Geometry Main

The 14th GTSS

GEOMETRY-TOPOLOGY SUMMER SCHOOL

İstanbul Center for Mathematical Sciences - Online (Youtube Channel)
July 29 - Aug 2, 2024





Daily schedule of talks

 

TIME              SPEAKER                  TITLE
July 29-Aug 2
3-4:15		 Federico Trinca
 Calibrated geometry in non-compact complete manifolds of exceptional holonomy
S : 1 , 2 , 3 , 4
July 29-Aug 2
4:30-6		 Jesse Madnick
 Minimal submanifolds in spheres
S : 1, 2 , 3 , 4


Scientific Commitee

 

Vicente Cortés University of Hamburg, Germany
Anna Fino University of Torino, Italy
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
İzzet Coşkun University of Illinois at Chicago, USA
Mustafa Kalafat Rheinische Friedrich-Wilhelms-Universität Bonn






Registration       Poster       Participants       Arrival


Information

The 14th GTSS will be held at the IMBM, İstanbul Center for Mathematical Sciences (Online)
which is established in the main (South) Campus of Boğaziçi University (Bosphorus).
There will be mini-courses of introductory nature, related to the Geometry-Topology research subjects.


Application

Graduate students, recent Ph.D.s and under-represented 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



Calibrated geometry in non-compact complete manifolds of exceptional holonomy

Similarly to the celebrated Calabi-Yau 3-folds in dimension 6, G2 and Spin(7) manifolds are certain 7-dimensional and 8-dimensional manifolds that appear in the Berger list of possible Riemannian holonomy groups. Such manifolds are called manifolds of exceptional holonomy, are Ricci-flat and admit special stable minimal submanifolds: calibrated submanifolds.

The tentative schedule for the mini-course is as follows:

Lecture 1: Calibrated geometry and manifolds of exceptional holonomy. (Preliminary).

First, we provide a basic introduction to Riemannian holonomy and calibrated geometry. Afterwards, we discuss the exceptional holonomy groups G2 and Spin(7), with particular attention to their calibrated submanifolds (associatives and coassociatives for G2 manifolds and Cayleys for Spin(7)).

Lecture 2: Non-compact complete manifolds of exceptional holonomy. (Preliminary)

First, we describe the complete non-compact G2 and Spin(7) manifolds constructed by Bryant and Salamon [2]. We then describe all the complete non-compact G2 manifolds of SU(2)^2xU(1)-symmetry, which we call FHN manifolds after [3].

Lecture 3: Calibrated submanifolds of the Bryant-Salamon manifolds. (Research)

We discuss calibrated submanifolds of the Bryant-Salamon manifolds constructed as vector subbundles or using cohomogeneity-one techniques (cf. [6,7,9])

Lecture 4: Calibrated submanifolds of the FHN manifolds. (Research)

We discuss calibrated submanifolds of the FHN manifolds using indirect cohomogeneity-one techniques [1].

References
  1. B. Aslan and F. Trinca. On G2 manifolds with cohomogeneity two symmetry. 2024.

  2. R. L. Bryant and S. Salamon. On the construction of some complete metrics with exceptional holonomy. 1989.

  3. L. Foscolo, M. Haskins, and J. Nordström. Infinitely many new families of complete cohomogeneity one G2-manifolds:
    G2 analogues of the Taub-NUT and Eguchi-Hanson spaces. 2021.

  4. R. Harvey and H. B. Lawson. Calibrated geometries. 1982.

  5. D. Joyce. Riemannian Holonomy Groups and Calibrated Geometry. 2007.

  6. S. Karigiannis and J. D. Lotay. Bryant–Salamon G2 manifolds and coassociative fibrations. 2021.

  7. S. Karigiannis and M. Min-Oo. Calibrated subbundles in non-compact manifolds of special holonomy. 2005.

  8. T. B. Madsen and A. Swann. Multi-moment maps. 2012.

  9. F. Trinca. Cayley fibrations in the Bryant–Salamon Spin(7) manifold. 2023.

Minimal submanifolds in spheres

I will discuss my work with Gavin B. and Uwe S. regarding the Morse index of quartic minimal hypersurfaces. Here's a tentative plan:

Lecture 1: Minimal submanifolds, the index, and calibrations (expository).

Lecture 2: Minimal submanifolds of spheres and semi-calibrations (expository + research).

Lecture 3: Isoparametric hypersurfaces in spheres (expository + research).

Lecture 4: The Morse index of quartic minimal hypersurfaces (research).

References
  1. Benjamin Aslan, Spiro Karigiannis, Jesse Madnick. Calibrated Geometry in Hyperkähler Cones, 3-Sasakian Manifolds, and Twistor Spaces.
    Canadian Journal of Mathematics (2024) 1-59. Available on the ArXiv at https://arxiv.org/abs/2308.11831.

  2. Gavin Ball, Jesse Madnick, Uwe Semmelmann. The Morse index of quartic minimal hypersurfaces.
    E-print 2023. Available on the ArXiv at https://arxiv.org/abs/2310.19404.

  3. Torralbo, Francisco; Urbano, Francisco. Index of compact minimal submanifolds of the Berger spheres.
    Calc. Var. Partial Differential Equations 61 (2022), no. 3, Paper No. 104, 35 pp.
    Available on the ArXiv at https://arxiv.org/abs/2110.08027.


Contact: s55mtasd@uni-bonn.de, ilaydabariss@gmail.com, berk.bozoglu@uni-bonn.de


Activities are supported by University of Bonn and Boğaziçi University