Teaching Statement and Statement of Research Plans - GWDG

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2 Index Teaching Statement: I enjoy teaching mathematics and gain great satisfaction from seeing my students learning different ways of thinking about mathematical notions and problems. My prefered method of teaching is to allow students the time so they can attain the fundamental understanding of a particular subject. This process of learning is completed by stressing the essential elements of mathematics. To understand Mathematics at the university level, one must develop the ability to think abstractly and also geometrically. It is also my belief that my students should recognize abstract algebraic notions and geomemetrical objects. The teaching philosophy that I adhere to is based on the theory, of how students want to learn mathematics. Students develop their mathematical dexterity best by building relationships between new definitions and their capacity of abstract understanding and geometrical imagination. Students must be active participants in the learning process by trying to design nontrivial questions. Further, they must be prepared to apply their knowledge in order to solve mathematical problems. I try to give in my lectures many examples and analogies that make the often abstract concepts more clear. The most important aspect of my teaching is to provide the core abilities mixed with new concepts that allow student to develop their problem solving skills. I belive that students can be provided with a deeper understanding of mathematics, in particular under graduate level of mathematics, if they know applications of certain contexts in the real world, such as computer science, communication networks or even in physics. Teaching Interest: -Calculus -Number theory (elementary course) -Galois theory and Finite fields -The first and second course in Algebra -The first and second course in Analysis -Differential Geometry (elementary course) -Complex functions and Analysis -General Topology -Discrete Mathematics -The first and second course in Graph theory -Seminars for graduate students
Index 3 Introduction In this short article, I describe briefly the projects that I am currently working on, and the projects that I’d like to work on in the near future. My primary research interest is in automorphic representations and arithmetic number theory. More specifically, I would like to use Bruhat-Tits Buildings based on non-archimedian local fields and associated automorphic forms and representations to solve certain problems in Number Theory, Combinatorics, and Computer Science. Most of my research is centered around Ramanujan (graphs) hypergraphs and associated Zeta and (L)-functions. Before giving a description of projects, I give an informal overview of my construction of Ramanujan hypergraphs. Ramanujan hypergraphs In my thesis (see : http://www.uni-math.gwdg.de/asarveni/index1.html or enclosed paper) I studied Ramanujan regular hypergraphs, generalizing the definition of Ramanujan graphs introduced by A.Lubotzky, R.Philips, P. Sarnak [13], [14]. There are closely related papers by A.Lubotzky B.Samuels and U.Vishne [15] ,and a beautiful simple construction by W. Lie [12]. The main difference between these and my construction is that I am using a very special kind of algebra, namely the skew polynomial ring F q d{τ} and its quotient field F q d(τ), as the main tools throughout my thesis to arrive at an explicit construction of Ramanujan hypergraphs. These skew polynomial rings also are very useful in other branches of Mathematics, such as Drinfeld Modules, or Invariant theory. Though the explicit construction is beautiful, it is not possible to understand this construction without technical arguments and a deep knowledge of the geometric nature of Buildings of type Ãd−1. I would like to illustrate the rich connections of the notion of Ramanujan hypergraph with several parts of mathematics, for example with the theory of Bruhat -Tits Buildings, with representation theory, with combinatorics, and with ‘non-commutative algebraic-geometry’ as the natural search for zeros of skew polynomials over finite fields. This viewpoint shows that my construction is more practical for applications to Computer Science and is the most natural extension of the notion of Ramanujan graph. Apart from the use of the arithmetic skew polynomial ring as specified above, the other important difference is in making use of Lafforgue’s and Arthur-Clozel’s results in the representation theory of automorphic forms. Below we quickly review my explicit construction (for proofs and further details see [21]). The explicit construction shows that the Cayley graph of a certain complicated group
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Teaching Statement and Statement of Research Plans - GWDG

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