an even dimensional compact orientable manifold to its global topology. Utilizing the vanishing theorems and the Frolicher spectral sequence, I show that the conjecture holds for compact complex manifolds with positive sectional, holomorphic bisectional and isotropic curvature under certain conditions. Finally, another interesting question is whether there exists a complex structure on S6 or not. By using a theorem of Gray that if S6 had a complex structure then its (0,1)-Hodge number must be strictly positive and by combining this with the vanishing of the irregularity result proved, David L. Johnson and myself prove that under the specified conditions for the vanishing of the irregularity, S6 can not have a complex structure by Gray's theorem.
||Log geometric techniques for open invariants in mirror symmetry|
We will discuss an algebraic-geometric approach to the symplectic Fukaya category via log Gromov-Witten theory and tropical geometry. Our main object of study will be the degeneration of elliptic curves, namely the Tate curve. We will also discuss a construction of a split-generating set of real Lagrangians using log geometric techniques. This is joint work with Bernd Siebert, with general ideas based on discussions of Bernd Siebert and Mohammed Abouzaid. The symplectic aspects we will overview is joint work in progress with Dmitry Tonkonog.