Combinatorial optimization has recently discovered new usages of probability theory, information theory, algebra, semidefinit programming, etc. This allows addressing the problems arising in new application areas such as the management of very large networks, which require new tools. A new layer of results make use of several classical methods at the same time, in new ways, combined with newly developed arguments.
After a brief panorama of this evolution, I would like to show the new place of the best-known, classical combinatorial optimization tools in this jungle: matroids, matchings, elementary probabilities, polyhedra, linear programming.
More concretely, I try to demonstrate on the example of the Travelling Salesman Problem, how strong meta-methods may predict possibilities, and then be replaced by better suited elementary methods. The pillars of combinatorial optimization such as matroid intersection, matchings, T-joins, graph connectivity, used in parallel with elements of freshmen's probabilities, and linear programming, appropriately merged with newly developed ideas tailored for the problems, may not only replace difficult generic methods, but essentially improve the results. I would like to show how this has happened with various versions of the TSP problem in the past years (see results of Gharan, Saberi, Singh, Mömke and Svensson, and several recent results of Anke van Zuylen, Jens Vygen and the speaker), essentially improving the approximation ratios of algorithms.
Many important question in the theory of surfaces and in algebraic geometry have been solved thanks to explicit constructions of algebraic surfaces as abelian coverings branched over special configurations of lines. After recalling the classical configurations (Pappus, Desargues, Fano, Hesse) I shall describe simple equations for such surfaces, as the Fermat, and Hirzebruch-Kummer coverings. As the configuration of lines becomes special some interesting geometry shows up, as in the case of the six lines of a complete quadrangle, relted to the Del Pezzo surface of degree 5 and its icosahedral symmetry. After mentioning many important such examples and applications, by several authors, I shall concentrate on a recent simple series of such surfaces, studied in my joint work with Ingrid Bauer and Michael Dettweiler, discussing new results and quite general open questions.
Anisotropic diffusion describes random walk with different diffusivities in different directions. The fully anisotropic formulation, sometimes called myopic random walk, is based on active random walk of individuals and it has a Fokker-Planck like of diffusion term. In this talk I give intuitive reasons for anisotropic diffusion and I present scaling limits of kinetic equations which, quite naturally, lead to the fully anisotropic formulation. I show how this framework can be used for the modelling of sea turtle navigation, wolf movement, and brain tumor spread.
The disk embedding problem is of fundamental importance in the study of topology of dimension four. We will discuss backgrounds on its significance and difficulty, including why dimension four is intrinsically different from other dimensions, and then present some recent advances toward the existence and non-existence of embedded disks.
A toric variety, which arose in the field of algebraic geometry, of dimension n is a normal algebraic variety with an algebraic action of a complex torus (ℂ*)n having a dense orbit.
For a given toric variety X, the subset consisting of points with real coordinates of X is called a real toric variety Xℝ. In particular, if Xℝ is compact and smooth, it is called a real toric manifold.
The formula for the integral cohomology ring of toric varieties (and their generalizations) have been well established. Interestingly, the formula is quite simple; according to the formula, the ring is obtained as a quotient of a polynomial ring generated by only degree 2 elements, and it has no torsion.
Nevertheless, only little is known about the topology of real toric manifolds.
The topological structures of real toric manifolds are more complicated than those of toric manifolds.
For instance, every real toric manifold is not a simply connected while every toric manifold is simply connected.
Hence, in general, it is difficult to compute topological invariants of real toric manifolds.
Only the formula of ℤ2-cohomology ring has been established by Davis-Januszkiewicz.
In this talk, we introduce the notion of real toric space as a generalization of a real toric manifold. We provide a formula of the rational cohomology ring of real toric spaces, and discuss the existence of arbitrary torsion in the integral cohomology. Furthermore, we propose several topological classification problems for real toric spaces.
Algebraic varieties over finite fields have their associated zeta functions. André Weil conjectured that these functions have a list of properties, including an analogue of the Riemann hypothesis, and these Weil conjectures were proved by Pierre Deligne in the 1970s. Deligne used the so-called l-adic étale cohomology theory, but it is told as a folklore that Alexander Grothendieck was not fully satisfied by this Fields Prize winning work of Deligne for not having proven the conjectures using algebraic cycles.
In this talk, I will first roughly sketch the above historical background, and then talk about how one could revisit the Weil conjectures through algebraic cycles, via 40 years' modern mathematical developments from the late 1970s to now, spanning from higher algebraic K-theory, crystalline cohomology, motivic cohomology, intersection theory, triangulated categories of motives, by Daniel Quillen, Pierre Berthelot, Spencer Bloch, Luc Illusie, Vladimir Voevodsky, Kiran Kedlaya, Hélène Esnault, etc. The main theorem is my joint work with Amalendu Krishna of the Tata Institute of Fundamental Research.