METU Mathematics Seminars
Fall 2012

October 4

Mustafa Kalafat

Tunceli University

Einstein-Hermitian 4-Manifolds of Positive Bisectional Curvature.

We show that a compact complex surface together with an Einstein-Hermitian metric of positive holomorphic bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini-Study metric up to rescaling. This result relaxes the Kaehler condition in Berger's theorem (1965). Joint work with C.Koca.

October 11

University of Wisconsin, Madison

Topology of Musical Data

Techniques for discovering topological structures in large data sets are now becoming practical. Though the data can some from anywhere (images, financial data, computational models, simulation, time series data, etc) this talk shows why the musical realm is a particularly promising arena in which to expect to find nontrivial topological features. While some familiarity with topological notions will be helpful, the emphasis is on applying computational tools (i.e., readily available computer programs) to help locate underlying (topological) structures in the data. The analysis is able to recover three important features in the musical application: the circle of notes, the circle of fifths, and the rhythmic repetition of timelines, often pictured in the necklace notation. Current work includes applying the methods to the study of Turkish maqams.

October 18

Middle East Technical University
Physics Department

Fuzzy Geometries and Quantum Physics

Fuzzy spaces appear as a special subclass of noncommutative spaces, which possess most of the symmetries of their commutative parents. For this reason and also in part due to motivations coming from string theory, they stand as an especially appealing ground for studying quantum field theories. In this talk, I will first sketch some elementary features of noncommutative and fuzzy spaces and explain in some simple instances what makes them useful for physicists. In the remaining part of the talk I will present some ideas from my recent research, where fuzzy spaces appear as extra dimensions in a certain gauge theory, and equivariant dimensional reduction ideas are applied to integrate over them.

November 8
(Arf Lecture)

University of California, Berkeley

Traces and Loops

This talk will focus on the interplay between two basic notions originating in algebra and topology. In algebra, there is the trace of a matrix, important for its equality with the sum of the eigenvalues of the matrix. In topology, there are the loops on a space, which play a central role in the computation of homotopy groups and in the structure theory of spaces. There is a well-developed understanding of the intimate relation between traces and loops coming from non-commutative geometry and mathematical physics. We will explain how modern formulations elucidate fundamental identities in geometry and representation theory.

November 15

Muazzez Simsir

Hitit University

Affine manifolds

A manifold is said to be affine flat if it admits local coordinate systems whose transition maps are affine transformations. For affine flat manifolds it is natural to ask the following question: "Among many Riemannian metrics that may exist on an affine flat manifold, which metrics are most compatible with the flat structure?" In this talk, I will explain that among all others the Kaehler affine metric provides the best compatibility. I will also recall the Kaehlerian manifolds, which are formally similar to the Kaehler affine manifolds noting that the Kaehlerian metric provides the best compatibility with the complex structure. In addition, if I have time I will describe affine harmonic maps which should be a useful tool for studying affine manifolds.

November 22

Sabanci University

A combinatorial construction of the series side of the Andrews-Gordon Identities

In 1961, Gordon found a family of partition identities generalizing the Rogers-Ramanujan Identities. The corresponding q-series identities were discovered by Andrews in 1974 using analytical techniques. More recently, Andrews revisited the problem, and gave several infinite families of similar identities. He also made a conjecture. We will describe a method to construct the series sides of the mentioned identities, and prove the conjecture if time allows. Open problems and directions for future research will also be discussed. The talk will be self contained.

November 29

Middle East Technical University
Northern Cyprus

Constructing Quantum Groups

Quantum groups are the group objects of noncommutative geometry. They are known to have applications in diverse areas of mathematics such as knot theory, repre- sentation theory and mathematical physics. The two main techniques for constructing quantum groups use deformations and bicrossed products. We will attempt to propose another method for constructing quantum groups and give some examples.

December 6

Emre Coskun

Middle East Technical University

Rationality of moduli spaces of vector bundles

The rationality question of the moduli spaces of vector bundles over an algebraic curve is a longstanding open problem. In this talk, we will talk about the progress toward the solution of this problem and some recent developments.

December 13

Bilkent University

Rank conjecture on free actions on products of spheres

A classical conjecture on free actions on products of spheres states that if an elementary abelian p-group of rank r acts freely on a product of k spheres, then r must be less than or equal to k. Recently, we proved this conjecture (joint with Osman Berat Okutan) under the condition that the average of the dimensions of the spheres is large compared to the differences between them. I will start with a survey of this problem and then explain some of the basic ideas behind the proof.

December 20

Hursit Onsiper

Middle East Technical University

Fundamental Group in Algebraic Geometry

We will try to explain how one defines the "true" fundamental group in algebraic geometry.

December 27

Universite Lyon I

Classification of conjugacy classes of elements of SL(2,Z) and "real" classes in SL(2,Z)

We present the classification of conjugacy classes in GL(2,Z) of elements of SL(2,Z). The classification is given in terms of the action of SL(2,Z) on the Poincare Disk endowed with the Farey tessellation. We then use this classification to describe elements of SL(2,Z) which can be written as a product of two involutions of GL(2,Z)-SL(2,Z). Elements admitting such a decomposition are called "real".

January 2,3 and 4
 

Matthias Creck from Bonn University will give a series of talks in our department joint with the Institute of Applied Mathematics.

Universitat Bonn

In the first lecture addressing a general audience I will quickly introduce error correcting codes. Mathematically they look completely uninteresting at the first glance (although very useful). That they are in contrast very interesting will follow from their relation to unimodular lattices, which will be discussed in the second part of the first lecture. At the end I will very briefly indicate a relation between manifolds and codes and explain a new construction of self dual codes motivated from topology.

I have not yet decided in which order I will present the content of the following lectures. But the following will be explained.
--> The details of the construction of codes from manifolds.
--> The details of the proof of the theorem concerning the realization of self dual codes by manifolds.
--> The formulation and proof of a result which characterizes those 3-manifolds which lead to doubly even codes.
--> The realization problem for arbitrary codes and the proof of partial results.
--> The relation of the construction of codes to equivariant cohomology.
--> A short report about a result by Manin about the distribution of codes and - based on this - a speculative discussion about "exceptional manifolds".

Codes, Arithmetics and Manifolds I

January 2, Gunduz Ikeda Seminar Room, 15:40.

Codes, Arithmetics and Manifolds II

January 3, Gunduz Ikeda Seminar Room, 15:40.

Codes, Arithmetics and Manifolds III

January 4, S209 IAM, 15:40.

About Matthias Kreck:

Matthias Kreck (1947) received his doctorate in Bonn, in 1972, under the supervision of Friedrich Hirzebruch, and was his assistant from 1970 to 1976. In 1977 he completed his habilitation in Bonn (Bordism groups of diffeo-morphisms). Then, he was working as a professor in the Wuppertal University, the University of Mainz, the University of Heidelberg, and currently, is a professor at the University of Bonn .

From 1994 to 2002 he was director of the Mathematical Research Institute in Oberwolfach. From 2007 until October 2011 he was the founding director of the Hausdorff Research Institute for Mathematics at the University of Bonn.

From 1990 to 1998 he was an editor of Mathematischen Annalen and from 1998 to 2002 for Archiv der Mathematik. Since 2000 he has been a member of the Heidelberg Academy of Sciences. In 2010 he was awarded the Cantor Medal. Among his PhD students is Peter Teichner,[3] presently a Director of the Max Planck Institute for Mathematics in Bonn.