Geometry Main

The 13th GTWS

GEOMETRY-TOPOLOGY WINTER SCHOOL

Nesin Mathematics Village, Şirince, İzmir
February 5-10, 2024





Schedule of talks

 

TIME              SPEAKER                  TITLE
Feb 5-9
9-10:30		 Tommaso Pacini Introduction to Gauge Theory
Feb 5-10
10:30-12		 Kotaro Kawai
 Mirror of Submanifolds
Lecture Notes
Feb 5-9
12-13:30		 Mustafa Kalafat
 Minimal submanifolds of Kähler manifolds
Lunch Time
Feb 5-9
15:30-16:45		 Özgür İnce Introduction to Riemann Surfaces
Cake Time
Feb 5-9
17-18:30		 Buket Can Bahadır Spectrum of the Laplace Operator on Riemannian manifolds

Scientific Commitee

 

Vicente Cortés University of Hamburg, Germany
İzzet Coşkun University of Illinois at Chicago, USA
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

 

Özgür İnce Sivas Cumhuriyet University
Mustafa Kalafat Rheinische Friedrich-Wilhelms-Universität Bonn



Register-TR       Poster       Participants       Arrival


Information

The 13th GTWS will be held at the Nesin Mathematical Village in Şirince, İzmir.
There will be mini-courses of introductory nature, related to the Geometry-Topology research subjects.
The venue is close to the Ancient City of Ephesus and/or Kuşadası Beach.

Wifi info: Passwords are asurbanibal at the Library (Kutuphane) networks, bakkalnmk for istasyon and terasbahce for kisbahcesi at the registration lobby.


Application

Graduate students, recent Ph.D.s and under-represented minorities are especially encouraged to the research school.
Partial financial support is available. Daily expenses including bed, breakfast, lunch, dinner is around 20€.

Please fill out the application form to attend to the research school.

Airport: İzmir Adnan Menderes Airport - ADB is the closest one.
For pick up, faculty is advised to get out of the terminal building from the domestic gate and wait there.
Also let us know before hand, your whatsapp number and/or Google-Hangouts address in case of a problem.
Small amounts of Turkish Lira may be needed in the village
which you can withdraw from ATM machines at the airport or tell the driver along the way.
In case you are coming to the Village independently, from the Airport you may take the regional trains to Selçuk,
possibly with a break at Tepeköy. Then from Selçuk take a taxi which costs about 10€.
Taxi Drivers: +90 232 892 3125 (Gar Taxi), +90 553 243 0835 (Volkan), +90 532 603 1518 (Ata).

Visas: Check whether you need a visa beforehand.




Abstracts


Introduction to Gauge Theory

For several decades, gauge theory has been one of the driving forces of Geometry and Analysis. Its problems and methods have been foundational for many other fields. It is also an important topic in Physics. This course will offer a brief introduction to the mathematical point of view on gauge theory and to some of its relationships to other parts of geometry. It should be of interest to students of both Geometry and Analysis.

Program

The course will attempt to cover the following topics.

GT 1: Smooth vector bundles, connections, curvature. The Yang-Mills functional.

GT 2: Flat bundles and connections. Holomorphic vector bundles.

GT 3: Overview of stability and of the Narasimhan-Seshadri theorem.

Some textbook and references for the subject are as follows.

References
  1. Donaldson, S. K.; Kronheimer, P. B. The geometry of four-manifolds.
    Oxford Mathematical Monographs. Oxford: Clarendon Press. ix, 440p. 1990.

  2. Donaldson, S. K. A new proof of a theorem of Narasimhan and Seshadri.
    Journal of Differential Geometry, 18 (2): 269-277. 1983.

Prerequisites: Basic knowledge of differential geometry and of complex analysis. An understanding of
(i) Riemannian Hodge theory and (ii) Riemann surfaces, line bundles and the first Chern class would also be useful.

Mirror of Submanifolds

The Strominger-Yau-Zaslow conjecture suggests that it will be important to consider the special Lagrangian torus fibration for the study of mirror symmetry of Calabi-Yau manifolds. I will first introduce the real Fourier-Mukai transform, which gives the explicit ``mirror" correspondence on the torus fibrations. By this method, we can define notions for Hermitian connections on a Hermitian line bundle arising from submanifolds, such as the "mirror" volume. Then I will explain that the ``mirrors" of minimal/calibrated submanifolds are defined and they indeed have similar properties to minimal/calibrated submanifolds (and other gauge theoretic objects).

Some references for the subject are as follows.

References
  1. K. Kawai and H. Yamamoto. The real Fourier–Mukai transform of Cayley cycles.
    Pure Appl. Math. Q. 17 (2021), no. 5, 1861–1898.

  2. K. Kawai and H. Yamamoto. Mirror of volume functionals on manifolds with special holonomy.
    Adv. Math. 405 (2022), Paper No. 108515.

  3. K. Kawai. A monotonicity formula for minimal connections.
    Available on the ArXiv at https://arxiv.org/abs/2309.11796

Prerequisites: Basics of Riemannian geometry, vector bundles and principal bundles.

Minimal submanifolds of Kähler manifolds

A minimal submanifold(or surface) is the one that locally minimizes its area or volume. This is equivalent to having zero mean curvature vector field. They are 2-dimensional analogue to geodesics, which are analogously defined as critical points of the length functional. Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface z = z(x, y) of least area stretched across a given closed contour. He derived the Euler–Lagrange equation for the solution He did not succeed in finding any solution beyond the plane. In 1776 Jean Baptiste Marie Meusnier discovered that the helicoid and catenoid satisfy the equation and that the differential expression corresponds to twice the mean curvature of the surface, concluding that surfaces with zero mean curvature are area-minimizing. In this lecture series we will give an introduction to some topics in modern minimal submanifold theory. The topics to be covered are as follows.

Syllabus
  1. Mean curvature vector field on a Riemannian submanifold.

  2. First and second variational formulae for the area/volume functional.

  3. Minimal surfaces in Kähler manifolds. Index and nullity.

  4. Relation to holomorphic cross sections of the normal bundle and Riemann-Roch Theorem.

  5. Spectrum of the Riemannian Laplacian on the round n-dimensional sphere.

  6. Complex projective space and Page space as examples.

Some textbook and references for the subject are as follows.

References
  1. Li, Peter. Geometric analysis.
    Cambridge University Press, 2012.

  2. Simons, James. Minimal varieties in riemannian manifolds.
    Ann. of Math. (2), 88:62–105, 1968.

Introduction to Riemann surfaces

Riemann surfaces are obtained by gluing together patches of the complex plane by holomorphic maps, whereas algebraic curves are one-dimensional shapes defined by polynomial equations. (Compact) Riemann surfaces and (complex smooth projective) algebraic curves are equivalent in a precise sense. This means we can study the same objects using both complex analysis and algebra.

Some textbook and references for the subject are as follows.

References:
  1. E. Girondo and G. Gonzalez-Diez. Introduction to compact Riemann surfaces and dessin d’enfants.
    LMS Students texts 79, Cambridge University Press, Cambridge, 2012.

  2. R. Miranda. Algebraic curves and Riemann surfaces.
    Graduate Studies in Mathematics.

  3. H. M. Farkas and I. Kra. Riemann surfaces.
    Graduate Texts in Mathematics.

  4. S. Donaldson. Riemann Surfaces.
    Oxford Graduate Texts in Mathematics.

Prerequisites: Complex Analysis and Set Point Set Topology, basic notion about algebraic curves.

Spectrum of the Laplace Operator on Riemannian manifolds

Main objective of this lecture series is to review the theoretical tools to calculate the spectrum of Laplace operator on a compact Riemannian manifold, and using these techniques to actually compute the spectra of torus and sphere. We also aim to briefly discuss the situation in real and complex projective spaces. If time permits, we would like to end the series by asking the inverse problem, namely, “Given a Riemannian manifold, does the spectrum determine geometrically, up to an isometry, the manifold itself?” Daily description is as follows.

SLO 1: Spectral theory of compact operators.

SLO 2: The Laplacian on a compact Riemannian manifold

SLO 3: Spectral theory for the Laplacian

SLO 4: Explicit calculation of the spectrum: Flat Tori

SLO 5: Explicit calculation of the spectrum: Spheres

SLO 6: Can one hear the holes of a drum? Inverse Problems

Some textbook and references for the subject are as follows.

References:
  1. Olivier Lablée. Spectral Theory in Riemannian Geometry.
    European Mathematical Society, 2015.

  2. Piotr Hajlasz. , Functional Analysis Lecture Notes.
    Available online at https://sites.google.com/view/piotr-hajasz/.

  3. Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine. Riemannian Geometry.
    Universitext, Springer, 2004.

Prerequisites: Functional Analysis, Differential Geometry.

Simply Easy Learning

Contact: cerenaydin@nesinkoyleri.org, ozgince@gmail.com


Activities are supported by Nesin Mathematical Village, Turkish Mathematical Society and University of Bonn