Department of Mathematics

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Mathematics Colloquium 2011

 ◊ : research seminar


 

  28 December, 2011 

Prof. Martin Hils – Universite Paris 7  

Finite rank expansions of fields

In this talk, we give an overview on the constructions of finite rank expansions of fields, in the algebraically closed and in the pseudofinite case. In particular, we will connect the so-called ‘bad fields’ with certain finiteness and uniformity properties of algebraic varieties.

 


  21 December, 2011 

Dr.Onur Gün – Weierstrass Institute  

Trap models and aging for spin glasses

Spin glasses are highly disordered systems and their out-of-equilibrium dynamics have age dependent decorrelation properties, a phenomena known as aging. First, I will explain the results on the aging properties of the mean field spin glass models on short time scales where Bouchaud’s REM-like trap models have been confirmed as a universal aging scheme. However, due to the famous Parisi ansatz, on longer time scales, the dynamics should live on a hierarchically organized energy landscape. As a first attempt to understand the effects of this hierarchical structure on the dynamics we introduce the so-called GREM-like trap models. In this model the energy landscape is given through a tree with L levels. We prove that there exists various time scales where aging happens with different limiting functions depending on how many levels of the tree has been equilibrated.(Joint work with V. Gayrard.)

 


  14 December, 2011

Dr.Saadet Öykü Yurttaş – Dicle University  

Study of pseudo-Anasov braids using Dynnikov's coordinates

Isotopy classes of orientation preserving homeomorphisms on the n-times punctured disk Dn are represented by braids. The aim of this talk is to study dynamical properties of pseudo-Anosov braids on Dn making use of the so-called Dynnikov coordinate system. The Dynnikov coordinate system gives a homeomorphism from the space of measured foliations MFn on Dn (up to a certain equivalence relation) to R2n-4\{0}, and restricts to a bijection from the set of integral laminations (disjoint unions of finitely many essential simple closed curves) on Dn to Z2n-4\{0}.
First, we shall introduce a new method for computing the topological entropy of each member of an infinite family of pseudo -Anosov braids making use of Dynnikov’s coordinates. The method is developed using the results in Thurston’s seminal paper on the geometry and dynamics of surface homeomorphisms and builds on, more recent work of Moussafir. To be more specific, the method gives a so-called Dynnikov matrix which describes the action of a given pseudo-Anosov braid β near its invariant unstable measured foliation [F;µ] on the projective space PSn, and the eigenvalue λ > 1 of this matrix gives the topological entropy of β. If time permits, we shall compare the spectra of Dynnikov matrices with the spectra of the train track transition matrices of a given pseudo-Anosov braid, and show that these matrices are isospectral up to roots of unity and zeros under some particular conditions.

 


  7 December, 2011

Rahmi Güven – Boğaziçi University  

Observations on Interactions between Geometry and Physics
The purpose of this talk is to give an overview of the amazing interactions between geometry and physics. The rich and old history of the subject forces one to be selective. Before the 20th century there was not a necessary distinction between physicists and mathematicians as well as between theoreticians and experimentalists, and the talk will start by focusing on certain basic developments which belong to that era. We shall then concentrate on interactions between general relativity, geometry and gauge theory which occurred during the 20th century. In 1960 E. Wigner had noted the ‘unreasonable effectiveness of mathematics in physics’, and the results obtained by theoretical physicists during the last three decades have tempted some of the eminent mathematicians to rephrase this comment as the ‘unreasonable effectiveness of physics in mathematics’. We shall give a list of these seminal results and expound on some of them. The talk will end by remarks on the current dispute about the role of string theory in physics.

 


 30 November, 2011

Giorgi Khimshiashvili – Ilia State University (Georgia)  

Geometry and topology of polygonal linkages
We’ll present a recent approach to the study of geometry of polygonal linkages based on investigation of critical points of various natural functions on the
moduli spaces of linkages. In particular, we’ll discuss in some detail the case of oriented area considered as a function on the moduli space of planar polygonal linkage. It will be shown that, for a generic linkage, the oriented area is a Morse function, its critical points are given by the cyclic configurations of the linkage, and their indices can be computed from the geometry and combinatorics of cyclic configuration.
These general results will be essentially complemented for quadrilateral and pentagonal linkages. In particular, we’ll show that, for each configuration of a planar quadrilateral linkage with pairwise distinct side-lengths (a, b, c, d), the cross-ratio of its vertices belongs to the circle of radius ac/bd centered at point (1,0). Moreover, we’ll present an analog of Poncelet porism for the discrete dynamical system on the planar moduli space of 4-bar linkage defined by the product of diagonal involutions, and discuss some related issues suggested by a beautiful link to the theory of discrete
integrable systems discovered by J.Duistermaat.
We’ll also describe a connection between certain extremal problems for configurations of a linkage and convex polyhedra obtained from its configurations using Minkowski 1897 theorem and present several related concrete results.

 


2 November, 2011

Şahin Koçak – Anadolu Üniversitesi  

Politopların iç tüp hacimleri
R^n deki konveks, kompakt bir kümeye en fazla x uzaklığında bulunan noktalar kümesinin hacmi x’in bir polinomu ile verilmektedir. 1840’da Steiner tarafından keşfedilen bu özellik, aradan geçen zaman içinde, Minkowski’nin karışık hacimler teorisi, Minkowski boyutu ve ölçümü teorisi, Hadwiger’in konveks kümeler üzerindeki değerlemeler teorisi, Federer’in pozitif erişimli kümeler teorisi ve eğrilik ölçümleri teorisi, Lapidus’un fraktallerin kompleks boyutları teorisi gibi birçok teoriye esin kaynağı oldu. Bu konuşmada bu teoremin evrimi hakkında bilgi vermek ve bu konuya (Andrei Ratiu ile birlikte yaptığımız) küçük bir katkıdan bahsetmek istiyorum.

 


26 October, 2011  ◊

Tolga Karayayla – Boğaziçi Üniversitesi  

Automorphism groups of rational elliptic surfaces with section
In this talk I will describe the structure of the automorphism groups of rational elliptic surfaces with section defined over the field of complex numbers. While generic fiber of the map from the surface to the base curve is a smooth elliptic curve, there are also singular fibers. The configuration of those singular fibers on the elliptic surface gives very important information about the automorphism group. I will explain how the automorphism groups of rational elliptic surfaces are classified with respect to the configurations of singular fibers. Generally, the automorphism group is the semi-direct product of the Mordell-Weil group and the subgroup of automorphisms preserving the zero section of the surface. The Mordell-Weil group, which is the group of sections, has been classified by Oguiso and Shioda. In the talk I will concentrate on how the latter subgroup is determined.

 


19 October, 2011

Anthony Lau – University of Alberta  

Fourier algebra and group von Neumann algebra
In 1964 in a celebrated paper, P.Eymard has associated to any locally compact group G a commutative Banach algebra A(G), known the Fourier algebra of G. In the case G is Abelian this algebra is isometrically isomorphic to the group algebra L1(H), where H is the dual group of G. The group von Neumann algebra, VN(G), is the dual of the algebra A(G). Since Eymard's paper these two algebras have become central objects of study in harmonic analysis. In this introductory talk, I will introduce these algebras and mention their basic properties to invite the interested students to this area of harmonic analysis.

 


12 October, 2011  ◊

Richard Gonzales – Boğaziçi Üniversitesi  

GKM theory of certain group embeddings

 


5 October, 2011

Şerban Stratila – Institute of Mathematics of the Romanian Academy 

The fundamental theorem of analysis
In this talk we shall present a new and elementary proof of the Fundamental Theorem of the Lebesgue integration theory.

 


 

10 August, 2011

Kellogg S. Stelle – Imperial College, London

Gravity and unification: Hints from microphysics and cosmology
Understanding the nature of gravity at a microscopic level remains a key problem in fundamental physics. In the struggle to reconcile Einstein's theory with the principles of quantum mechanics, many different paths have been explored, some in the search for relevant analogies to the problems posed by gravity, but taken in the context of particle physics. As the subject has developed, it seems more and more that the microworld of particle physics and the macroworld described by gravity are intimately interrelated, with hints of a fundamental unification of all fundamental physics.

 


 

8 June, 2011

Theodora Bourni – Max Planck Institute for Gravitational Physics, Potsdam

Curvature estimates for surfaces with bounded mean curvature
We consider a surface in R^3. The differential of the normal to the surface, known as the second fundamental form A, is a very important tool for studying the geometry of the surface. In this talk I will first describe how estimates on A give us information about the curvature of the surface; in particular when |A| is bounded the surface cannot bend too sharply. Then I will discuss some results concerning estimates for the norm of the second fundamental form, |A|, for surfaces with bounded mean curvature (i.e. for which the trace of A is bounded). In particular I will show that for an embedded geodesic disk with bounded L^2 norm of |A|, |A| is bounded at interior points, provided that the W^{1,p} norm of its mean curvature is sufficiently small, p>2. This is joint work with Giuseppe Tinaglia.

 


 

25 May, 2011

Aurelian Gheondea – Bilkent Üniversitesi

Some noncommutative Lebesgue-Radon-Nikodym decompositions
We survey and present some older and some recent results on Lebesgue decompositions and Radon-Nikodym derivatives in the context of operator valued completely positive maps on C∗-algebras.

 


18 May, 2011

Kemal Ilgar Eroğlu – Bilgi Üniversitesi

Measure and dimension results for sums of Cantor sets
Arithmetic sums of Cantor sets in the real line is a problem of interest in dynamical systems. In particular, it is useful to have an estimate on the ”size” of these sums. Depending on the context, a ”big” sum may mean one with positive Lebesgue measure or large Hausdorff dimension. In this talk we will go over the past and recent progress on the Lebesgue measure and Hausdorff dimension problem of arithmetic sums, briefly explaining the methods of approach.

 


11 May, 2011

Burak Erdoğan – University of Illinois at Urbana-Champaign

Global smoothing for Korteweg de Vries equation with periodic boundary conditions
In this talk we will review the low regularity well-posedness results of the KdV equation introducing various methods used in the theory. We will also discuss a recent result on the smoothing properties of the equation which states that for initial data in Sobolev spaces (H^s, s>-1/2) the difference of the linear and nonlinear evolutions always belongs to a higher index Sobolev space. This is a joint work with Nikos Tzirakis.

 


 

4 May, 2011

Ferruh Özbudak – ODTÜ

Quadratics forms in even characteristics and maximal/minimal curves over finite fields
We present some of our results on quadratic forms of codimension 2 in even characteristic and maximal/minimal curves over finite fields. This is a report on our joint studies with Elif Saygi and Zulfukar Saygi.

 


 

20 April, 2011

Ayşe Altıntaş – Yıldız Teknik Üniversitesi

Examples of finitely determined map-germs
A holomorphic map-germ is said to be finitely determined with respect to an equivalence relation if it is determined by its Taylor series expansion up to a finite degree. The equivalence relation that interests us the most is Right-Left equivalence which is induced by an action of the groups of local diffeomorphisms of the source and the target on the space of holomorphic map-germs from n-space to p-space. Classification of finitely determined map-germs, or even finding examples, becomes harder as the dimensions and corank get higher. In this talk, I will present a theorem for generating new examples from old and list three "series" of finitely determined map-germs of corank 2 from 4-space to 5-space. I will finish by a discussion on the geometry of the examples.

 


 

13 April, 2011

Burak Özbağcı – Koç Üniversitesi

Milnor fillable contact structures are universally tight
We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tight contact structures that are not deformations of taut (or Reebless) foliations.This is a joint work with Yanki Lekili.

 


 

6 April, 2011

Ergün Yalçın – Bilkent Üniversitesi

Fusion systems and constructions of finite group actions on products of spheres
Some of the most interesting problems in group action theory are about finite group actions on products of spheres. I will first give an overview of problems and conjectures in this area and explain some of the tools that are used for constructing group actions on products of spheres. Then, I will describe a recent construction that we have done with Ozgun Unlu using fusion systems. Fusion systems are abstract models for the p-local structure of a finite group and recently have been studied intensively. Using fusion systems in the constructions of finite group actions is one of the few explicit applications of abstract fusion systems.

 


 

16 March, 2011

Hüsnü Erbay – Işık Üniversitesi

Global existence and blow-up results for nonlocal nonlinear wave equations
TIn this talk, I will summarize our recent results obtained in [1-3] for nonlocal nonlinear wave equations. In those studies three different problems which model longitudinal, transverse and anti-plane shear motions, respectively, were considered and the regularizing effect of the nonlocal behavior was discussed. In each case the wave equation involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. Some well-known examples of nonlinear wave equations are special cases of the present equations for suitable choices of the kernel function. We establish global existence of solutions of the equations assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are provided.
[1] N. Duruk, H.A. Erbay, A.Erkip, Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity 23, 107-118 (2010).
[2] N. Duruk, H.A. Erbay, A. Erkip, Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations, Journal of Differential Equations 250, 1448-1459 (2011).
[3] H.A. Erbay, S. Erbay, A. Erkip, The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials, Nonlinearity (in press).

 


 

9 March, 2011

Ali Ülger – Koç Üniversitesi

Sets of uniqueness for the trigonometric series
The Modern Mathematics was born as a consequence of the efforts spent to solve the following problem. Consider the trigonometric series ∑_{n=-\infty}^{n=+\infty} c_n e^{inx} . If, for each x in [0, 2\pi], this series converges to zero, is then this series identically zero? i.e. c_n = 0 for all n ∈ Z? Cantor did not only solve this problem in the affirmative, he also proved that there are infinite sets E ⊆ [0, 2\pi] such that, if this series converges to zero for each x ∈ [0, 2\pi] \ E, then the series is still identically zero. Such a set E is said to be a "set of uniqueness for the trigonometric series". In spite of the efforts of many outstanding mathematicians (Cantor, H. W. Young, Lebesgue, Menshov, Rajchman, Barry, Zygmund, Salem, Kahane, Piatetski-Shapiro, Körner,...) no characterization of the sets of uniqueness is obtained so far. In this talk I will survey some of the results about sets of uniqueness problem and present a recent contribution of mine to this problem.