Department of Mathematics

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Mathematics Colloquium 2012

 


  26 December, 2012 

Flavio d'Alessandro, University of Rome La Sapienza   

Synchronizing automata and the hybrid Cerny-Road coloring Problem

The synchronization problem for a deterministic n-state automaton consists in the search of an input-sequence, called synchronizing word, such that the state attained by the automaton, when this sequence is read, does not depend on the initial state of the automaton itself. If such a sequence exists, the automaton is called synchronizing. If the automaton is deterministic and complete, a well-known conjecture by Cerny claims that it has a synchronizing word of length not larger than (n 1)^2. In this talk, we outline some classical and new results on this problem and an application to the Road coloring problem on finite oriented colored graphs.

 


  28 November, 2012 

Alp Bass, Sabancı University   

How many rational points can a high genus curve over a finite  
field have?

A In this talk we will be interested in the question of how many rational points a high genus curve over a finite field can have. We will introduce several approaches to this problem and present a recent result (joint work with Beelen, Garcia, Stichtenoth) over all non-prime finite fields.

 


  16 November, 2012 

Willam D. Gillam, ETH Zurich   

Degeneration techniques in algebraic geometry and topology

A basic idea in mathematics is to study a complicated object by breaking it into simpler pieces. For example, one can study a topological space by expressing it as a union of closed subspaces. In algebraic geometry this approach is not as directly available because the spaces of interest are typically "irreducible" (and the notion of "union" is more subtle); one must first "degenerate" a smooth space to a reducible space, then make sense of the objects of interest on the reducible space, which is typically singular. After providing a general overview of degeneration techniques in various geometric contexts, I will illustrate the general philosophy by explaining how one can study the symmetric products of a 2-manifold X by cutting X along a circle.

 


  7 November, 2012 

Muazzez Simsir, Hitit University  

Affine manifolds and harmonic maps

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, I will describe affine harmonic maps which should be a useful too for studying affine manifolds.

 


  10 October, 2012 

  Giray Okten - Florida State University 

The story of RASRAP

(Pseudo)random sequences are useful in many applications, in particular, in Monte Carlo algorithms. However, for certain problems such as numerical integration or global optimization, low-discrepancy(u.d. mod 1) sequences can give estimates with lower error. Rasrap is a hybrid sequence obtained by randomizing certain features of a low- discrepancy sequence. I will discuss theoretical and numerical properties of Rasrap, touching on topics such as von Neumann - Kakutani transformation, genetic algorithms, and GPU computing.

 


  3 October, 2012 

  Joel Spencer from Courant Institue, NYU

Discrete Percolation

The random graph with n vertices and edge probability p undergoes a discrete phase transition when p reaches 1/n. Slightly earlier (as originally shown by Paul Erdos and Alfred Renyi) the graph consists of tiny components while slightly later a giant component has emerged.
Today we see this is the major example of Discrete Percolation. We examine other graph processes which sometimes show similar behavior. A notion of susceptibility allows differential equations to be used. Analogues to Bond Percolation in Mathematical Physics are given. We step ‘inside the phase transition’ to see fine behavior. Connections are given to the Galton-Watson birth process with percolation when the average number of births is one.

 


  19 September, 2012 

  Robert Boltje - University of California Santa Cruz

The (double) Burnside ring of a finite group

For a finite group G, the Burnside ring B(G) is defined as the Grothendieck group of the category of left G-sets. It is an interesting invariant of the group G and is used extensively in the representation theory of G. The main tool to analyze its ring structure is the mark homomorphism which imbeds it into a direct product of copies of the integers, its so-called ghost ring. More recently, the double Burnside ring B(G;G), the Grothendieck group of finite (G;G)-bisets (sets with a left and right G-action), has become an important algebraic tool in representation theory and algebraic topology. We give a ghost ring construction for the double Burnside ring. This is joint work with Susanne Danz.

 


  6 June, 2012 

Olcay Coşkun - Boğaziçi University

Tower Tableaux

The group of symmetries of n letters is a very rich source for the theory of combinatorics. One of the basic problems regarding the symmetric group is the description of its elements, namely permutations, in terms of finite words. In this talk, we address this problem by introducing a new combinatorial object called tower diagrams. This is joint work with Müge Taşkın Aydın.

 


  9 May, 2012 

Ayşe Berkman - Mimar Sinan Güzel Sanatlar Üniversitesi

Multiply transitive group actions

Sharply multiply transitive group actions are rare for higher degrees in the finite setting, and non-existent in the infinite setting. However, when we loosen the definition in a model-theoretically obvious way, the most natural ac- tions (such as the action of the multiplicative subgroup of a field on its additive subgroup, or GLn (F) on F n ) fit this new description. In my talk, I shall outline possible classification projects of similar settings in the finite Morley rank case and mention some concrete results. (At the beginning, I shall give the necessary background in model theory and algebra; for example, Morley rank and sharp transitivity will be defined.)

 


  18 April, 2012 

Sinan Ünver - Koç University

Periods and motives

I will talk about a general conjectural framework of Grothendieck (later pursued by Bloch, Beilinson, Voevodsky etc.) in which certain arithmetic and geometric questions can be reduced to problems in linear algebra. In order to give the flavor of the subject without assuming a huge amount of background, I will follow the influential paper of Kontsevich and Zagier

 


 11 April, 2012

F. Alberto Grünbaum - University of California, Berkeley

Soliton mathematics as a unifying force

The study of nonlinear partial differential equations of mathematical physics such as those of Korteweg-deVries, Toda, nonlinear Schroedinger, etc starting around 1970 has given a unifying push to several parts of mathematics such as linear algebra, algebraic geometry, the Painleve property, inverse scatter- ing, isomonodromic deformations, and many others. All of these equations exhibit solitons, a nonlinear version of the superposition principle going back at least to Fourier in the case of linear equations.

 

 

  28 March, 2012

Ofer Zeitouni - Weizmann Institute of Science  

Maxima of Gaussian free fields and branching random walks

The Gaussian free field on a (rooted, finite, connected) graph is the Gaussian field whose covariance is given by the Green function of a continuous
time simple random walk, killed at hitting the root. We are interested in the fluctuations of the maximum of the field. When the graph is a box in the lattice Zd, the fluctuations of the maximum are of interest in the critical case d = 2.

 


  21 March, 2012 

Aybike Çatal-Özer – Istanbul Technical University

A massive S-duality in four dimensions

An important research problem in string theory is the fate of string dualities, when twists and/or flux is introduced in string compactifications. We start
by giving a general overview of the subject. Then we explain our current work, where we establish an S-duality relation between the two massive/gauged theories obtained from twisted compactifications of heterotic and IIA string theories down to four dimensions.