The 6th GTSS
GEOMETRYTOPOLOGY SUMMER SCHOOL
Feza Gürsey Institute  Online (Youtube Channel,
Slides)
August 214, 2021
First Week
Second Week
Scientific Commitee
Vicente Cortés 
University of Hamburg, Germany

İzzet Coşkun 
University of Illinois at Chicago, USA

Ljudmila Kamenova 
Stony Brook University, USA

Lei Ni 
University of California at San Diego, USA

Tommaso Pacini 
University of Torino, Italy 
Gregory Sankaran 
University of Bath, UK 
Misha Verbitsky 
IMPA, Brasil

Organizing Commitee
Craig van Coevering 
Bosphorus University

İlhan İkeda 
Bosphorus University

Mustafa Kalafat 
Nesin Mathematical Village

Registration
Poster
Participants
Arrival
Information
The 6th GTSS will be held at the
FGE,
Feza Gürsey Center for Physics and Mathematics (Online)
which is established in the Kandilli (North) Campus of Boğaziçi University (Bosphorus).
The first photo on top of this webpage demonstrates the view of the venue.
There will be about 6 minicourses of introductory nature, related to the GeometryTopology research subjects.
Thursday is holiday.
In the middle of the week (i.e. on Thursday) there is an excursion on the Bosphorus.
Do not forget to get your swimsuit with yourself. (For the real events only.)
Application
Graduate students, recent Ph.D.s and underrepresented minorities are
especially encouraged to the summer school.
Daily expenses including bed, breakfast, lunch, dinner is around 20€.
You may stay at cheap
hostels around Taksim(Nightlife) square and take the
subway to the Campus easily.
We are trying to arrange accomodation on Campus as well.
Please fill out the
Registration form
in order to attend the "Online" summer school. Registration is free but mandatory.
Airport: İstanbul Airport  IST is the closest one. Take a
bus from the airport to the 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Alternatively you can use Sabiha Gökçen Airport  SAW and take a
bus from there to 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Navigate the link above for more detailed arrival and venue information.
Visas: Check whether you need a visa beforehand.
Abstracts
Holomorphic curves in the 6sphere
The content of each lecture is as follows.
Lecture 1: G2 Geometry & Associative 3folds.
This lecture is expository, meant as an introduction for nonexperts.
We discuss Riemannian holonomy in general, and then focus on G2 holonomy and G2structures.
We then discuss the two most important classes of submanifolds in G2 geometry:
the associative 3folds and coassociative 4folds.
Lecture 2: NearlyKahler Geometry & Holomorphic Curves.
This lecture is expository, meant as an introduction for nonexperts.
We introduce the class of nearlyKahler 2nmanifolds by way of the GrayHervella classification.
We then specialize to (strict) nearlyKahler 6manifolds,
explaining their relationship with G2holonomy cones,
and briefly reviewing the known compact examples.
Finally, we discuss the two most important classes of submanifolds in nearlyKahler 6manifolds:
the holomorphic curves and Lagrangian 3folds.
Lecture 3: The Jacobi Spectrum of Closed Holomorphic Curves in S^6.
This lecture is a research talk.
Recall that minimal surfaces are areaminimizing to first order,
but not necessarily to secondorder.
The extent to which a minimal surface is (or isn’t) areaminimizing to
secondorder is encoded by its Jacobi operator.
However, for a given minimal surface, computing the spectrum of the Jacobi operator 
i.e., the eigenvalues and their multiplicities  is generally nontrivial.
We will discuss a class of minimal surfaces in the round S^6 known as "nulltorsion" holomorphic curves.
By a remarkable theorem of Bryant, extended by Rowland,
every closed Riemann surface may be conformally embedded in S^6 as a nulltorsion holomorphic curve.
In the case of low genus,
we give a simple explicit formula for the multiplicity of the first Jacobi eigenvalue.
Moreover, for all genera, we give a simple lower bound for the nullity.
We expect that these results will have implications for the deformation theory of asymptotically conical associative 3folds in R^7.
Lecture 4: FreeBoundary Holomorphic Curves in NearlyKahler 6Manifolds.
This lecture is a research talk.
We will discuss holomorphic curves with boundary in nearlyKahler 6manifolds,
analyzing two different boundary conditions.
First, we show that a holomorphic curve in a geodesic ball B of S^6 that meets the boundary of B orthogonally must be totallygeodesic.
This implies rigidity results for reflectioninvariant holomorphic curves in S^6 and associative cones in R^7.
Second, we consider holomorphic curves with boundary on a Lagrangian in a nearlyKahler 6manifold.
By deriving a second variation formula for area,
and exploiting a suitable analogue of RiemannRoch,
we obtain a topological lower bound on the Morse index.
References:
 R. Bryant  Submanifolds and Special Structures on the Octonians.
J. Diff. Geom. (1982).
 N. Ejiri  The Index of Minimal Immersions of S^2 into S^{2n}.
Mathematische Zeitschrift. (1983).
 A. Fraser and R. Schoen  Uniqueness Theorems for Free Boundary Minimal Disks in Space Forms.
Int. Math. Res. Not. (2015).
 A. Gray and L. M. Hervella  The Sixteen Classes of Almost Hermitian Manifolds and their Linear Invariants.
Ann. Mat. Pura. Appl. (1980).
 F. Reese Harvey and H. Blaine Lawson  Calibrated Geometries.
Acta Mathematica. (1982).
 D. Joyce  Riemannian Holonomy Groups and Calibrated Geometry.
Oxford University Press. (2007).
 J. Madnick  The Second Variation of NullTorsion Holomorphic Curves in the 6Sphere.
ArXiv:2101.09580 (Jan 2021) 35 pages.
 J. Madnick  FreeBoundary Problems for Holomorphic Curves in the 6Sphere.
Arxiv:2105.10562 (May 2021). 17 pages.
 S. Montiel and F. Urbano  Second Variation of Superminimal Surfaces into SelfDual Einstein 4Manifolds.
Trans. AMS. (1997).
Constant mean curvature(CMC) surfaces and integrable systems
These lectures give a detailed introduction to the theory of constant mean curvature surfaces
in space forms via gauge theoretic methods and the theory of integrable systems. The main
goal is to explain recent results about the construction of new examples and the derivation of
geometric quantities in terms of complex analytic data.
Lecture 1: Minimal and CMC surfaces in R^{3} and S^{3}.
We start with a short summary about minimal
surfaces in R3. We explain the induced Riemann surface structure, the Hopf differential and
the spinorial Weierstrass representation.
In the second part, we discuss surfaces of constant mean curvature in R3 and S3. We derive
the gaugetheoretic description of these surfaces [7], and explain the Lawson correspondence.
We discuss some important examples [8, 6].
Lecture 2: Holomorphic bundles and at connections. We introduce holomorphic bundles over Rie
mann surfaces from various points of view. We study at connections over holomorphic bundles.
In particular, we discuss unitary at connections and the famous NarasimhanSeshadri theorem
 this will enable us to deal with intrinsic closing conditions of CMC surfaces in space forms.
We introduce the moduli spaces of at connections and moduli spaces of polystable holomorphic bundles.
Lecture 3: Loop group factorisation methods. We define the associated family of at connections of
a CMC surface in R3 and S3 and study its basic properties. We construct minimal and CMC
surfaces from certain maps into the moduli space of at connections satisfying intrinsic and
extrinsic closing conditions [3, 4]. The proof uses a loop group factorisation. We discuss the
DPW method  the special case of families of holomorphic connections [1, 2].
Lecture 4: Recent results and examples. We discuss recent results about the existence and properties
of minimal and CMC surfaces [5, 6]. In particular, we investigate the area of the Lawson
surfaces in terms of their genus.
A survey: Gerding, A.; Pedit, F.; Schmitt, N. Constant mean curvature surfaces an integrable systems perspective.
Harmonic maps and differential geometry, 7–39, Contemp. Math., 542, AMS 2011.
References
 J. Dorfmeister, F. Pedit, and H. Wu, Weierstrass type representation of harmonic maps into symmetric
spaces.
Comm. Anal. Geom. 6 (1998), no. 4, 633668.
 S. Heller, Higher genus minimal surfaces in S3 and stable bundles.
J. Reine Angew. Math., Volume 685 (2013), pp 105122.
 S. Heller, A spectral curve approach to Lawson symmetric CMC surfaces of genus 2.
Math. Annalen, Volume 360, Issue 3 (2014), pp 607652.
 L. Heller, S. Heller, N. Schmitt, Navigating the Space of Symmetric CMC Surfaces.
Journal of Differential Geometry, Vol. 110, No. 3 (2018), pp. 413455.
 L. Heller, S. Heller, M. Traizet, Area Estimates for High genus Lawson surfaces via DPW.
Arxiv:1907.07139
 A. Bobenko, S. Heller, N. Schmitt, Constant mean curvature surfaces based on fundamental quadrilaterals.
Math. Physics, Analysis and Geometry, Preprint: arxiv: 2102.03153
 N.J. Hitchin, Harmonic maps from a 2torus to the 3sphere.
J. Differential Geom. 31 (1990), 627710.
 H. B. Lawson. Complete minimal surfaces in S3.
Ann. of Math. (2) 92 (1970), 335374.
Topics in Algebraic Curves
This will be lectures on selected topics on algebraic curves.
The daily description will be as follows.
Lecture 1: (Mon 2821) Rational curves in projective space
In this lecture, I will describe spaces of rational curves in projective space. I will discuss the stratification of the space according to the splitting type of the restricted tangent bundle and the normal bundle. I will show that the general curve has a balanced normal bundle and following Sacchiero describe the possible splitting types of the normal bundle. Following joint work with Eric Riedl, I will exhibit some pathological behavior of the stratification by the splitting type of the normal bundle.
Lecture 2: (Tue 3821) Rational curves on hypersurfaces
In this lecture, I will discuss the space of rational curves on hypersurfaces in projective space. I will mention several conjectures concerning these spaces and prove some special cases of these conjectures. Following joint work with Eric Riedl and Geoffrey Smith, I will discuss the separable rational connectedness of hypersurfaces and more generally complete intersections in certain homogeneous spaces.
Lecture 3: (Wed 4821) Algebraic hyperbolicity of very general hypersurfaces of large degree
In this lecture, I will discuss bounds on the genus of a curve in a very general hypersurface of degree at least 5 in projective 3space. Following joint work with Eric Riedl, I will show that a very general surface of degree at least 5 is algebraically hyperbolic.
Lecture 4: (Fri 6821) Normal bundles of BrillNoether space curves
In this lecture, I will discuss the stability of the normal bundle of general BrillNoether space curves. Following joint work with Eric Larson and Isabel Vogt, I will show that a general BrillNoether space curve of degree d and genus g at least 2 has a stable normal bundle if and only if (d,g) is not equal to (5,2) or (6,4).
References:
 I. Coskun, E. Larson and I. Vogt, Stability of normal bundles of space curves
 I. Coskun and E. Riedl, Normal bundles of rational curves in projective space,
Mathematische Zeitschrift vol 288 (2018), 803827.
 I. Coskun and E. Riedl, Normal bundles of rational curves on complete intersections
Communications in Contemporary Mathematics 21 no. 2 (2019).
 I. Coskun, Izzet and E. Riedl,
Algebraic hyperbolicity of the very general quintic surface in P^3
Adv. Math. 350 (2019), 13141323.
 I. Coskun and G. Smith, Very free rational curves in Fano varieties
 I. Coskun and J. Starr, Rational curves on smooth cubic hypersurfaces
International Mathematics Research Notices Article RNP102 (2009)
 J. Harris, M. Roth and J. Starr, Rational Curves on hypersurfaces of low degree
J. Reine Angew. Math., 571 (2004), 73106
 Riedl, Eric and Yang, David.
Rational curves on general type hypersurfaces
Journal of Differential Geometry, 116 no.2 (2020), 393403.
 Riedl, Eric and Yang, David.
Kontsevich spaces of rational curves on Fano hypersurfaces
J. Reine Angew. Math. 748 (2019), 207225.
Interplays of Complex and Symplectic Geometry
I will present  mostly using examples  how complex and symplectic structures relate on
compact manifolds.
I will show how symplectic forms taming complex structures are intimately related to a special
type of Hermitian metrics,
known in the literature also as "pluriclosed" metrics. Then I will present some results on the
symplectic CalabiYau problem in dimension 4
and on balanced metrics in relation to the StromingerHull system.
A tentative schedule:
1. Kähler geometry. Examples and nonexamples.
2. Pluriclosed metrics
3. Symplectic CalabiYau problem
4. Balanced metrics and the StromingerHull system
References
 L. Alessandrini, G. Bassanelli, Plurisubharmonic currents and their extension across analytic subsets,
Forum Math. 5 (1993), 291316.
 L. Bedulli, L. Vezzoni, A parabolic flow of balanced metrics,
J. Reine Angew. Math. 723 (2017), 79–99.
 E. Buzano, A. Fino, L. Vezzoni, The CalabiYau equation on the KodairaThurston manifold, viewed as an S1bundle over a 3torus,
J. Differential Geom. 101 (2015), no. 2, 175195.
 E. Buzano, A. Fino, L. Vezzoni, The CalabiYau equation for T^2bundles over the nonLagrangian case,
Rend. Semin. Mat. Univ. Politec. Torino 69 (2011), no. 3, 281298.
 S.K. Donaldson, Twoforms on fourmanifolds and elliptic equations,
Inspired by S.S. Chern, 153–172, Nankai Tracts Math. 11, World Scientific, Hackensack N.J., 2006.
 A. Fino, Y.Y. Li, S. Salamon, L. Vezzoni, The Calabi–Yau equation on 4manifolds over 2tori,
Trans. Amer. Math. Soc. 365 (2013), no. 3, 15511575.
 N. Enrietti, A. Fino, L. Vezzoni, Tamed symplectic forms and SKT metrics,
J. Symplectic Geom. 10 (2012), no. 2, 203–223. Correction: J. Symplectic Geom. 17 (2019), no. 4, 1079–1081.
 A. Fino, G. Grantcharov, Properties of manifolds with skewsymmetric torsion and special holonomy,
Adv. Math. 189 (2004), no. 2, 43950.
 A. Fino, G. Grantcharov, L. Vezzoni, Solutions to the HullStrominger system with torus symmetry,
arXiv:1901.10322.
 A. Fino, M. Parton, S. Salamon, Families of strong KT structures in six dimensions,
Comment. Math. Helv. 79 (2004), no. 2, 317–340.
 A. Fino, F. Paradiso, Balanced Hermitian structures on almost abelian Lie algebras,
arXiv:2011.09992.
 A. Fino, N. Tardini, L. Vezzoni, Pluriclosed and Strominger Kählerlike metrics compatible with abelian complex structures,
arXiv:2102.01920.
 A. Fino, A. Tomassini, Blowups and resolutions of strong Kähler with torsion metrics,
Adv. Math. 221 (2009), no. 3, 914935.
 J.X. Fu, S.T. Yau, The theory of superstring with flux on nonKähler manifolds and the complex MongeAmp`ere equation,
J. Differential Geom. 78 (2008), no. 3, 369428.
 P. Gauduchon, La 1forme de torsion d'une variété hermitienne compacte. (French)
[Torsion 1forms of compact Hermitian manifolds], Math. Ann. 267 (1984), no. 4, 49518.
 P. Gauduchon, Hermitian connections and Dirac operators.
Boll. Un. Mat. Ital. B (7) 11 (1997), no. 2, suppl., 257288.
 E. Goldstein, S. Prokushkin, Geometric model for complex nonKähler manifolds with SU(3) structure,
Comm. Math. Phys. 251 (2004), no. 1, 6578.
 M. L. Michelsohn, On the existence of special metrics in complex geometry,
Acta Math. 149 (1982), no. 34, 261295.
 D. Phong, S. Picard, X. Zhang, Anomaly flows,
Comm. Anal. Geom. 26 (2018), no. 4, 9551008.
 J. Streets, G. Tian, A parabolic flow of pluriclosed metrics,
Int. Math. Res. Not. IMRN 2010, no. 16, 31013133.
 C. Taubes, Clifford Henry Tamed to compatible: symplectic forms via moduli space integration,
J. Symplectic Geom. 9 (2011), no. 2, 161250.
 V. Tosatti, B. Weinkove, S.T. Yau, Taming symplectic forms and the Calabi–Yau equation,
Proc. London Math. Soc. 97 (2008), no. 2, 401424.
 V. Tosatti, B. Weinkove, The Calabi–Yau equation on the Kodaira–Thurston manifold,
J. Inst. Math. Jussieu 10 (2011), no. 2, 437447.
 M. Verbitsky, Rational curves and special metrics on twistor spaces.
Geom. Topol. 18 (2014), no. 2, 897909.
 B. Weinkove, The CalabiYau equation on almostKähler fourmanifolds,
J. Differential Geom. 76 (2007), no. 2, 317349.
 S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I,
Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.
Topics in locally conformally Kähler (LCK) geometry
In this series of lectures I will introduce locally conformally Kähler (lcK)
metrics, give explicit examples of compact manifolds admitting such structures and study their topological and cohomological properties. We shall
focus in the latter part of the course on Vaisman manifolds.
References
 F. A. Belgun, On the metric structure of nonKähler complex surfaces,
Math. Ann. 317 (2000), 140.
 M. Brunella, Locally conformally Kähler metrics on Kato surfaces,
Nagoya Math. J., 202 (2011), p. 7781.
 P. Gauduchon, Le theoreme de l'excentricite nulle,
C. R. Acad. Sci. Paris Ser. AB 285 (1977), no. 5, A387A390.
 P. Gauduchon, L. Ornea, LCK metrics on Hopf surfaces,
Ann. Inst. Fourier, 48 (1998), 11071127.
 N. Istrati, A. Otiman, M. Pontecorvo, On a class of Kato manifolds,
Int. Math. Res. Not. (IMRN), Vol. 2021, No. 7, pp. 53665412.
 M. de Leon, B, Lopez, J.C. Marrero, E. Padron, On the computation of the
LichnerowiczJacobi cohomology,
J. Geom. Phys. 44 (2003), 507522.
 K. Oeljeklaus, M. Toma, NonKähler compact complex manifolds associated to
number fields,
Ann. Inst. Fourier, Grenoble, 55 (2005), no. 1, 161171.
 L. Ornea, M. Verbitsky, A report on locally conformally Kähler manifolds,
Contemporary Mathematics 542, 135150, 2011.
 A. Otiman, M. Toma, Hodge decomposition for Cousin groups and OeljeklausToma manifolds,
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XXII (2021), 485503.
 K. Tsukada, Holomorphic forms and holomorphic vector fields on compact generalized
Hopf manifolds,
Compositio Mathematica, vol.93, no.1(1994), p.122.
 I. Vaisman, On locally conformal almost Kähler manifolds,
Israel J. Math. 24 (1976), no. 34, 338351.
 I. Vaisman, On locally and globally conformal Kähler manifolds,
Trans. AMS 262(1980), 533542.
Yamabe Invariant of Complex Surfaces
The Yamabe invariant is a realvalued diffeomorphism invariant which arises from Riemannian geometry.
Somewhat surprisingly, the Yamabe invariant of a complex surface is intimately related to complex geometric properties of the surface.
LeBrun used SeibergWitten theory to show that the sign of the Yamabe invariant of a compact Kähler surface is determined by its Kodaira dimension.
Recent work has investigated the extent to which this relationship persists in the nonKähler setting.
The relevant papers that will be discussed are as follows.
References:
 LeBrun  Kodaira Dimension and the Yamabe Problem.
 Albanese  The Yamabe invariants of Inoue surfaces, Kodaira surfaces, and their blowups.
 Albanese & LeBrun  Kodaira Dimension and the Yamabe Problem, II.
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