The unreasonable power of the lifting property in elementary mathematics
This volume contains the abstracts of the talks given at the 2015 European Summer Meeting of the Association for Symbolic Logic—Logic Colloquium ’15—taking place on August 3 - 8, 2015, in Helsinki, Finland. The Colloquium is co-located with the 15th International Congress of Logic, Methodology and Philosophy of Science, CLMPS, and the SLS Summer School in Logic. As usual, the talks at the Logic Colloquium consist of invited plenary lectures, invited talks in the various special sessions, tutorials, and, last but not least, contributed papers. Abstracts of all of these presentations are given here. The book of abstracts is an important part of the Logic Colloquium, since it not only gives the authors the opportunity to present their topic and main results, but also helps the Colloquium participants choose which of the many parallel sessions to attend at any given time. I would like to thank my colleagues in the Program Committee for a very constructive and pleasant period of collaborative effort, which has resulted in, we think, an excellent program of invited speakers, representing current frontline research in the main areas of logic. We are also particularly happy that there are so many contributed papers at this year’s Logic Colloquium. Finally, I want to thank all the members of the Organizing Committee, and in particular its chair Jouko Väänänen, for their expert and smooth organization of this important event.
A perpetual change of foundations observed in the real history of the discipline is not a historical accident but an essential feature of foundations. I distinguish between the progress of mathematics and renewal of its foundations and show how the latter contributes to the former with some historical examples. I also describe a mechanism of renewal of foundations, which has to do with needs of mathematics education, and provide an account of robustness of mathematical facts and arguments surviving through the change of their foundations. I outline my vision of today's situation and argue for the renewal of standard structuralist Bourbaki-style set-theoretic foundations in favor of new Category-theoretic foundations, which are linked to Structuralism historically and dialectically but imply a very different philosophical view on mathematics.
Предсказан новый тип безмассовых дираковских фермионов в кристаллических 3D топологических изоляторах(ситуация 3D->2D). Спектр имеет четырехкратное вырождение в вершине 2D-зоны Бриллюэна (точка M) и двукратное - в окрестности точки M. В 3D топологических изоляторах кристаллическая симметрия совместно с инвариантностью к инверсии времени допускает четырехкратно вырожденные дираковские конусы, которые отсутствуют в классификации топологических особенностей в работе R.-J.Slager, A.Mesaros, V.Juriycic, J.Zaanen, Nature Phys. 9, 98 (2013). Гамильтониан последней не содержит в себе дираковских особенностей с более чем двукратным вырождением. Поэтому ее топологическая классификация является неполной. Продольное магнитное поле в безспиновом случае сохраняет безмассовый закон дисперсии фермионов и не снимает четырехкратного вырождения. В спинорном случае магнитное поле снимает четырехкратное вырождение, оставляя только двукратное, и приводит к появлению щели в спектре фермионов.
The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal’s Γ-spaces. The formalism of topological operads generalises well to different categories yielding such notions as (Formula presented.)-algebras in chain complexes, while the Γ-space approach faces difficulties. In this paper we discuss how, by attempting to extend the Segal approach to arbitrary categoires, one arrives to the problem of understanding “weak” sections of a homotopical Grothendieck fibration. We propose a model for such sections, called derived sections, and study the behaviour of homotopical categories of derived sections under the base change functors. The technology developed for the base-change situation is then applied to a specific class of “resolution” base functors, which are inspired by cellular decompositions of classifying spaces. For resolutions, we prove that the inverse image functor on derived sections is homotopically full and faithful.
The book contains the reports of the member of the congress from the different countres. They consider the idea of the symmety in the science and in the art.
The paper examines the significance of Platonic Solids in different parts of modern science.