Department of Mathematics

Mathematics Colloquium 2015




30 December, 2015

E. Mehmet Kiral, Texas A&M University    

The Voronoi formula and double Dirichlet series 

Abstract: A Voronoi formula is an identity where on one side, there is a weighted sum of Fourier coefficients of an automorphic form twisted by additive characters, and on the other side one has a dual sum where the twist is perhaps by more complicated exponential sums. It is a very versatile tool in analytic studies of L-functions. In joint work with Fan Zhou we come up with a proof of the identity for L-functions of degree N. The proof involves an identity of a double Dirichlet series which in turn yields the desired equality for a single Dirichlet coefficient. The proof is robust and applies to L-functions which are not yet proven to come from automorphic forms, such as Rankin-Selberg L-functions. The first two thirds of my talk should be accessible to a general audience.


23 December, 2015

Hatice Boylan, Istanbul University    

2*2 matrices of determinant 1 and its finite dimensional representations 

Abstract: Already the group SL_2(Z) of 2*2 matrices with integral entries and determinant 1 provides beautiful and interesting mathematics and still deep open problems. We shall give an overview over various aspects of this group, indicate its connections to geometry and number theory, and shall finally arrive at the theory of its finite dimensional representations. In the third part of my talk I shall discuss extensions of the theory to Hilbert modular groups, i.e. groups of 2*2 matrices and determinant 1 with entries in the ring of integers of a given number field. In particular I shall describe a few results of my own work in this domain. The first two thirds of my talk should be accessible to a general audience.


16 December, 2015

Ergun Yalcin, Bilkent University    

Finite group actions on spheres 

Abstract: Studying symmetries of topological spaces is a classical topic in algebraic topology. Finite groups which can act freely on a sphere are completely classified but there are many unanswered questions related to free actions on products of spheres or about actions on spheres with nontrivial isotropy subgroups. In this talk I will give a survey of well-known results in this area and present some new results from our joint work with Ian Hambleton related to rank two finite group actions on homotopy spheres with rank one isotropy.


9 December, 2015

Ugur Madran, Izmir University of Economics    

Recent developments in invariant theory 



25 November, 2015

Tolga Etgü, Koc University    

Simply laced Dynkin diagrams 

Abstract: I will talk about a truly exceptional class of trees which shows up in various areas of mathematics.


18 November, 2015

Laurence J. Barker, Bilkent University    

Algebras associated with Green functors, and algebras that ought to be associated with Green functors 

Abstract: In the representation theory of finite groups, the following situation occurs frequently: for each finite group G, there is a ring A(G) and, for each subgroup H of G, there are induction and restriction maps between A(H) and A(G). Examples of such rings A(G) are Burnside rings, character rings, trivial source rings. James Green, in 1971, initiated an axiomatic approach to describing many features common to such situations. Subsequent developments have often ignored the ring structures. However, Serge Bouc, in 2010, described a rich algebraic scenario, called the theory of Green biset functors, which does involve the ring structures. In fact, in his scenario, there are three multiplicative operations. We shall discuss a generalization that accommodates cases previously excluded, such as cohomology rings and modular character rings. Nevertheless, some similar situations of interest, such as the theory of the linear category associated with the bifree trivial source bimodules, still evade capture. This talk is partially supported by Istanbul Center for Mathematical Sciences (IMBM).


11 November, 2015

Kivanc Ersoy, Mimar Sinan Fine Arts University    

Finite groups with splitting automorphisms 

Abstract: Thompson proved that a finite group with a fixed point free automorphism of prime order is nilpotent. Kegel proved the same result for splitting automorphisms of prime order. Problems about splitting automorphisms of finite solvable groups are studied by Jabara, Kurzweil, Meixner and Khukhro. In this talk we will prove the following results:

Theorem: (E.) A finite group with a splitting automorphism of odd order is solvable.

Theorem: (E. -Gupta) For any natural number n, a finite group with a splitting automorphism of order 2^n is solvable.


4 November, 2015

Matthew Gelvin, Bilkent University    

Minimal characteristic bisets of fusion systems 

Abstract: Fusion theory is the study of the structure of finite groups from the point of view of a fixed prime number. It connects the worlds of finite groups, algebraic topology, and modular representation theory. In this talk, we will be introduced to the notion of a fusion system–an organizing frame-work for a group’s p-local data and a sort of algebraic object in its own right. We will also discuss the Martino-Priddy conjecture, one of the highlights of the field, which serves as a bridge between algebra and topology. In the course of outlining the proof of this conjecture, we will see the need for an object that contains the same data as the fusion system but is more structured in a way that allows us to mimic fundamental group-theoretic constructions in an "elementwise" fashion. One such structure is the minimal characteristic biset of the fusion system, whose characterization and basic properties are the result of joint work with Sune Reeh. We will conclude with a survey of these results and consider some open questions they raise.


21 October, 2015

Baris Coskunuzer, Koc University    

Geometrik topoloji ve bir milyon dolar 

Ozet: Bu konusmada, geometrik topoloji alaninin temel yapi ve sorularini tanitip, ardindan Perelman’in cozdugu bir milyon dolar odullu Poincare Sanisi’ndan bahsedecegiz.


14 October, 2015

Ipek Tuvay, Mimar Sinan Fine Arts University    

The effect of Clifford theory on multiplicity modules 

Abstract: Clifford theory is a technique which reduces blocks of a finite group to blocks of central extensions of its quotient subgroups. The first step in this technique is reduction to the inertial subgroup. In this talk, I will describe the effect of this step on multiplicity modules and Külshammer-Puig classes. The Puig category, which carries more information than the fusion system of a block, is the main tool that is used to deliver these results. This is a joint work with Laurence Barker.


7 October, 2015

Alexandre V. Borovik, University of Manchester    

One-way algebra 

Abstract: I will give a survey of recent results in probabilistic methods of computational algebra (and, more, specifically, group theory) concerned with the categories of explicitly defined finite algebraic structures (groups, rings, fields, projective spaces, etc.) and efficiently computable homomorphisms between them. The motivation for some results comes from cryptography, for others from needs of practical computations. This is an intriguing world where most natural morphisms are likely to be non-invertible, and where we do not know whether finite fields of prime order are unique up to (efficiently computable both ways!) isomorphisms. However, it is a surprisingly rich and beautiful theory with unexpected links with some classical areas of mathematics. (Joint work with ¸Sükrü Yalçınkaya)


10 June, 2015

Kellogg Stelle, Imperial College    

Spherically symmetric solutions in higher derivative gravity 

Abstract: Quantum corrections to gravity, either to general relativity, or to its field theory extensions such as supergravity, or to string theory, all produce quadratic curvature terms in the effective action. The seminar will present what is now known about the effect such terms have on the most basic family of gravitational solutions, i.e. spherically symmetric solutions. This reveals a rich set of new solution types, including non-Schwarzschild black holes, wormholes and solutions without horizons.


13 May, 2015

Burak Ozbagci, Koc University    

Open books and contact structures 

Abstract: I will explain the correspondence between open books and contact structures and give examples of contact open books with 4-dimensional exotic pages I recently constructed in a joint work with Otto van Koert.


6 May, 2015

Christophe Ritzenthaler, University Rennes 1    

Plane quartics 

Abstract: We will explore several aspects of plane quartics (smooth or not): geometry, arithmetic and applications to cryptography.


29 April, 2015

Zafeirakis Zafeirakopoulos, National and Kapodistrian University of Athens    

Polyhedral Omega: A linear Diophantine system solver 

Abstract: Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decomposition and Barvinok’s short rational function representations. In this way, we connect two branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega by far the simplest algorithm for solving linear Diophantine systems available to date. This is joint work with Felix Breuer.


8 April, 2015

George Moutsopoulos, Boğazici University    

Non-geometric backgrounds of string theory 

Abstract: Due to the hidden symmetries T-duality and U-duality, string theory admits backgrounds that are locally smooth but globally cannot be described by a manifold. I will describe how they arise and their description in supergravity.


1 April, 2015

Hakan Gunturkun, Gediz University    

Linear tropicalizations 

Abstract: Tropical geometry and Berkovich spaces have been used to tackle many problems in classical algebraic geometry. The relation between them has important applications in both fields. In this talk I will give gentle introductions to both concepts. I will also give some details about higher dimensional tropical geometry and show a natural relation between tropicalizations of a variety and analytification of it in the sense of Berkovich. Then I will introduce linear tropicalizations that gives a similar relation and has additional properties. I will talk about some applications of it to complex line arrangements and give a solution to transverse intersection problem in tropical geometry. This is a joint work with Özgür Kisisel.


25 March, 2015

Henning Stichtenoth, Sabanci University    

Asymptotic bounds for codes 

Abstract: Error-correcting codes are a mathematical tool for detecting and correcting errors that occur in information transmission. The basic parameters of a code are its information rate and its relative minimum distance. These parameters describe the error-correcting capability of the code, and they satisfy some more or less obvious bounds. In this talk I will discuss old and new results on such bounds. A basic tool will be algebraic curves over finite fields having many rational points. Following an idea of Goppa, such curves can be used for the construction of powerful codes.


4 March, 2015

Richard Gonzales, Heinrich-Heine-Universitat Dusseldorf    

Understanding algebraic varieties through group actions 



25 February, 2015

Peter Paule, Research Institute for Symbolic Computation, Johannes Kepler University Linz    

The concrete tetrahedron in algorithmic combinatorics 

Abstract: Donald Knuth introduced a course “Concrete Mathematics” that has been taught annually at Stanford University since 1970. The course, and the accompanying book coauthored with Ron Graham and Oren Patashnik, was originally intended as an antidote to “Abstract Mathematics.” In the 1990s, Doron Zeilberger’s “holonomic systems approach to special functions identities” inspired a further wave in this “concrete evolution”: the development of computer algebra methods for symbolic summation, generating functions, recurrences, and asymptotic estimates. The book “The Concrete Tetrahedron,” by Manuel Kauers and the speaker, describes basic elements of this tool-box and can be viewed as an algorithmic supplement to “Concrete Mathematics” by Graham, Knuth, and Patashnik. The talk introduces to some of these methods. A major application concerns the computer-assisted evaluation of relativistic Coulomb integrals, joint work with Christoph Koutschan and Sergei Suslov.


11 February, 2015

Kagan Kursungoz, Sabanci University    

Radial limits of partial theta and similar series 

Abstract: Consider unilateral series in a single variable where the exponent is a polynomial, and the coefficients are periodic. Quadratic polynomials correspond to partial theta series. Such series converge inside the unit disk. We compute limits and asymptotic expansions of those series as the variable tends radially to a root of unity. The method is suitable for automation and part of the computations is based on the idea of the q-integral.


14 January, 2015

Olgur Celikbas, University of Connecticut    

Vanishing of Tor, and depth properties of tensor products of modules