A term coming from nowhere - The Dirichlet problem on perforated domains
Tuesday, 22.10.19, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Titel folgt
Thursday, 24.10.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Titel folgt
Thursday, 24.10.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
Tree-Forcing Notions
Thursday, 24.10.19, 17:00-18:00, Hörsaal II, Albertstr. 23b
During the 1960s Cohen and Solovay introduced and developed the method of\nforcing, which soon became a key technique for building various models of\nset theory. In particular such a method was crucial for answering questions\nconcerning the use of the axiom of choice to construct non-regular objects\n(such as non-Lebesgue measurable sets, non-Baire sets, ultrafilters) and to\nanalyse possible sizes of several types of subsets of reals (such as\ndominating and unbounded families, and other so-called cardinal\ncharacteristics).\nOne of the key ideas in both cases is the notion of a tree-forcing, i.e.\na partial order consisting of a specific kind of perfect trees. In this\ntalk, after a brief historical background, we will focus on some results\non Silver, Miller and Mathias trees. We will also see applications of\ninfinitary combinatorics and tree-forcing in the context of\ngeneralized descriptive set theory and the study of social welfare\nrelations\non infinite utility streams.\n
The construction problem for Hodge numbers
Friday, 25.10.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
To a smooth complex projective variety, one often associates its Hodge diamond, which consists of all Hodge numbers and thus collects important numerical invariants. One might ask which Hodge diamonds are possible in a given dimension.\n\nA complete classification of the possible Hodge diamond seems to be out of reach, since unexpected inequalities between the Hodge numbers occur\nin some cases. However, I will explain in this talk that the above construction problem is completely solvable if we consider the Hodge numbers modulo an arbitrary integer. One consequence of this result is that every polynomial relation between the Hodge numbers in a given dimension is induced by the Hodge symmetries. This is joint work with Stefan Schreieder.
Lifting BPS States on K3 and Mathieu Moonshine
Monday, 28.10.19, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
The elliptic genus of K3 is an index for the 1/4-BPS states of its sigma-model. At the torus orbifold point there is an accidental degeneracy of such states. We blow up the orbifold fixed points and show that this fully lifts the accidental degeneracy of the 1/4-BPS states with dimension h=1. Thus, at a generic point near the Kummer surface the elliptic genus measures not just their index, but counts the actual number of these BPS states. Finally, we comment on the implication of this for symmetry surfing and Mathieu moonshine.
Flutters and Chameleons
Wednesday, 30.10.19, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Let K be the collection of infinite sets of natural numbers. A colouring c\nof K with a finite number of colours is Ramsey if for some infinite A in K and every infinite subset B of A, c(B) = c(A).\nA non-Ramsey colouring is one for which no such A exists.\nSolovay in a famous paper published in 1970 used a strongly inaccessible cardinal to construct a model of ZF + DC in which various principles hold which contradict AC:\n\n\nLM: every set of real numbers is Lebesgue measurable;\n\nPB: every set of real numbers has the property of Baire;\nUP: every uncountable set of real numbers has a perfect subset.\nTwo other principles to be considered are\n\nRAM: all colourings are Ramsey\nNoMAD: there is no maximal infinite family of pairwise almost disjoint infinite sets of natural numbers.\n\n\nThe speaker showed in 1968 that in Solovay's model, RAM holds, and in 1969 that if one started from a Mahlo cardinal, NoMAD would hold in the corresponding Solovay model.\nIt is natural to ask whether these large cardinals are necessary; the inaccessible is necessary for UP (Specker) and LM (Shelah) but not for PB (Shelah).\nMore recently Toernquist has shown that NoMAD holds in Solovay's original model, and Shelah and Horowitz have extended his work to show that even that inaccessible is unnecessary to get a model of NoMAD. Toernquist and Schrittesser have very recently shown that NoMAD follows from RAM plus a uniformisation principle.\n\nBut it has been open for fifty years whether RAM requires an inaccessible.\nThis talk will be chiefly about flutters and chameleons, which are non-Ramsey sets with elegant properties, constructed using weak forms of AC; surprisingly their existence has been found to follow from various Pareto principles of mathematical economics, as described in this week's colloquium talk by Giorgio Laguzzi.\n\nTheir relation to feeble filters will also be discussed: that every free filter on the set of natural numbers is feeble follows from RAM (Mathias 1973) but not from NoMAD, (Shelah and Horowitz in a second recent paper). A filter is feeble if it is meagre in the Cantor topology; equivalently, if some finjection projects it to the Frechet filter.\n\n\n
Thursday, 31.10.19, 17:00-18:00, Hörsaal II, Albertstr. 23b