# Department of Mathematics

Mathematics Colloquium 2010

 ◊ : research seminar 6 January, 2010 Sibel Özkan – Michigan Technological University Generalization of the Erdös-Gallai inequality A sequence of non-negative integers is said to be graphic if it can be realized by a simple graph. P. Erdös and T. Gallai gave necessary and sufficient conditions for a sequence of non-negative integers to be graphic. Here, their result is generalized to multigraphs with a specified multiplicity. This both generalizes and provides a new proof of a result in the literature by Chungphaisan. 3 March, 2010 Mohan Bhupal – Orta Doğu Teknik Üniversitesi Milnor open books of links of some rational surface singularities In this talk, I will describe Legendrian surgery diagrams for canonical contact structures on the links of certain rational surface singularities. Also I will describe an infinite family of contact 3- manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. support norm) of that canonical contact structure, for each member of this family. 10 March, 2010 Stephan Garcia – Pomona College Hidden symmetries in everyday operators What do a 2 x 2 matrix, a Jordan block, a complex Hankel matrix, the adjacency matrix of a graph, $\int_0^x f(y)\,dy$, and the Fourier transform have in common? They each enjoy hidden symmetries (can you find them?) which are part of a general theory, only recently developed by the speaker and his collaborators. This talk should be accessible to graduate students and advanced undergraduates. (Partially supported by NSF Grant DMS-0638789). 17 March, 2010 Gizem Karaali – Pomona College Quantization and superization Differential geometry and Lie theory have traditionally provided the mathematical framework for our most intuitive physical theory: classical mechanics. However, as is well-known, in the last century physicists have developed newer theories which incorporate different kinds of symmetries, and bold concepts like the uncertainty principle have arisen that need to be addressed mathematically. Mathematical physicsists' response has been a constant search for methods of quantization and superization, thus allowing the integration of older techniques into these newer, broader theories. This talk will explain one part of this story in more detail. In particular we will describe super quantum group theory, an eclectic collection of theorems and conjectures whose development is very much still in progress, but one that promises a solution to some foundational questions in mathematical physics. The mathematical background needed is limited, the physical background needed is none; the main prerequisite for this talk is a curious mind which is willing to accommodate some occasional vague language. 24 March, 2010 Türker Bıyıkoğlu – Işık Üniversitesi Extremal eigenvalues of graphs The fundamental graph properties e.g. coloring, diameter, isomorphism and connectivity are closely related (or bounded) to the eigenvalues of matrix representations of these graphs (e.g. adjacency matrix or Laplacian matrix of the graph). The sharp eigenvalue bounds for such graph invariants depend on the extremal eigenvalues. Extremal graph eigenvalue problem is finding a graph in a given graph class that has the minimum (or maximum) eigenvalue for a given matrix representation. I shall talk about these connections between graph properties and eigenvalues of graphs. I shall present results, methods, difficulties and possible further research topics on extremal graph eigenvalue problems. This talk is intended for general audience. No specialist knowledge is required. 31 March, 2010 Matias Courdurier – Pontificia Universidad Catolica de Chile Computed tomography and inversion formulas with incomplete measurements In Computed Tomography, as well as other imaging applications, the goal is to reconstruct a function in two or three dimensions from knowledge of its line integrals, also known as the ray- transform of the function or Radon Transform in the two dimensional case. In this talk we will present how the ray-transform appears as a model for the measurements in Computed Tomography. We will describe the mathematical setting in which this object is studied and quickly overview the main classic results, including inversion formulas. We will analyze in detail the non-locality characteristic of the classic formulas and present a different approach, which allows to obtain inversion formulas for cases of incomplete measurements. We will finish by studying how this approach can be exploited to obtain new results in particular cases of practical interest. 7 April, 2010 Robin Wilson – Open University, UK Leonhard Euler -- 300 years on Leonhard Euler (born 1707) was the most prolific mathematician of all time, and worked in a wide variety of areas, ranging from the very 'pure' ¯ the theory of numbers, the geometry of a circle and musical harmony ¯ via such areas as infinite series, logarithms, the calculus and mechanics, to the practical ¯ optics, astronomy, the motion of the Moon, the sailing of ships, and much else besides. In this illustrated lecture I shall explore some of these topics in a historical context. 21 April, 2010 Cem Güneri – Sabancı Üniversitesi Some applications of curves over finite fields I plan to present two applications of algebraic curves over finite fields to coding theory. One is related to the study of weights in cyclic codes, which is one of the classical problems of coding theory. The second is the construction of the so-called algebraic geometry codes. Both of these applications require answers to some challenging problems about curves. We will try to present some of these problems. 12 May, 2010 Sinem Çelik Onaran – Boğaziçi Üniversitesi Legendrian knots and open book decompositions Due to Alexander, it is well known that every closed oriented 3-manifold has an open book decomposition. In this talk, we will define open book decompositions and discuss various examples in detail. Further, we will discuss the importance of the open books in manifold theory, in paticular in contact geometry. After a brief introduction on contact 3-manifolds, we will focus on a class of knots in contact 3-manifolds called Legendrian knots. We will define a new invariant for Legendrian knots using open book decompositions and we will discuss applications of this invariant. This talk aims to reach general audience. 3 June, 2010 Atilla Yılmaz – UC Berkeley Large deviations for random walk in a random environment I will talk about large deviations for nearest-neighbor random walk in an i.i.d. environment on $\mathbb{Z}^d$. There exist variational formulae for the quenched and the averaged rate functions $I_q$ and $I_a$, obtained by Rosenbluth and Varadhan, respectively. $I_q$ and $I_a$ are not identically equal. However, when $d\geq4$ and the walk satisfies the so-called (T) condition of Sznitman, they are equal on an open set $A_{eq}$. For every $\xi$ in $A_{eq}$, there exists a positive solution to a Laplace-like equation involving $\xi$ and the original transition kernel of the walk. This solution lets us define a new transition kernel via the h-transform technique of Doob. This new kernel corresponds to the unique minimizer of Varadhan's variational formula at $\xi$. It also corresponds to the unique minimizer of Rosenbluth's variational formula provided that the latter is slightly modified. In other words, when the limiting average velocity of the walk is conditioned to be equal to $\xi$, the walk chooses to tilt its original transition kernel by an h-transform. 30 June, 2010 Ali Göktürk – Brown University When do Teichmuller geodesics fellowtravel WP geodesics?  The coarse geometry of the Teichmuller spaces have strong connections with combinatorial spaces like the curve complex and the pants complex. These spaces provide combinatorial models for the Teichmuller and Weil­Petersson metrics on Teichmuller spaces. Using these models I prove the following: Given any two points on the Teichmuller space of five-punctured sphere or twice­punctured torus, the Teichmuller geodesic and the WP geodesic connecting these points fellowtravels each other in the WP metric. If time permits, I will explain why this fact is not true for higher complexity surfaces. 2 July, 2010 Robert Boltje – UC Santa Cruz On the depth of subgroups  TBA 5 July, 2010 Başak Gürel – Vanderbilt University Action and index spectra and periodic orbits of hamiltonian systems   The action and index spectra of a Hamiltonian diffeomorphism and their behavior under iterations carry important information about the periodic orbits of the diffeomorphism. For instance, on a closed, symplectically aspherical manifold, the boundedness of the minimal action- index gap implies the Conley conjecture asserting that such a diffeomorphism has simple periodic orbits of arbitrarily large period. In this talk, based on joint works with Viktor Ginzburg, we will discuss some results along these lines and, time permitting, touch upon their proofs. 7 July, 2010 Yevhen Zelenyuk – University of the Witwatersrand, Johannesburg Regular idempotents in βG   Every idempotent ultrafilter p on a group G determines a Hausdorff left translation invariant maximal topology on G in which p converges to the identity. We say that p is regular if this topology is regular and p is uniform. We show that for every infinite group G, there exists a regular idempotent ultrafilter on G. As a consequence we obtain that for every infinite cardinal k, there exists a homogeneous regular maximal space of dispersion character k, which is the answer to an old difficult question. Another consequence tells us that the topology of the real line can be refined to a translation invariant regular maximal topology of dispersion character continuum. 21 July, 2010 Baver Okutmuştur – Université de Pierre et Marie Curie Finite volume methods for nonlinear hyperbolic conservation laws on manifolds   The first part of presentation is devoted to the study of finite volume methods for conservation laws on manifolds. We study first an approach based on a metric on Lorentzian manifolds. Our main result establishes the convergence of monotone and first-order finite volume schemes for a large class of (space and time) triangulations. Next, we consider another approach based on differential forms. We establish a new version of the finite volume methods which only requires the knowledge of family of n-volume form on an (n + 1)-manifold. The second part is concerned with error estimates for finite volume methods and the implementation of a model of relativistic compressible fluids. We consider first nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1 -error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. Next, we consider the hyperbolic balance laws posed on a curved spacetime endowed with a volume form, and, after imposing a natural Lorentz invariance property we identify a unique balance law which can be viewed as a relativistic version of Burgers equation. Numerical experiments demonstrate the convergence of the proposed finite volume scheme. 3 August, 2010 Allan P. Fordy – University of Leeds Integrable maps and Poisson algebras derived from cluster algebras   We consider a class of map, recently derived in the context of cluster mutation [1]. Not all of these are integrable, but we specialise to some particular families which are. We discuss invariant Poisson brackets, along with the Poisson algebra of special families of functions associated with some of these maps. For one particular family, a bi-Hamiltonian structure is derived and used to construct a sequence of Poisson commuting functions and hence show complete integrability [2]. One of the most important open questions regards the identification and classification of the integrable cases of maps obtained through the construction of [1]. This classification is the main theme of [3]. [1] A.P. Fordy and R.J. Marsh, Cluster mutation-periodic quivers and associated laurent sequences (2009). Preprint arXiv:0904.0200v2 [math.CO]. [2] A.P. Fordy, Mutation-periodic quivers, integrable maps and associated Poisson algebras (2010). Preprint arXiv:1003.3952v1 [nlin.SI]. [3] A.P. Fordy and A.N.W. Hone, Integrable maps and Poisson algebras derived from cluster algebras (2010). In preparation. 22 September, 2010 Kazım Büyükboduk – Koç Üniversitesi Zeta-functions and arithmetic   The purpose of this expository talk is to discuss a fundamental theme in Number Theory: The relation between zeta-functions (objects of analytic nature) and certain objects (which we generally call them Selmer groups) of arithmetic nature. Kummer was first to recognize the arithmetic significance of the special values of the classical zeta-function, using which he was able to deduce an important portion of the "Fermat's Last Theorem". Kummer's ideas were much later generalized by Ribet and Wiles (in a certain sense) to conclude with the full proof. An important portion of this talk will be devoted to explaining Kummer's ideas, and if time permits, say a few words about their influence in modern number theory. 13 October, 2010 Zafer Ercan – Bolu İzzet Baysal Üniversitesi Some recent results on Banach Stone Theorem for Banach lattice valued continuous functions For a topological space X, C(X) denotes the space of real valued continuous functions on X. Banach Stone type theorems deal with relations between C(X) and C(Y) which imply that X and Y are homeomophic. In this talk some old and new results on Banach Stone theorems will be given. In particular vector valued Banach Stone type theorems will be discussed. 20 October, 2010 Christophe Eyral – University of Aarhus A short introduction to Lefschetz theory on the topology of algebraic varieties   In his book 'L’analysis situs et la géométrie algébrique', S.Lefschetz proved two essential results on the topology of algebraic varieties: the Hyperplane Section Theorem and so-called 'Second Lefschetz Theorem'. These results give a comparison between the homology groups of a non-singular irreducible projective variety X in CPn and the homology groups of a generic hyperplane section of X. Precisely, the Hyperplane Section Theorem says that, for a generic hyperplane L, the natural map Hq (L ∩ X) → Hq (X) is an isomorphism for q ≤ dim X − 2 and an epimorphism for q = dim X − 1. The Second Lefschetz Theorem describes the kernel of the map Hdim X−1(L ∩ X) → Hdim X−1(X)  in terms of 'vanishing cycles' that appear in a generic pencil of hyperplanes. In this talk, I will discuss generalization of these theorems to quasi-projective varieties and homotopy groups. 3 November, 2010 Aybike Özer – İstanbul Teknik Üniversitesi Deformations of string and M-theory backgrounds via dualities We start with a brief overview of string and M-theory and the AdS/CFT correspondence for the general audience. Then we explain how some string and M-theory backgrounds, relevant for the AdS/CFT correspondence can be deformed by using a stringy symmetry: T-duality. This talk is based on our joint work with Nihat Sadık Değer, Class.Quant.Grav.26:245015, 2009. 10 November, 2010 ◊ Aslı Pekcan On Darboux integrability of semi-discrete hyperbolic type We study a differential-difference equation of the form t_x(n+1)= f(t(n) , t(n+1),t_x(n)) with unknown t=t(n,x) depending on x and n. The equation is called Darboux integrable if there exist functions F (called an x-integral) and I (called an n-integral), both of a ﬁnite number of variables x, t(n) , t(n+1) , t(n+2) , ... ,t(n-1) , t(n-2),...,t_x(n) , t_xx(n) ,..., such that D_x F=0 and DI=I, where D_x is the operator of total differentiation with respect to x and D is the shift operator: D p(n)=p(n+1). The Darboux integrability property is reformulated in terms of characteristic Lie algebras that give an effective tool for classiﬁcation of integrable equations. 1 December, 2010 Metin Arık – Boğaziçi Üniversitesi Fizik Construction of a Hopf algebra as the quantum invariance group of an algebra Let A be an algebra which is a left module of a Hopf Algebra H such that there exists an algebra homomorphism from A to tensor product of H and A. We define H to be a quantum invariance group of A. A familiar simple case is when A is C^n considered as a free commutative algebra generated by n indeterminates and H is GL(n,C) considered as a commutative Hopf Algebra generated by n^2 indeterminates. The quantum invariance group concept is important from the point of view of quantum physics. We will construct the quantum invariance groups of the d-dimensional fermion algebra and the d-dimensional boson algebra. These examples are important in that they do not contain a deformation parameter. 15 December, 2010 Michèle Vergne – Institut de Mathématiques de Jussieu Atiyah-Bott index theorem and locally polynomial functions Let M be a compact complex manifold with a holomorphic line bundle L. If the circle group S^1 acts as a group of symmetries on M,L, we can associate two invariants to this situation: 1) a locally polynomial measure on the real line (the Duistermaat-Heckman measure); 2) a function on Z associated to the representation of S^1 in the space of holomorphic sections of L. We will discuss these two invariants and relate them. 22 December, 2010 Nick Bezhanishvili – Imperial College London Topological bisimulations Modal logic has numerous applications in computer science and mathematics. Modal language is an expressive and yet decidable formalism for reasoning about relational structures, thus useful for computer science. From the mathematical point of view, algebraic models of modal logic are Boolean algebras with operators which via the Stone and Jonsson-Tarski duality correspond to topological relational structures. Topological relational structures are relational structures equipped with a compact, Hausdorff and zero-dimensional topology. In this talk I will connect these two approaches via the notion of a topological bisimulation. A bisimulation is a relation between two relational structures satisfying some special properties. If there exists a bisimulation between two relational structures, then these structures are indistinguishable in the modal language. The theory of bisimulations of relational structures is by now very well established and forms a core part of the general theory of modal logic. In this talk, I will introduce bisimulations of topological relational structures and discuss their topological, categorical and model-theoretic properties. 29 December, 2010 ◊ Ali Göktürk – Brown University Comparison of Teichmüller geodesics and Weil-Petersson geodesics Let S be a surface with genus g and n punctures and let x(S) = 3g + n denote the complexity of the surface S. In this talk we prove that in the Teichmüller space T(S), Teichmüller geodesics and Weil-Petersson geodesics with the same pair of end points are fellow-travelers with respect to the Weil-Petersson metric if and only if x(S)<6. More precisely, we show that if x(S)<6 then there is a constant N such that for any X,Y in T(S), the Teichmüller geodesic connecting X to Y lies in an N-neighborhood (in the Weil-Petersson metric) of the Weil-Petersson geodesic connecting X to Y. On the other hand we show the opposite of the above statement for surfaces with x(S)>5. 5 January, 2011  ◊ Elçim Elgün Idempotents in the Eberlein compactification of locally compact abelian groups The Eberlein Algebra E of a locally compact group G is defined to be the uniform closure of the Fourier-Stieltjes algebra of G. E(G) satisfies certain invariance properties, hence for any locally compact group G, Eberlein compactification G^E of G can be constructed as the spectrum of E(G). We proved that for any locally compact abelian G, G^E contains at least 2^c idempotents. This is an indication of the complexity of the structure of the semigroup G^E. In this talk, we will study Z^E, the Eberlein compactification of (Z,+). We will construct a family of 2^c many idempotents in Z^E, which gives the exact cardinality of the lattice of idempotents in Z^E.