Introduction to contact dynamics
Monday, 2.7.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
This talk will be an introduction to contact geometry and dynamics.\nIn particular, I will explain the relation between Hamiltonian systems and contact dynamics.\nThe main examples are level sets of Hamiltonians from classical mechanics and contact manifolds, ob which the Reeb flow induces an S^1-action with reasonably nice quotients.
Soergel's Endomorphismensatz for perverse sheaves
Tuesday, 3.7.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
A fundamental result obtained by Soergel in his celebrated work\non the BGG category O is the description of the endomorphism algebra of\nthe projective cover of a simple Verma module, in terms of the\n"coinvariant algebra". Through Beilinson-Bernstein localization, this\nresult can also be stated in terms of perverse sheaves on flag varieties,\nwith complex coefficients. In this talk I will explain a new "topological"\nproof of this result, completely in the setting of perverse sheaves, which\nin fact applies for arbitrary fields of coefficients and leads to a\ndescription of the category of perverse sheaves in terms of commutative\nalgebra. This is joint work with Roman Bezrukavnikov.\n\n
Tuesday, 3.7.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Commutators of simply-connected o-minimal groups
Wednesday, 4.7.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Groups definable in an o-minimal expansion of a real closed field can be seen as a non-standard version of a Lie group (if the real closed field is the real field then such a group is actually a Lie group). For example, algebraic groups over a real closed field are o-minimal groups. In fact, the behaviour of o-minimal groups rests in between algebraic groups and Lie groups. The definability of the derived subgroup is a good example of this dichotomy.\n\nThe commutator subgroup of an algebraic group is again algebraic. However, the commutator of a Lie group may not be a Lie subgroup (there are even solvable counterexamples). The commutator of an o-minimal group may not be definable; in previous work with Jaligot and Otero, we proved that it is so if the group is solvable. \nCommutators play an important role in any category of groups; they played a crucial role in Conversano-Onshuus-Starchenko's characterisation of which solvable Lie groups are definable in an o-minimal expansion of the real field. \n\nIn this talk I will present an overview of these results and provide new insights concerning commutators of simply-connected o-minimal groups.\n\n
The rigidity theorem for motives of non-archimedean analytic spaces
Friday, 6.7.18, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
I will give a quick introduction to the notions of motive and of motivic sheaves. Then, after recalling the main ideas of non-Archimedean analytic geometry I will define the category of motivic sheaves of non-Archimedean analytic spaces. Finally, I will state the Rigidity Theorem in this context and if time permits I will briefly sketch the main ideas about its proof and mention some applications.
Global Serre dualities
Monday, 9.7.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Serre equivalences are important autoequivalences of k-linear categories appearing in different fields of mathematics. In this talk we will ask the following question. In which way are Serre equivalences compatible with k-linear functors? For this we first review the situation in the case of algebraic geometry, where some compatibilty results are known. This motivates us to introduce the notion of a "global Serre duality", which is an abstract framework encoding the naturality of Serre equivalences. Afterwards we show the existence of global Serre dualities in the case of (abstract) representation theory. In interesting special cases, we obtain explicit descriptions of Serre equivalences. This last step will require some techniques from abstract cubical homotopy theory. This is part of an on-going project with Moritz Groth.\n
Vector-valued automorphic functions and their Fourier series
Monday, 9.7.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Connecting Creative Minds – Trinational, European, Global
Tuesday, 10.7.18, 15:45-16:45, Großer Hörsaal der Physik, Hermann-Herder-Straße 3a
\n - Was meint Connecting Creative Minds?\n - Wie wollen wir dieses Motto – im Großen wie im Kleinen – erfolgreich leben?\n - Mit welchen Vorhaben können wir künftig die Kreativität aller Mitglieder unserer Universität noch stärken?\n\nAlle Wissenschaftlerinnen und Wissenschaftler, alle Studierenden, alle Mitarbeiterinnen und Mitarbeiter der Fakultät sind herzlich eingeladen, diese Fragen\ngemeinsam mit dem Rektor zu diskutieren und die Gesamtstrategie der Universität weiterzuentwickeln.
Baumforcings und Ideale
Wednesday, 11.7.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Die Topologie des Cantor-Raumes, die durch endliche 0-1-Folgen erzeugt wird,nennen wir die Standardtopologie. Bezüglich dieser Topologie ist eine Teilmenge genau dann nirgends dicht, wenn es zu jeder endlichen Folge eine\nendliche Fortsetzung gibt, die zu keinem Element der Teilmenge Anfangsstück ist. Da jeder Cohen-Baum durch eine endliche Folge definiert wird, können wir das Obige als Definition des Ideals der Cohen-nirgends dichten\nMengen nehmen. Das \(\bsigma\)-Ideal der Cohen-mageren Mengen\nbesteht aus abzählbaren Vereinigungen Cohen-nirgends dichter Mengen. Diese beiden Definitionen können wir nun auf beliebige Baumforcings übertragen und uns fragen, ob die beiden Ideale übereinstimmen.\n\nIn dem Vortrag möchte ich nachweisen, dass im klassischen Mathiasforcing die beiden Ideale der Mathias-nirgends dichten Mengen und Mathias-mageren Mengen übereinstimmen, wenn wir jedoch das Mathiasforcing für stark unerreichbares \(\bkappa\) verallgemeinern, diese Gleichheit verlieren.
Tilting modules for reductive algebraic groups and their quantum counterparts.
Thursday, 12.7.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Modules which have filtrations by Weyl modules as well as by dual Weyl modules\n\nare called tilting modules. Starting with this definition and a few basic properties I will in this\n\ntalk survey some of the results in representation theory where they have played a key role.\n\n
Two-point functions in differential geometry and the Lawson conjecture
Thursday, 12.7.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
The Lawson conjecture from 1970 states the following:\n\nConjecture (Lawson, 1970). Any embedded minimal torus in the 3-dimensional unit-sphere is the Cliford torus.\n\n\n\nIn 2013, Brendle confirmed the validity of this conjecture thereby complementing the case of genus zero due to Almgren. Although Lawson himself provided many crucial ingredients used in the final proof by Brendle, the missing\npiece in the completion of the proof was finally given in form of a sophisticated use\nof the maximum principle on the surface T: Instead of trying to come up with an\nauxiliary function on T, an auxiliary function on the product T × T is constructed,\nwhich contains much more geometric information at points where a maximum is\nachieved. This method followed up similar techniques used earlier in the context of\nnon-collapsing for curvature flows by Andrews and Huisken. This auxiliary\nfunction led to the conclusion that the second fundamental form on T must have\nconstant length and due to an earlier result of Lawson, Brendle was able to conclude\nthat T must be the Clifford torus up to isometries.\nIn this talk we present this powerful method of two-point functions and sketch\nits various mentioned applications.
Sheaves on the alcoves and modular representations
Friday, 13.7.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
I will give an overview of some recent results obtained jointly with Martina Lanini. While trying to understand the intricacies of the combinatorial category of Andersen, Jantzen and Soergel we came up with a new category that consists of ordinary sheaves on the space of alcoves of an affine Weyl group. I will show how this category provides new methods and tools for the problem of determining rational characters of algebraic groups in positive characteristics.
Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications
Friday, 13.7.18, 12:00-13:00, Bibliothek Angewandte Mathematik, R216, RZ, Hermann-Herder-Strasse 10
We propose a new variational model in weighted Sobolev spaces with\nnon-standard weights and applications to image processing. We show that\nthese weights are, in general, not of Muckenhoupt type and therefore the\nclassical analysis tools may not apply. For special cases of the\nweights, the resulting variational problem is known to be equivalent to\nthe fractional Poisson problem. The trace space for the weighted Sobolev\nspace is identified to be embedded in a weighted \(L^2\) space. We propose\na finite element scheme to solve the Euler-Lagrange equations, and for\nthe image denoising application we propose an algorithm to identify the\nunknown weights. The approach is illustrated on several test problems\nand it yields better results when compared to the existing total\nvariation techniques.\n
Festvortrag: "Das brauche ich nicht zu lernen, das habe ich erlebt!"
Friday, 13.7.18, 15:00-16:00, Großer Hörsaal Physik, Hermann-Herder-Str. 3a
im Anschluss:\nGemütliches Beisammensein im Rahmen des Sommerfestes der Fakultät im Innenhof des Physikalischen Instituts\n\nFür weitere Informationen und ein Programm finden Sie, klicken Sie bitte auf den beigefügten Link.
The Weyl Denominator Identity in light of the structure of root systems
Monday, 16.7.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
The Weyl Denominator Identity arises usually as a special case of the Weyl Character Formula of a complex semisimple Lie algebra. Even though it prominently features the roots of the Lie algebra, the original proof provides limited insight into the structure of the root system.\nI will therefore present a direct proof of the Denominator Identity. This alternative approach explicitly uses connections between the roots and the Weyl Group and might provide a different perspective on the Denominator Identity as well as the structure of the root system.
Systoles of \(\bmathbb{C}P^n\) - freedom and rigidity
Tuesday, 17.7.18, 10:15-11:15, Raum 414, Ernst-Zermelo-Str. 1
The \(k\)-systole of a closed Riemannian manifold \((M,g)\) is an\ninvariant that captures the 'size' of the \(k\)-dimensional homology of \(M\). A\nclassic question in systolic geometry is whether there are relations between\ndifferent systoles, which hold for every possible choice of \(g\), or if they are\nfree to vary independently when changing the metric. It is, for example, a well\nknown result due to C. Loewner that for every metric on the 2-Torus the length\nof the shortest noncontractible loop is bounded from above by a constant times\nthe total volume.\nIn this talk we will focus on complex projective space and ask whether or not\nit is possible to realize an arbitrary set \(a_1,\bldots,a_n\) of positive real\nnumbers as the systoles of \(\bmathbb{C}P^n\). We discuss techniques to modify\nsystoles and present some partial results they provide in the general setting.\nAfterwards we focus on the case of \(\bmathbb{C}P^2\), where we are able to give a\npositive answer, stating and motivating general results on systolic freedom in\nthe process.
TBA
Tuesday, 17.7.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Stationäre Unabhängigkeit und die Erweiterung partieller Automorphismen
Wednesday, 18.7.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Die Automorphismengruppe \(G\) einer abzählbaren Struktur erster Ordnung ist, ausgestattet mit der Topologie punktweiser Konvergenz, eine polnische Gruppe. Insbesondere wenn die Struktur als Limes einer Klasse endlicher Strukturen entsteht, gibt es sehr enge Verbindungen zwischen den kombinatorischen Eigenschaften jener Klasse und den topologischen Eigenschaften von \(G\).\n\nIn diesem Rahmen wird vor allem der Zusammenhang der Eigenschaft des kleinen Index (SIP) und der Existenz generischer Automorphismen (AG) auf der topologischen Seite und der Erweiterungseigenschaft partieller Automorphismen (EPPA) und der Ramsey-Eigenschaft auf der kombinatorischen Seite untersucht.\n\nDes Weiteren tendieren solche Limesstrukturen häufig dazu, einen Begriff der stationären Unabhängigkeit (SIR) zwischen ihren endlichen Unterstrukturen definieren zu lassen, welche ebenfalls topologische und algebraische Eigenschaften von \(G\) determiniert.\n\nIn diesem Überblicksvortrag möchten wir die wichtigsten existierenden Konzepte und Zusammenhänge präsentieren und suggestive Fragen zwischen dem Zusammenhang der kombinatorischen Eigenschaften EPPA und der Existenz einer stationären Unabhängigkeit aufwerfen.
Thursday, 19.7.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
reserviert von Frau Mildenberger
Variations on the theme of moment graphs
Friday, 20.7.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Naturally arising as the 1-skeletons of torus actions on\n(nice) complex projective algebraic varieties, moment graphs were\noriginally introduced by Goresky, Kottwitz and MacPherson to compute\nequivariant cohomology of such varieties. In this talk, I will review\nsome applications of moment graph theory, starting from the equivariant\ncohomology of the flag variety, and the representation theory of a\ncomplex finite dimensional simple Lie algebra. Time permitting, I will\nalso discuss some ongoing joint work with Tomoyuki Arakawa on a certain\nclass of modules ("admitting a Wakimoto flag") for an affine Kac-Moody\nalgebra at a negative level.