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Title:  Quasi-optimality in the backward Euler-Galerkin method for linear parabolic problems

Abstract:  We analyse the backward Euler-Galerkin method for linear parabolic problems, looking for quasi-optimality results in the sense of Céa's > Lemma. We study first the spatial discretization, proving that the H1-stability of the L2- projection is also a necessary condition for quasi-optimality. Regarding the discretization in time with backward Euler, we prove that the error is equivalent to the sum of the best errors with piecewise constants for the exact solution and its time derivative. Concerning the case when the spatial discretization is allowed to change with time, we bound  the error with the best error and an additional term, which vanishes if there are not modifications of the spatial dicretization and it is consistent with the example of non convergence in Dupont '82. We combine these elements in an analysis of the backward Euler-Galerkin method and derive error estimates in case the spatial discretization is based on finite elements.

All are welcome!

Title:  Algebraic K-theory and crossed objects

Abstract:  After reviewing the classical lower K-groups, Milnor's K_2, and Quillen's plus construction (stopping for examples along the way), we will look at definitions of crossed modules and crossed complexes. After showing that certain K-groups can be regarded as these crossed objects, we will see how this might give insight into explicit descriptions of the plus construction in terms of generators and relations of the Steinberg group.

Title:  Terraces, Latin squares, and the Oberwolfach problem

Abstract:  A terrace is an arrangement of the elements of a finite group in which differences between adjacent elements adhere to certain restrictions. We introduce terraces and a number of related objects, including R-terraces and directed terraces, and discuss conjectures concerning the groups for which we can construct terraces. We also consider applications of terraces to problems in the areas of combinatorial design and graph theory - namely, the construction of row-complete Latin squares and solutions to some particular cases of the Oberwolfach problem.

Title:  Homogeneous Gorenstein Ideals and Boij Söderberg Decompositions

Abstract:  This talk consists of two parts. Part one revolves around a construction for homogeneous Gorenstein ideals and properties of these ideals. Part two focuses on the behavior of the Boij-Söderberg decomposition of lex ideals. Gorenstein ideals are known for their nice duality properties. For codimension two and three, the structures of Gorenstein ideals have been established by Hilbert-Burch and Buchsbaum-Eisenbud, respectively. However, although some important results have been found about Gorenstein ideals of higher codimension,  there is no structure theorem proven for higher codimension cases. Kustin and Miller showed how to construct a Gorenstein ideals in local Gorenstein rings starting from smaller such ideals. We discuss a modification of their construction in the case of graded rings. In a Noetherian ring, for a given two homogeneous Gorenstein ideals, we construct another homogeneous Gorenstein ideal and so we describe the resulting ideal in terms of the initial homogeneous Gorenstein ideals. Gorenstein liaison theory plays a central role in this construction. For the second part, we talk about Boij-Söderberg theory which is a very recent theory. It arose from two conjectures given by Boij and Söderberg and their proof by Eisenbud and Schreyer.

It establishes a unique decomposition for Betti diagram of graded modules over polynomial rings. We focus on Betti diagrams of lex ideals which are the ideals having the largest Betti numbers among the ideals with the same Hilbert function. We describe Boij-Söderberg decomposition of a lex ideal in terms of Boij-Söderberg decomposition of some related lex ideals.