Nonlocal Aspects in Geometric Analysis
Thursday, 7.1.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
A classical problem in the geometric calculus of variations is the\nproblem of finding particularly “nice” representatives within a class\nof geometric objects.\nThese representatives are usually achieved by finding critical points\nof an energy functional acting on these objects.\nIn this talk we will have a look at certain curvature energies, such\nas the M\b"obius energy which acts on knots. The fundamental question\nis: Given a knot (isotopy) class, can we find a minimizer (or other\ncritical points) in this class, and is this minimizer smooth?\n\nQuite surprisingly, these questions are intrinsically related to the\ntheory of fractional order Dirichlet-type energies (For knot energies:\nThey are related to fractional versions of geodesics in the 2-sphere).\nWe will have a look at this relations and at some of these problems\nwhich are only partially understood and require abstract tools from\nharmonic analysis.\n
Non-archimedean links of singularities
Friday, 8.1.16, 10:15-11:15, Raum 404, Eckerstr. 1
I will introduce a non-archimedean version of the link of a singularity. This object will be a space of valuations, a close relative of non-archimedean analytic spaces (in the sense of Berkovich) over trivially valued fields. \nAfter describing the structure of these links, I will deduce information about the resolutions of surface singularities. \nIf times allows, I will then characterize those normal surface singularities whose link satisfies a self-similarity property. The last part is a current work in progress with Charles Favre and Matteo Ruggiero.
Existenz und Eindeutigkeit der stochastischen Allen-Cahn-Gleichung
Tuesday, 12.1.16, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
On (spinoral) Yamabe equations on noncompact manifolds
Tuesday, 12.1.16, 16:00-17:00, Raum 125, Eckerstr. 1
We study generalizations of the Yamabe problem and its\nspinorial sybling to noncompact manifolds. We examine general properties\nof (sybling) Yamabe equations and there link to the corresponding\nvariational problems. In particular, we investigate the (non-)existence\nof solutions for certain model spaces.
Von der Praxis in die Theorie der Finanzmathematik: Eine sprunghafte Angelegenheit
Thursday, 14.1.16, 16:15-17:15, Hörsaal II, Albertstr. 23b
Thursday, 14.1.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Numerische Analysis: Kleine Tricks statt großer Maschinen
Thursday, 14.1.16, 17:15-18:15, Hörsaal II, Albertstr. 23b
Classifying line bundles over rigid analytic varieties
Friday, 15.1.16, 10:15-11:15, Raum 404, Eckerstr. 1
Hitchin and Calabi-Yau integrable systems
Monday, 18.1.16, 16:15-17:15, Raum 404, Eckerstr. 1
Computational Complexity Theory of Metric Spaces
Tuesday, 19.1.16, 08:00-09:00, Raum 404, Eckerstr. 1
Computational Complexity Theory (P/NP) and Recursive Analysis each extend\nclassical Recursion Theory - in different directions: The first with regard to\nefficient tractability, the latter from the discrete to the continuous setting\nof real numbers and functions, encoded suitably. We unify and generalize both\nto sigma-compact metric spaces and function spaces thereon and function spaces\nthereon. This constitutes a major step towards our vision of a logical\nfoundation of numerical computing ranging from provably optimal algorithms via\nsound programming semantics to verification.\n\nProbevorlesung: Die Unentscheidbarkeit der erststufigen Theorie von (N,+,•,0,1)
Definierbare Gruppen in Erweiterungen algebraisch abgeschlossener Körper
Tuesday, 19.1.16, 10:15-11:15, Raum 404, Eckerstr. 1
Poizat zeigte, dass in der Theorie algebraisch abgeschlossener Körper\ndefinierbare Gruppen mit algebraischen Gruppen übereinstimmen. In diesem\nVortrag werden wir definierbare Gruppen etlicher Expansionen algebraisch\nabgeschlossener Körper betrachten und eine Charakterisierung dieser liefern. \nDie Algebraizitätsvermutung verbindet Ideen und Methoden aus der Klassifikation\neinfacher endlicher Gruppen und aus der Algebraischen Geometrie. Im\nZusammenhang mit dieser Vermutung wurde ein sogenannter schlechter Körper der\nCharakteristik null konstruiert, der in positiver Charakteristik die Existenz\nunendlich vieler Mersenne Primzahlen widerspricht. Wir werden eine\nvollständige Beschreibung definierbarer Gruppen im obengenannten schlechten\nKörper geben und zeigen, dass jede in dieser Struktur definierbare einfache\nGruppe algebraisch ist. \n\nProbevorlesung: Die Unentscheidbarkeit der erststufigen Theorie von (N,+,•,0,1)
Harnack inequalities in curvature flows
Tuesday, 19.1.16, 16:15-17:15, Raum 125, Eckerstr. 1
We give an overview over the state of research in the theory of Harnack inequalities for extrinsic geometric flows, such as the mean curvature flow. We also discuss some applications to the classification of ancient solutions.
Eine Logik für uniforme Analysis
Wednesday, 20.1.16, 08:00-09:00, Hörsaal II, Albertstr. 23b
Analysis kann man nicht nur in den reellen oder komplexen Zahlen betreiben,\nsondern allgemeiner in sogenannten lokalen Körpern (topologische Körper, die\nlokal kompakt sind). Lokale Körper sind aus Sicht der Logik erster Stufe gut\nverstanden; insbesondere kann man mit dem Transferprinzip von Ax-Kochen/Ershov\n(aus den 60ern) erste-Stufe-Aussagen zwischen gewissen lokalen Körpern K\nübertragen. Ich werde einen logischen Formalismus vorstellen, der es\nermöglicht, auch analytische (nicht-erste-Stufe-) Aussagen uniform in diesen\nKörpern auszudrücken und zwischen verschiedenen K zu übertragen. Dies hat\ninteressante Anwendungen in der Darstellungstheorie.\n\nProbevorlesung: Die Unentscheidbarkeit der erststufigen Theorie von (N,+,•,0,1)
Mathias forcing associated to filters and its applications
Wednesday, 20.1.16, 10:15-11:15, Raum 404, Eckerstr. 1
We shall present topological characterizations of filters F on the set of\nnatural numbers such that the Mathias forcing M(F) associated to F adds no\ndominating reals or preserves ground model unbounded families. These\ncharacterizations have a number of applications. For instance, they imply that\nfor an analytic filter F, M(F) adds no dominating reals iff F is a countable\nunion of compact sets, thus answering a question of M. Hrusak.\n\nProbevorlesung: Die Unentscheidbarkeit der erststufigen Theorie von (N,+,•,0,1)
Splitting, bounding and almost disjoint families
Wednesday, 20.1.16, 13:00-14:00, Raum 404, Eckerstr. 1
The cardinal characteristics of the continuum describe various combinatorial, topological,\nor measure theoretic properties of the real line. They are usually defined as the minimum\nsize of a family of reals satisfying certain property and take cardinal values between א₁, and\n𝔠. For example, a maximal almost disjoint family is an infinite family of infinite subsets of\nℕ whose elements have pairwise finite intersections and which is maximal among all such\nfamilies under inclusion. The almost disjointness number 𝔞 is defined as the minimum size of\na maximal almost disjoint family.\n\nWe will consider some of the classical cardinal characteristics of the continuum, like 𝔞, 𝔰, 𝔟, 𝔡\nand 𝔠, and see how the study of their possible constellations has influenced the development\nof various forcing techniques. Among those are the first appearance of creature forcing,\na method of iteration known as matrix-iterations, as well as Shelah’s template iteration\ntechniques. Many of the above techniques have found further applications in the study of the\ncardinal invariants associated to measure and category. We will conclude our discussion with\nsome open questions.\n\nProbevorlesung: Die Unentscheidbarkeit der erststufigen Theorie von (N,+,•,0,1)
Applications of the projective Fraisse limit construction
Wednesday, 20.1.16, 15:15-16:15, Hörsaal II, Albertstr. 23b
The projective Fraisse construction, a dualization of the Fraisse construction\nfrom model theory, was introduced several years ago in a paper by Irwin and\nSolecki. We show how to use this construction to obtain the following results:\n1. Compute the universal minimal flow of the homeomorphism group of the Lelek\nfan, a compact connected metric space with many symmetries. (This is joint work\nwith Dana Bartosova.) 2. Show that the homeomorphism group of the Cantor space\nhas ample generics, that is, it has a comeager conjugacy class in every\ndimension.\n\nProbevorlesung: Die Unentscheidbarkeit der erststufigen Theorie von (N,+,•,0,1)
Applications of the projective Fraisse limit construction
Wednesday, 20.1.16, 15:15-16:15, Raum 404, Eckerstr. 1
The projective Fraisse construction, a dualization of the Fraisse construction\nfrom model theory, was introduced several years ago in a paper by Irwin and\nSolecki. We show how to use this construction to obtain the following results:\n1. Compute the universal minimal flow of the homeomorphism group of the Lelek\nfan, a compact connected metric space with many symmetries. (This is joint work\nwith Dana Bartosova.) 2. Show that the homeomorphism group of the Cantor space\nhas ample generics, that is, it has a comeager conjugacy class in every\ndimension.\n\nProbevorlesung: Die Unentscheidbarkeit der erststufigen Theorie von (N,+,•,0,1)
Thursday, 21.1.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Degenerations of polarized Calabi-Yau manifolds
Friday, 22.1.16, 10:00-11:00, Raum 404, Eckerstr. 1
I will present a joint work with Mattias Jonsson in which we rely on non-Archimedean geometry to study the limit behavior of the volume forms of Ricci-flat Kähler metrics in a degenerating family.
Mock Modular Forms in Mathematics and Physics
Monday, 25.1.16, 16:15-17:15, Raum 404, Eckerstr. 1
Limits of \balpha-harmonic maps
Tuesday, 26.1.16, 16:15-17:15, Raum 125, Eckerstr. 1
In a famous paper, Sacks and Uhlenbeck introduced a perturbation of the\nDirichlet energy, the so-called \balpha-energy E\balpha, \balpha>1, to construct non-trivial\nharmonic maps of the two-sphere in manifolds with a non-contractible\nuniversal cover. The Dirichlet energy corresponds to \balpha= 1 and, as \balpha\ndecreases to 1, critical points of E\balpha are known to converge to harmonic\nmaps in a suitable sense.\nHowever, in a joint work with Andrea Malchiodi and Mario Micallef,\nwe show that not every harmonic map can be approximated by critical\npoints of such perturbed energies. Indeed, we prove that constant maps\nand the rotations of S^2 are the only critical points of E_\balpha for maps from\nS^2 to S^2 whose \balpha-energy lies below some threshold, which is independent\nof \balpha (sufficiently close to 1). In particular, nontrivial dilations (which are\nharmonic) cannot arise as strong limits of \balpha-harmonic maps. We shall\nalso discuss similar results for other perturbations of the Dirichlet energy.
Thursday, 28.1.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Frobenius splittings in birational geometry
Friday, 29.1.16, 10:15-11:15, Raum 404, Eckerstr. 1
Frobenius splittings in birational geometry
Friday, 29.1.16, 10:15-11:15, Raum 404, Eckerstr. 1
Due to the absence of the Kawamata-Viehweg vanishing theorem, the classification of algebraic varieties in positive characteristic, as of very recently, has been seen as an insurmountable task. Recent progress in the field has been inspired by the discovery of Frobenius-split varieties. In my talk, I will discuss connections between the geometry of projective varieties and properties of the Frobenius action, focusing particularly on surfaces.