Title of the course: Introduction to Gauge Theory

Title of the instructor: Professor

Instructor’s Name: Tommaso Pacini

Institution: University of Torino, Italy

Dates: February 5 (monday), 2024-February 9 (friday), 2024

Prerequisites: Some knowledge of Differential Geometry

Level: Advanced Undergraduate
Abstract:

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.

Textbook or/and course webpage:

Donaldson, S. K.; Kronheimer, P. B. The geometry of four-manifolds.

Oxford Mathematical Monographs. Oxford: Clarendon Press. ix, 440p.

1990.

Donaldson, S. K. A new proof of a theorem of Narasimhan and Seshadri.

Journal of Differential Geometry, 18 (2): 269-277. 1983.

Language: English

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Title of the course: Minimal Submanifolds of Kähler manifolds

Instructor: Assoc. Prof. Mustafa Kalafat

Institution: Bonn University

Prerequisites: Multivariable Calculus, Linear Algebra

Level: Graduate, advanced undergraduate

Abstract:
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.

Textbook/Reference:

1. Li, Peter. Geometric analysis. Cambridge University Press, 2012.
2. James Simons. Minimal varieties in riemannian manifolds. Ann. of Math. (2), 88:62–105, 1968.

Language: EN

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Title of the course: Riemann surfaces and Algebraic Curves

Title of the instructor: Dr.

Instructor’s Name: Ozgur Ince

Institution: Sivas Cumhuriyet University

Prerequisites: Complex Analysis and Set Point Topology, basic notion

about algebraic curves

Level: Graduate, Advanced undergraduate

Abstract:
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.

Textbook or/and course webpage:

– E. Girondo and G. Gonzalez-Diez, Introduction to compact Riemann

surfaces and dessin d’enfants, LMS Students texts 79, Cambridge

University Press, Cambridge, 2012.

– R. Miranda, Algebraic curves and Riemann surfaces, Graduate Studies

in Mathematics

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

– S. Donaldson, Riemann Surfaces, Oxford Graduate Texts in Mathematics

Language: English

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Title of the course:  Spectrum of the Laplacian Operator on Riemannian Manifolds

Title of the Instructor: Ms.

Instructor:  Buket Can Bahadır

Institution: METU

Prerequisites:  Functional Analysis, Differential Geometry

Level:  Graduate, advanced, undergraduate

Abstract or a daily curriculum:

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

Textbook or/and course webpage:

1. Olivier Lablée, – Spectral Theory in Riemannian Geometry, European Mathematical Society, 2015.
2. Piotr Hajlasz, Functional Analysis Lecture Notes, online at https://sites.google.com/view/piotr-hajasz/
3. Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, Riemannian Geometry (Universitext), Springer, 2004

Language: EN/TR

Title of the course: Introduction to Mirror of Submanifolds

Title of the instructor: Associate Professor

Instructor’s Name: Kotaro Kawai

Institution: Yanqi Lake Beijing Institute of Mathematical Sciences and Applications

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

Level: Graduate, Advanced Undergraduate

Abstract:

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).

Language: English

Textbook or/and 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