Department of Mathematics

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Mathematics Colloquium 2014

 

 

 


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24 December, 2014

Mehmet Haluk Sengun, University of Sheffield    

Asymptotics of torsion homology of hyperbolic 3-manifolds 

Abstract: Hyperbolic 3-manifolds have been studied intensely by topologists since the mid1970’s. When the fundamental group arises from a certain number theoretic construction (in this case, the manifold is called “arithmetic”), the manifold acquires extra features that lead to important connections with number theory. Accordingly, arithmetic hyperbolic 3-manifolds have been studied by number theorists (perhaps not as intensely as the topologists) with different motivations. Very recently, number theorists have started to study the torsion in the homology of arithmetic hyperbolic 3-manifolds. The aim of the first half of this introductory talk, where we will touch upon notions like “arithmeticity”, “Heckeoperators”, will be to illustrate the importance of torsion from the perspective of number theory. In the second half, I will present new joint work with N.Bergeron and A.Venkatesh which relates the topological complexity of homology cycles to the asymptotic growth of torsion in the homology. I will especially focus on the interesting use of the celebrated “Cheeger-Mueller Theorem” from global analysis.

 


17 December, 2014

Debasis Sen, Indian Institute of Technology Kanpur, India    

Realizing homotopy group actions 

Abstract: A homotopy action of a group G on a topological space X is a group homomorphism from G to the group of homotopy classes of self-homotopy equivalences of X. George Cooke described an obstruction theory for realizing a homotopy action of a finite group G on a space X by strict action. However, the resulting G-space is only determined up to a homotopy equivalence which is a G-map (Borel equivalence), and in this sense every G-space is equivalent to a free one. So the more delicate aspects of equivariant topology are not visible in this way. A more informative approach to equivariant homotopy theory, due to Bredon, studies G-spaces X up to G-homotopy equivalence, that is, G-maps having G-homotopy inverses. The purpose of this talk is to define a notion of homotopy action of a finite group in Bredon equivariant homotopy theory, and describe an associated inductive procedure for realizing such an action by a strict one. (This is a joint work with Prof. David Blanc)

 


26 November, 2014

Ozgur Martin, Mimar Sinan Fine Arts University    

Disjoint supercyclic weighted shifts 

Abstract: After a brief introduction to the dynamics of linear operators, we will consider the notion of disjointness in hypercyclicity and supercyclicity. We will mainly focus on dynamics of weighted shifts on sequence spaces.

 


19 November, 2014

Dieter Van Den Bleeken, Boðaziçi University    

Blackholes under microscope 

Abstract: The study of black hole thermodynamics has been one of the most important guidances for research on quantum gravity. I will present a broad introduction into the subject, reviewing some of the basic motivations, questions and results in the field. I will then move on and discuss my recent work on a microscopic model for extremal D4D0 black holes using geometric and topological techniques.

 


12 November, 2014

Michel Lavrauw, University of Padova, Italy    

Finite non-associative division algebras: semifields 

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22 October, 2014

Serkan Sutlu    

Hopf-cyclic cohomology of quantum enveloping algebras 

Abstract: Given a coalgebra coextension, we introduce a filtration whose associated spectral sequence computes the Hochschild cohomology of the coextension. We then use such a spectral sequence to compute the Hopf-cyclic cohomology of a quantized enveloping algebra of a Lie algebra. Time permitting, we will also discuss the Hopf-cyclic cohomology of the h-adic Drinfeld-Jimbo quantum enveloping algebras. (Based on a joint work with A. Kaygun.)

 


15 October, 2014

Kazim Buyukboduk, Koc University    

An Asymptotic Birch and Swinnerton-Dyer Conjecture 

Abstract: The conjecture of Birch and Swinnerton-Dyer (BSD) is one of the Clay Millennium problems that links the arithmetic invariants of an elliptic curve to its analytic invariants. Most of this talk will be devoted to explaining the content of this conjecture and stating an *asymptotic* variant. As time permits, I will sketch a proof of this asymptotic BSD conjecture for CM elliptic curves (and more generally, for CM elliptic modular forms) along the anticyclotomic tower. The proof relies on the Iwasawa theoretic study of the Beilinson-Kato elements and a reciprocity law that relates them to relevant L-functions.

 


4 June, 2014

Song-Sun Lin, National Chiao Tung University, Hsinchu, Taiwan    

Pattern generation problems arising in multiplicative integer systems 

In this talk, I would like to discuss the pattern generation problems which arise from multiplicative integer systems. We investigate the systems by using a method that was developed for studying pattern generation problems in symbolic dynamical systems. The entropy of general multiplicative systems can thus be computed. A multi-dimensional decoupled system is investigated in three main steps. (I) Identify the admissible lattices of the system; (II) compute the density of

 


28 May, 2014

Thomas Scanlon, UC Berkeley    

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14 May, 2014

Alp Eden, Boðaziçi University    

Cumhuriyetin Ýlk Matematikçileri: Kerim Erim, Ratip Berker ve Cahit Arf

Konuþmamda 1930’lu yýllarýn ortalarýndan 1950’li yýllarýn sonuna kadar Türkiye’de Matematiðin geliþimine katkýda bulunan, öðrenciler yetiþtiren, kurumlarýn geliþmesinde aktif rol oynayan ve yaptýðý araþtýrmalarla Türkiye’nin yurtdýþýnda tanýnmasýný saðlayan üç bilim adamýnýn kariyerlerinden deðiþik kesitler sunacaðým.

 


7 May, 2014

Ryan O'Donnell,  Carnegie Mellon University     

Testing surface area

Suppose we have “black-box” access to a set A in Rn ; i.e., we can “query” points x in Rn and learn whether or not they are in A. We would like to estimate the surface area of A. If we only make a few queries, it will be impossible to tell the difference between a nice set A (like a ball) and a nice-set-with-an-extremely-high-surface-area-tentacle-of-volume-epsilon-glued-on. So we content ourselves with a weaker goal: we want to ?nd a slight deformation A' of A satisfying vol(AΔA') ≤ ε, as well as an accurate estimation of the surface area of A'. We give an algorithm solving this problem in either Euclidean or Gaussian space and making roughly 1/ε queries. Surprisingly, the number of queries does not depend on the dimension n.

 


30 April, 2014

Stephan Garcia,  Pomona College     

Supercharacters and their super powers

The Mordell-Lang Conjecture (MLC) concerns finitely generated subgroups of abelian varieties, however, there are analogous statements in the “non-compact” case: namely the case of the (cartesian powers of) multiplicative group, Gm. (Sometimes, MLC is formulated for semiabelian varieties to include this case.) In very heuristic terms, in the multiplicative group case, MLC says “the addition doesn’t give new information about finitely generated multiplicative groups of fields.” (This will be made clear in this talk.) We isolate the conlusion of MLC as an abstract property for subgroups of Gm. It turns out that many groups other than finitely generated ones have this property. Moreover, some of them satisfy a uniform version of it. Vaguely, this corresponds to the function field field case of MLC. In this talk, after making everything above more accurate, we prove that certain finitely generated groups have this “uniform” Mordell-Lang Property.

 


16 April, 2014

Ayhan Günaydýn,  Mimar Sinan Fine Arts University     

Uniform Versions of the Mordell-Lang Conjecture for Multiplicative Groups

The theory of supercharacters, which generalizes classical character theory, was recently developed in an axiomatic fashion by P. Diaconis and I.M. Isaacs, based upon earlier work of C. André. When this machinery is applied to abelian groups, a wide variety of applications emerge. In particular, we develop a broad generalization of the discrete Fourier transform along with several combinatorial tools. This perspective illuminates several classes of exponential sums (e.g., Gauss, Kloosterman, and Ramanujan sums) that are of interest in number theory. We also consider certain exponential sums that produce visually striking patterns of great complexity and subtlety. (Partially supported by NSF Grants DMS-1265973, DMS-1001614, and the Fletcher Jones Foundation)

 


02 April, 2014

Bruno Poizat,  Lyon 1      

Supergeneric subsets of a group

We define in an arbitrary group a filter of large subsets, closed under translations, which apparently has remained unnoticed till now, and we study some of its basic properties. This talk will be an introduction to a paper which has been recently published in the Journal of Algebra.