TU Berlin, Germany
Optimization, Complexity and Invariant Theory
Invariant and representation theory studies symmetries by means of group actions and is a well established source of unifying principles in mathematics and physics. Recent research suggests its relevance for complexity and optimization through quantitative and algorithmic questions. The goal of the talk is to give an introduction to new algorithmic and analysis techniques that extendconvex optimization from the classical Euclidean setting to a general geodesic setting. We also pointout surprising connections to a diverse set of problems in different areas of mathematics, statistics, computer science, and physics.
Patrice Ossona de Mendez
CAMS, Paris, France
First-order transductions of graphs
Logical methods in Computer Science have a long history, as witnessed e.g. by the relative longevity of SQL in relational database management. More recently, Courcelle’s theorem, which combines second-order logic and tree decompositions of graphs, showed that a many NP-complete algorithmic problems in graph theory can be solved in polynomial time on graphs with bounded tree-width (and even on graphs with bounded clique-width). At the heart of the latter result is the notion of monadic second-order transductions, which are a way to encode a graph within a structure using coloring and monadic second-order logic formulas. In this presentation we consider first-order transductions, for which the formulas have to be first-order formulas. As a counterpart for this strong restriction, many algorithmic problems become fixed parameter tractable when restricted to nowhere dense classes, which include classes excluding a topological minor thus, in particular, classes of planar graphs and classes of graphs with bounded degrees.
In this setting, the main challenge is to extend results obtained in the sparse setting (for bounded expansion classes and nowhere dense classes) to the dense setting, in a similar way the results about monadic second-order model checking have been extended from classes with bounded tree-width to classes with bounded clique-width.
On the fluted fragment
The fluted fragmentis a recently rediscovered decidable fragment of first-order logic whose history is dating back to Quine and the sixties of the 20th century. The fragment is defined by fixing simultaneously the order in which variables occur in atomic formulas and the order of quantification of variables; no further restrictions concerning e.g. the number of variables, guardedness or usage of negation apply. In the talk we review some motivation and the history of the fragment, discuss the differences between the fluted fragment and other decidable fragments of first-order logic, present its basic model theoretic and algorithmic properties, and discuss recent work concerning limits of decidability of its extensions.