Seminar on Geometry 

Schedule of talks

Abstracts/Notlar

H-prensibi 1: PDEs and PDRs in differential geometry.
1. J-holomorphic curves 2. Seiberg-Witten (SW) equations 3. Yau's Monge-Ampere equations.
Jets, Ampleness.

H-prensibi 2: Tekrar. Örnekler: Immersion and submersion relations and their coordinate ampleness. Ampleness criterion for directed immersions.

H-prensibi 3: TBA

Geometric PDEs 1: Hölder, Sobolev spaces.

Geometric PDEs 2: Sobolev spaces, weak derivatives. Properties. Interrior and global approximations by smooth functions: Mollifiers.

Geometric PDEs 3: Extensions of functions in Sobolev spaces. Riemannian measure.

Geometric PDEs 4: Traces of functions in Sobolev spaces, i.e. correct way of extending a function to the boundary.

Geometric PDEs 5: Examples. Heaviside function. Weak derivatives. Cantor function. Greens function, etc.

Geometric PDEs 6: Sobolev inequalities: Gagliardo-Nirenberg-Sobolev Inequality

Geometric PDEs 7: Morrey's Inequality. Proof.

Geometric PDEs 8: Proof of Morrey's Inequality. General Sobolev inequalities

Geometric PDEs 9: Compactness Theorems: Rellich-Kondrachov Theorem: Compactness of W^k,p in L^q.

Geometric PDEs 10: Proof of Rellich-Kondrachov Theorem: Application of Arzela-Ascoli thm. Precompactness(Relatively compactness).

Geometric PDEs 11: Applications and adaptations to the Riemannian Manifolds. Norms.

Geometric PDEs 12: Sobolev spaces on Riemannian manifolds.

Geometric PDEs 13: Sobolev embeddings on compact Riemannian manifolds. Volume condition on balls. Examples. Gromov-Hausdorff convergence of metric spaces.

Geometric PDEs 14: Gromov precompactness theorem. Cheeger's finiteness type theorems. Sobolev embeddings for complete manifolds.

Geometric PDEs 15: Green's functions on Riemannian manifolds.

Şahin 1: We give the classical definition of a toric variety involving the torus action and provide examples to illustrate it.
We introduce two important lattices that play important roles in the theory of algebraic tori and demonstrate how they arise naturally in the toric case. Finally, we introduce affine toric varieties determined by strongly convex rational cones.

Şahin 2: We introduce fans and the (abstract) toric variety determined by a fan via gluing affine toric varieties defined by the cones in the fan. We include some examples and conclude with the correspondence between orbits of the torus action and the cones in the fan.

Toric Varieties 3: We will revise the material on toric varieties with emphasis on examples and introduce some new concepts as time permits: Examples. Fans to Manifolds. Tori. Hirzebruch surfaces.

Toric Varieties 4: Blow ups. Resolution of Singularities.

Toric Varieties 5: More on resolution of singularities. Weighted projective spaces and relation to Hirzebruch surfaces.
Torus action. Orbits.

Toric Varieties 6: Examples on finding the orbits in complex projective space. (Weil) and Cartier divisors.
Principle divisors. Injections. Invariant divisors.

Toric Varieties 7: Projective descriptions of Hirzebruch surfaces. C^* actions. Minitwistorspaces.

Toric Varieties 8: Projective descriptions of Hirzebruch surfaces. C^* actions. Minitwistorspaces. 2.

Toric Varieties 9: Projective descriptions of Hirzebruch surfaces. C^* actions. Minitwistorspaces. 3.

Toric Varieties 10: Homology and cohomology. Poincare Homomorphism. Characteristic classes.

Alex : In this very introductory talk I will try to discuss the interplay between such concepts as embedded toric resolutions of singularities via Newton polygons, Viro’s combinatorial patchworking, and tropical geometry.

Şahin 3 : We start with the definition of normal, very ample and smooth polytopes. We next define the projective toric variety X_A determined by a finite set A of lattice points. When A is the lattice points of a polytope P we demonstrate that X_A reflects the properties of P best if P is very ample. We also define the normal fan of P and discuss the relation between the corresponding "abstract" variety X_P and the embedded variety X_A.

Alex 2 : (Viro's patchworking) This is a continuation of my previous talk. After a brief introduction to Hilbert’s 16th problem, I will try to outline the basic ideas underlying Viro’s method of patchworking real algebraic varieties.

Şahin 4 : The aim of this talk is to introduce the so called homogeneous coordinate ring of a normal toric variety. We will see how Chow group of Weil divisors turn this ring into a graded ring. Finally we show that every normal toric variety is a categorical quotient.

Şahin 5 : After the promised example of "bad" quotient, I will review the correspondence between subschemes of a normal toric variety and multigraded ideals of its homogeneous coordinate ring.

CY 1 : Holonomy Groups: Berger's classification.

CY 2 : First, we will discuss connections over vector bundles and parallel transport to define the holonomy group of a bundle connection. Then, Riemannian holonomy groups, which has stronger properties compared to holonomy group of an arbitrary bunle connection, arise as a special case. Furthermore, we will talk about reducible Riemannian manifolds and symmetric spaces in order to understand Berger's classification of holonomy groups.

Sarı 1 : Literatürde yeni bir kavram olan (2n+s)-boyutlu genelleştirilen Kenmotsu manifoldlar tanıtılacaktır.
Konunu temelini teşkil eden kontak manifoldlar ve çatılandırılan manifoldlar hakkında kısa bir giriş yapılıp
genelleştirilen Kenmotsu manifoldların temel özellikleri verilecektir. A.Vanlı ile ortak çalışmadır.

Sözen : In the present talk, we will give the definition and basic facts about the topological invariant, Reidemeister torsion.
We will also present our topological applications of the torsion to Hitchin component and pleated surfaces.

Kronberg : For elliptic curves over the rational numbers the possible torsion subgroups are known due to the famous result by Mazur. In an earlier result Ogg found parametrized infinite families for all possible torsion subgroups. For hyperelliptic curves nearly nothing about the rational torsion structure is known. There are some resluts by Cassels, Elkies, Flynn and Leprévost who give single curves with large torsion. We will show the construction of families of hyperelliptic curves with p-torsion for small primes p. Furthermore we will give an overview over the attempts that did not work.

M.Tosun : Any simple elliptic singularity of type ~D_5 can be obtained by taking the intersection of the nilpotent variety and the 4-dimensional "good slices" in the semi-simple Lie algebra sl(2,C) + sl(2,C). We describe these new slices purely by the structure of the Lie algebra.
We also construct the semi-universal deformation spaces of ~D_5-singularities by using the 4-dimensional "good slices".

Ono 1 : Kaoru Ono is visiting the department. He will give a series of talks (more or less independent of each other): Monday, Wednesday and Thursday. All talks will be at 15:40 in the Ikeda Seminar Room.
Monday: In the middle 1980's, Andreas Floer initiated what is now called Floer theory for Lagrangian submanifolds. After explaining the idea of Floer theory for Lagrangian, I would like to present a general framework of the theory. Based on joint works with K. Fukaya, Y.-G. Oh and H. Ohta)

Ono 2 : Wednesday: For a compact Kahler toric manifolds, we have a family of Lagrangian tori in it. Namely, the inverse images of points in the interior of the moment polytope. I would like to explain Floer theory for these Lagrangian tori and discuss some of its implications. This is based on joint works with K. Fukaya, Y.-G. Oh and H. Ohta.

Ono 3 : Thursday: Lagrangian submanifolds are important objects in symplectic geometry. They are not so easily displaced by Hamiltonian isotopies. Sometimes, no Hamiltonian isotopy can displace a Lagrangian submanifold. Such a Lagrangian manifold is called non-displaceable. I would like to discuss some criteria for non-displaceablity with examples.

ConfK 1 : We show that a compact complex surface which admits a conformally Kähler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g becomes a multiple of the Fubini-Study metric.

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