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Meeting ID: 258 676 9419

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Linear Diophantine Systems, Integer Partitions, Polyhedra and  Applications


Speaker: Zafeirakis Zafeirakopoulos, Gebze Technical University


Abstract: Polyhedral Omega is an algorithm for solving linear Diophantine systems, 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.  It 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 decompositions and Barvinok’s short rational function representations. This synthesis of ideas makes Polyhedral Omega by far the simplest algorithm for solving linear Diophantine systems available to date.


After presenting the algorithm, we will see some applications in number theory and computer science. Finally, we will discuss how to generalize Polyhedral Omega, in order to solve families of problems parametrized by some integer parameter.


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Deformation theory of the Lafforgue variety


Speaker: Kosta Psaromiligkos, Université Clermont Auvergne


Abstract: In this series of talks we will construct the Lafforgue variety, an affine scheme equipped with an open dense subscheme parametrizing the simple modules of a non-commutative algebra that is a finite module over a finitely generated center. Our main applications and source of examples will be in the theory of Hecke algebras. We will also study how the Lafforgue variety varies under deformation of algebras, and in particular we prove in the case the center is regular a conjecture stated by Aubert, Baum and Plymen in 2007 on the reducibility loci of affine Hecke algebras.


In the first talk, we will introduce Hecke algebras and the type of questions we will consider, as well as relevant algebraic geometric notions for the second talk. In the second talk, we will construct the Lafforgue variety and study its deformation theory (the latter is work in progress).


Program of talks:


Stable and singular equivalences of finite dimensional algebras


Speaker: Georgios Dalezios, Aristotle University of Thessaloniki


Abstract: The plan is to give two talks on homological methods in the representation theory of finite dimensional associative algebras, culminating in recent new results on stable and singular equivalences. The first talk will be introductory, offering a crash course on concepts such as quiver representations and Morita equivalences (classical and derived). In the second talk, we will go beyond the class of derived equivalences and discuss singularity categories and singular equivalences, in relation to Gorenstein rings and Cohen-Macaulay modules. Effort will be put in making things comprehensive to a broad algebraic audience.


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Upcoming Series of Talks:


Polynomial invariants of finite group schemes 

Speaker: Kostas Karagiannis, University of Manchester

Abstract: It was previously shown by Peter Symonds that when a finite group G acts on a polynomial ring S over a field of prime characteristic, then only finitely many isomorphism classes of indecomposable G-modules occur as summands of S, and that the Castelnuovo-Mumford regularity of the invariant subring S^G is at most zero.

The main purpose of this series of talks will be to present joint work with Peter Symonds that generalizes the above results when one replaces the acting object by a finite group scheme. Considerable effort will be put into introducing the audience to the concepts and tools necessary to make sense of the problems and present the arguments of the relevant proofs. On the side of the acting object, this includes a crash course on finite group schemes over fields and their representations; on the side of the object acted upon the focus will be on the  complex and local cohomology. In any case, every effort will be made to restrict the prerequisites to basic familiarity with representation theory of finite groups, affine algebraic geometry and homological methods in commutative algebra. 

Program of talks:


Previous Talks:

For talks of previous years see here.