Department of Mathematics

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Mathematics Colloquium 2013

 

11 December, 2013          

Raytcho LazarovTexas A & M University    

Galerkin FEM for time-dependent inhomogeneous fractional diffusion equations

We consider an initial boundary value problem for the inhomogeneous time-dependent fractional diffusion equations with homogeneous Dirichlet boundary condition in a bounded convex polyhedral domain. We introduce and study the semi-discrete approximations based on the standard Galerkin and lumped mass finite element methods. We derive almost optimal estimate when the right hand side of the equationf(x,t) is in the space L((0,T), Hs), -1 < s ≤  1. For lumped mass method, the optimal L2-norm error estimate is established for meshes that have a symmetry property. Finally, we present numerical experiments that confirm the theoretical results. (This is based on my joint research with Bangti Jin, Joseph Pasciak, and Zhi Zhou, all from Texas A&M University) .

 


4 December, 2013 

 

Özgün ÜnlüBilkent University    

Equivariant Homotopt Diagrams

In this talk we will discuss constructing topological spaces by gluing basic building blocks to each other. In a joint work with Aslı Güçlükan ˙Ilhan, we explain how such gluing data can be obtained from equivariant homotopy diagrams when certain obstructions vanish. Then we explain how the combinatorial nature of these constructions, determined by the gluing data, can be exploited to understand the homotopy types of these topological spaces. Given a finite group G, the main goal will be to construct finite G-CW-complexes homotopy equivalent to a product of spheres with isotropy groups in a given family of subgroups of G.

 


27 November, 2013 

Evgenii BashkirovFatih University    

On subgroups of the group GL7 over a field that contain a Chevalley group of type G2 over a subfield

Let k be a field of characteristic  2 ,3 and let K be an algebraic field extension of k. In this talk, we describe subgroups of the general linear group GL7(K) that contain the 7-dimensional representation G2(k) of the Chevalley group of type G2 over k provided k contains more than 11 elements.

 


20 November, 2013 

Ehud HrushovskiThe Hebrew University of Jerusalem    

Between Model Theory and Combinatorics

I’ll discuss some connections and parallels between these two subjects.

 


 30 October, 2013 

Peter J. SternbergIndiana University    

Rotating Vortex Solutions to Gross-Pitaevskii on S2

After introducing the Ginzburg-Landau energy and some of its main features, I will discuss the conservative flow associated with it, namely the Gross-Pitaevskii system (a version of nonlinear Schrodinger). The focus is the construction of rotating periodic solutions that possess vortices (zeros of the complex-valued solution that carry non-zero degree). This is achieved using techniques from the calculus of variations. The work is joint with Michael Gelantalis.

 


 23 October, 2013 

Emre MengiKoç University    

Numerical Optimization of Eigenvalues of Hermitian Matrix-Valued Functions

Optimizing a prescribed eigenvalue of a differential operator or a matrix-valued function depending on parameters has a long history dating back to Euler. The prescribed eigenvalue of interest typically exhibits not only Lipschitz continuity, but also remarkable analyticity properties. We present a numerical approach, based on these analytical properties, for unconstrained optimization of a prescribed eigenvalue of a Hermitian matrix-valued function depending on a few parameters analytically. In the second part, we present a numerical approach for optimization problems with linear objectives and an eigenvalue constraint on the matrix-valued function. The common theme for numerical approaches is the use of global under-estimators for eigenvalue functions, that are built on the variational properties of eigenvalues over the space of Hermitian matrices.

 


 2 October, 2013 

Garth DalesLancaster University (United Kingdom)   

Finitely-generated maximal left ideals in Banach algebras

Let A be a unital Banach algebra. It is well-known that, under the hypothesis that all closed left ideals in A are algebraically finitely-generated as left ideals, necessarily A is a finitedimensional algebra. We investigate whether the same conclusion holds under the weaker assumption that all maximal left ideals of the unital Banach algebra A are algebraically finitely-generated. We prove that this is true whenever A is a commutative unital Banach algebra and whenever A is a unital C-algebra. We then try to prove the conjecture for every Banach algebra of the form B(E); this is the Banach algebra of all bounded linear operators on a Banach space E. We succeed for many, but not all, Banach spaces E. We have to look at some recently developed “exotic” Banach spaces E.

 


3 July, 2013 

Raffaella Servadei, University of Calabria (Italy)  

Elliptic nonlocal fractional equations

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  12 June, 2013 

Çağrı KarakurtUniversity of Texas at Austin  

Rational Points on Modular Curves and Obstruction Problems

Studying rational points on an algebraic curve is one of the main problems in number theory. In this talk, I will start with defining elliptic and modular curves and very briefly mention the role they played in the proof of Fermat’s last theorem. Then I will present my results about points on certain twists of the classical modular curve. Many of these twisted curves violate the Hasse principle. In the cases that the genus of the curve is bigger than one, some of these violations are explained by the Mordell-Weil sieve method which will be defined. If time permits, I will mention genus one cases. In these cases it is possible to study the Hasse principle violations via local-global trace obstructions.

 


  12 June, 2013 

Ekin ÖzmanUniversity of Texas at Austin   

Exotic smooth structures and corks of 4-manifolds

A differentiable manifold is said to admit an exotic smooth structure if there is another manifold which is homeomorphic but not diffeomorphic to it. Understanding this phenomenon in dimension 4 is subtler than the other dimensions, since it requires tools from geometric analysis. Corks are certain submanifolds that show wonderful topological properties. In this talk I’ll outline a program of constructing exotic smooth structures on 4-manifolds using their corks by incorporating Heegaard-Floer homology and symplectic/contact geometry.

 


  15 May, 2013 

Faruk TemurUniversity of Illinois at Urbana-Champaign   

Restriction Estimates for the Fourier Transform

For a function f in L1(Rn), the Fourier transform f^ is continuous, and thus can be meaningfully restricted to measure zero subsets of RnBut if f in L(Rn), the Fourier transform f^ can be an arbitrary L(Rn) function, hence this type of restriction is not possible. The restriction problem asks what happens for functions in Lp (Rn) for some 1 < p < 2. Ever since E. Steindiscovered that the answer was affirmative for specific cases, this problem has been subject of intense investigation. In this talk we will describe the problem and its variants, point out its connections to other areas of analysis, and review recent developments.

 


  19 April, 2013 

Enver ÖzdemirNanyang Technological University (Singapore)   

Detecting prime numbers?

The main ingredient of the most crypto systems is prime numbers. For a given large size number n, the Rabin-Miller test can determine if the number is composite or not for most of the time. However, if the test fails to say n is composite, it might not be safe to consider n a prime number. In order to determine if n is really a prime number, there are mainly two algorithms being used in practice but one of them is not deterministic and the other one is not practical all the time. The method that I discuss in this talk is similar to the Miller-Rabin test in some sense. Basically, it provides a powerful way to determine a given integer n is composite or not. Therefore, it might be considered a compositeness test. On the other hand, with the reasons I will explain, the test is also capable of determining if n is prime or not in an efficient way.

 


  17 April, 2013 

Zubeyir Cinkir, Zirve University  

Metrized Graphs for Arithmetic Geometry?

In the first part of the talk, we aim to give a panoramic view of the role of metrized graphs in arithmetic geometry. In the second part, we talk about the research problems linked to the metrized graphs, and about some of the progress we made on them.

 


  3 April, 2013 

Ludomir Newelski, The Mathematics Institute of University of Wroclaw   

Topological Methods in Model Theory?

Topological methods have been applied in model theory from its beginnings. With time they become more and more sophisticated. In the talk I will survey some of these methods, including the Cantor-Bendixson rank in Morley Categoricity theorem, meager sets in meager forking and, finally, topological dynamics in model theory of groups.

 


  27 March, 2013 

Mohan Ravichandran, Istanbul Bilgi University   

Diagonals of operators and convexity in operator algebras

The relationship between a self-adjoint matrix and its diagonal is completely described by the classical Schur-Horn theorem. Recently, I showed that an analogous theorem holds for selfadjoint operators in von Neumann algebras. Such theorems are important tools in understanding convexity in operator algebras and turn out to have interesting applications, some of which I will describe. A natural next step would be to seek multivariable generalizations but as William Arveson discovered a few years ago, the most pleasing aspect of the one variable case - the presence of convexity, fails to hold in the multivariable case. A careful analysis of this problem throws up some interesting operator algebraic, combinatorial and geometric questions.