Seminar on Geometry 

Schedule of talks

Abstracts/Notlar

E-PDEs 1: Second order elliptic equations. Existence of weak solutions. Lax-Milgram theorem.

E-PDEs 2: Proof of Lax-Milgram theorem.

E-PDEs 3: Energy estimates and first existence theorem for weak solutions.

E-PDEs 4: Second and Third theorem for weak solutions. Fredholm alternative.

E-PDEs 5: Interior and boundary regularity.

E-PDEs 6: Maximum Principle.

E-PDEs 7: Harnack's inequality.

Maxwell 1: Einstein-Maxwell equations on a Riemannian manifold.

Maxwell 2: Decomposing the 2-form F. Constant Scalar curvature Kähler metrics.

Maxwell 3: Linearization of curvature tensors. Ricci and scalar curvature. Variational characterization of Einstein metrics.

Maxwell 4: Variational characterization.

Maxwell 5: Ricci form of a Kähler Metric. Closedness. Critical points of the Einstein-Hilbert action functional under conformal deformations are constant scalar curvature metrics. Critical points under all variations are Einstein metrics.

Maxwell 6: Linearization of the Einstein-Maxwell equations.

Maxwell 7: Weitzenböck formula for p-forms. Holomorphic Weitzenböck formula.

Maxwell 8:

Maxwell 9: Introduction to Cohomogeniety 1-manifolds: Homogenous spaces. Symmetric spaces. Isometry groups of some basic spaces. Semi-direct products. Examples: Rotationally symmetric surfaces. Page Metric.

Holonomy 1: Principal bundles. Associate bundles. Reduction of the structural group. Spin structures. Relation to G-Structures.

Holonomy 2: Holonomy groups and parallel spinors.

Holonomy 3: Ricci-flat curvature and topology. A-hat genus.

Holonomy 4: Geometric measure theory: Rectifiable and Integral Currents. Compactness. Tangent cones.

Holonomy 5: Calabi-Yau manifolds.

Holonomy 6: Special Lagrangian Geometry.

Holonomy 7: Fiber bundle of positive (or definite) 3-forms on a 7-manifold. Abundance of G_2 structures.

Holonomy 8: Epsilon ε-notation. Topology of space of positive/definite 3-forms. G_2 representation theory. Orthogonal decomposition of forms. Intrinsic torsion forms.

Holonomy 9: Hyper-Kähler manifolds.

Holonomy 10: More on cohomology and topology of G_2 manifolds. Full G_2 finiteness, examples. Refined Betti numbers. Nontriviality of the first Pontryagin class for full G_2 metrics.

Holonomy 11: Quaternion-Kähler manifolds. Introduction to the Quaternionic hyperbolic space: HH^n.

Caner 1: The resolution of Calabi's Conjecture by S.-T. Yau in 1977 is considered to be one of the crowning achievements in mathematics in 20th century. Although the statement of the conjecture is very geometric, Yau's proof involves solving a non-linear second order elliptic PDE known as the complex Monge-Ampere equation. An immediate consequence of the conjecture is the existence of Kähler-Einstein metrics on compact Kähler manifolds with vanishing first Chern class (better known as Calabi-Yau Manifolds). In this expository talk, I will start with the basic definitions and facts from geometry to understand the statement of the conjecture, then I will show how to turn it into a PDE problem, and finally I will highlight the important steps in Yau's proof.

Caner 2: Einstein's Equations on Compact Complex Surfaces: After a brief review of Einstein's Equations in General Relativity and Riemannian Geometry, I will talk about one of my results: The only positively curved Hermitian solution to Einstein's Equations (in vacuo) is the Fubini-Study metric on the complex projective plane.

Caner 3: Extremal Kähler metrics are introduced by Calabi in 1982 as part of the quest for finding "canonical" Riemannian metrics on compact complex manifolds. Examples of such metrics include the Kähler-Einstein metrics, or more generally, Kähler metrics with constant scalar curvature. In this talk, I will start with an expository discussion on extremal metrics. Then I will show that, in dimension 4, these metrics satisfy a conformally-invariant version of the classical Einstein-Maxwell equations, known as the Bach-Maxwell equations, and thereby are related to physics (conformal gravity) in a surprising and mysterious way.

Info Geometry: We will describe statistical models and give some examples. Then, we will define and give examples of Fisher information metric of a statistical model.

Cscc 1: Yamabe Problem.

Cscc 2: Green's function rescaling trick to get scalar-flat structure. Glued metrics on connected sums. Combining constant and positive scalar curvature.

Cscc 3: Combining two metrics of constant scalar curvature. Using tube (cylinder) in between. Under LCF at a point assumption.

Cscc 4: Estimating the scalar curvature on the connected sum. Uniform bound. (2.4 & 2.5)

Cscc 5: Constant scalar curvature on the connected sum by rescaling. Using strong maximum principle.

Cscc 6: Constant negative curvature: + & - turns into -. - & - turns into -. (For ones having at least one LCF point.)

AS-1: Mirror duality in a Joyce manifold. Previously, they defined a notion of dual Calabi-Yau manifolds in a G_2 manifold, and described a process to obtain them. In this paper, they apply this process to compact G_2 manifold, constructed by D. Joyce and as a result they obtain a pair of Borcea- Voisin Calabi-Yau manifolds, which are known to be mirror duals of each other.

AS-2: G_2 metrics on S^3xR^4. Reference: Elementary constructions of metrics with holonomy G2 slides by Simon Salamon G_2 metrics on Λ_(M). M: self-dual & Einstein.

AS-3: Deformations of G_2 manifolds.

ø-Free 1-2 : Bu ders dizisinde,

İbrahim Ünal - h-Principle and ø-free Embeddings in Calibrated Manifolds. Preprint.

makalesi, gerekli altyapıya giriş ile birlikte anlatılacaktır. Konunun uzmanıolmayanların da soruları cevaplanarak genel dinleyiciye hitap edilmesi amaçlanmaktadır:

Abstract: We prove that the h-principle holds for f -free embeddings for coassociative, Cayley and quaternionic calibrations. As a result, we show that for coassocative calibration ∗j, an oriented smooth closed 4-manifold N4 can be embedded into a G_2-manifold M^7 as ∗j-free if the Euler characteristic of N, c (N) is zero.

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