Deformations of Lagrangian Q-submanifolds
Monday, 5.6.23, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
Positively graded symplectic Q-manifolds encompass a lot of well-known mathematical structures, such as Poisson manifolds, Courant algebroids, etc. Lagrangian submanifolds of them are of special interest, since they simultaniously generalize coisotropic submanifolds, Dirac-structures and many more. In this talk we set up their deformation theory inside a symplectic Q-manifold via strong homotopy Lie algebras.
Numerical computations and thermodynamically complete models for inelastic behaviour in solids
Tuesday, 6.6.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk, I will introduce some aspects of the mathematical modelling of inelastic solids, placing particular emphasis on models that are compatible with the second law of thermodynamics. In particular, I will describe a recent model from [Cichra, Pr?ša; 2020] and discuss its numerical approximation via the finite element method. One of the advantages of the approach considered here is that it is not necessary to introduce additional concepts, such as the plastic strain. Moreover, as a consequence of the thermodynamically consistent derivation, one is able to compute the evolution of the temperature field without additional complication. I will also showcase an application of this modelling approach to the Mullins effect, for which up to date there had been no simple yet fully coupled thermo-mechanical model.
Gluing spaces with Bakry-Emery Ricci curvature bounded from below
Monday, 12.6.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
In this talk I will explain the Bakry-Emery Ricci tensor and the metric gluing construction between two (weighted) Riemannian manifolds along isometric parts of their boundary. When the (weighted) Riemannian manifolds admit a lower bound for the (Bakry-Emery) Ricci curvature, I will present a necessary and sufficient condition such that the metric glued space has synthetic Ricci curvature bounded from below.
Ultrafilters, congruences, and profinite groups
Tuesday, 13.6.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
It is a well-known fact, with important applications in additive\ncombinatorics and Ramsey theory, that the usual sum of integers may\nbe extended to the space of ultrafilters over Z, yielding a compact\nright topological semigroup. The analogous construction also goes\nthrough for the product.\n\nRecently, B. Šobot introduced two (ternary) notions of congruence on\nthe space above. I will talk about joint work with M. Di Nasso, L.\nLuperi Baglini, M. Pierobon and M. Ragosta, in which the study of\nthese congruences led us to isolate a class of ultrafilters enjoying\ncharacterisations in terms of tensor products, directed sets,\nprofinite groups, and more.\n\n\n
Elementare Differentialgeometrie zum Anfassen
Tuesday, 13.6.23, 19:30-20:30, Hörsaal II, Albertstr. 23b
Laver Trees in the Generalized Baire Space
Wednesday, 14.6.23, 10:30-11:30, Raum 125, Ernst-Zermelo-Str. 1
n this talk, we present some results in the context of the generalized Baire space kappa^kappa. We prove that any suitable generalization of Laver forcing to the space κappa^κappa, for uncountable regular κappa, necessarily adds a Cohen κappa-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. This is a joint work with Yurii Khomskii, Marlene Koelbing and Wolfgang Wohofsky.
Young mathematicians in Geometry and Analysis
Thursday, 15.6.23, 11:00-12:00, Raum 226, Hermann-Herder-Str. 10
Hodge numbers of moduli spaces of principal bundles on curves
Friday, 16.6.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
The Poincaré series of moduli stacks of semistable G-bundles on curves has been computed by Laumon and Rapoport. In this joint work with Melissa Liu, we show that the Hodge-Poincaré series of these moduli stacks can be computed in a similar way. As an application, we obtain a new proof of a joint result of the speaker with Erwan Brugallé, on the maximality on moduli spaces of vector bundles over real algebraic curves.
Adaptive Testing: Bandits find correct answers fast
Friday, 16.6.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
Testing is the task of finding out which of several possible actions leads to the best outcome by repeatedly trying actions and observing their random effects. A company may want to find which web page A or B generates the most interaction with its clients. Clinical trials try to determine which drug quantity has the best efficiency-toxicity trade-off.\n\nIn the sequential testing framework, an agent repeatedly selects one of the actions and observes a random outcome. The agent wants to find the action with the best mean outcome as quickly as possible and with high certainty. A simple strategy is to try each action in turn until enough information is gathered. Bandit algorithms instead select their future actions based on past observations: they adapt to the data as it comes. This adaptive behavior makes them stop faster.
The energy technique for BDF methods
Tuesday, 20.6.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Abstract The application of the energy technique to numerical methods with very good stability properties for parabolic equations, such as algebraically stable Runge–Kutta methods or\nA-stable multistep methods, is straightforward. The extension to high order multistep methods requires some effort; the main difficulty concerns suitable choices of test functions. We\ndiscuss the energy technique for all six backward difference formula (BDF) methods. In the\ncases of the A-stable one- and two-step BDF methods, the application is trivial. The energy\ntechnique is applicable also to the three-, four- and five-step BDF methods via Nevanlinna–\nOdeh multipliers. The main new results are: i) No Nevanlinna–Odeh multipliers exist for the\nsix-step BDF method. ii) The energy technique is applicable under a relaxed condition on\nthe multipliers. iii) We present multipliers that make the energy technique applicable also to\nthe six-step BDF method. Besides its simplicity, the energy technique for BDF methods is\npowerful, it leads to several stability estimates, and flexible, it can be easily combined with\nother stability techniques.
Elliptic Lie Theory : state of art
Friday, 23.6.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
After introducing the so-called elliptic root system, \nI will explain some motivations. The aim of this talk \nis to present the state of art on this class of \nroot systems.
Toolbox for the Analysis of Motor Dynamics during Unrestrained Behavior
Friday, 23.6.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
Movement is the primary means of an organism interacting with its environment. To study neural processes underlying movement, neuroscientists often pursue the reductionist approach of reducing the variability of the task to a few controllable factors. Controlled but ethologically artificial paradigms delineate animal behavior into individual variables. Strongly limiting the animal's behavior via head-fixation reduces the number of uncontrolled variables in the design. However, it also limits our ability to understand the naturalistic dynamics of movement. Recent studies indicate that signals related to motion are spread throughout the whole brain, even in head-restrained animals. Spontaneous movements outside the task content influence the ongoing neural activity. These findings highlight the enormous contribution of behavior to neural activity and indicate that our interpretation of ongoing processes might be confounded. Thus, there is a drive in neuroscience to study neural processes in more naturalistic environments and freely moving conditions. Measuring animal behavior in such situations is not straightforward. Furthermore, many scientific tools for electrophysiology and optogenetic modulation were initially developed for acute experiments and need to be adopted for chronic, freely moving use. In this work, we developed multiple complementary tools to help study neural processes in freely moving animals. We designed and characterized different multifunctional techniques for combining electrophysiology and optogenetics. A fluidic probe could deliver viral vectors to the recording site; a multifiber approach enabled ultra-precise 3D interrogation of neural circuits. To measure unconstrained movements, we developed FreiPose, a versatile framework to capture 3D motion during freely moving behavior, and combined the movement tracking of rats with electrophysiology. Using a modeling strategy, we described the ongoing neural activity as a combination of simultaneous multiplexed coding of multiple body postures and paw movement parameters. A virtual head-fixation approach was devised of those models to distinguish paw movements from general movement information. Using encoding models of neural activity, we clamped body and head movements to obtain the impact of the paw movements on neuronal activity. Consequently, a large fraction of neurons in the motor cortex was uncovered to be tuned to paw trajectories. This tuning was previously masked by the influence of the varying body posture information. We conclude that measuring the movements of freely moving animals is an essential step toward understanding the underlying neural dynamics. Adding precise descriptions of ongoing behavior into computational models of neural activity will enable us to describe motifs of neural population activity related to sensorimotor integration and decision-making.
Generic nilpotent groups and Lie algebras
Friday, 23.6.23, 14:30-15:30, Hörsaal II, Albertstr. 23b
We will present ongoing work on the model-theoretic classification of generic nilpotent groups and Lie algebras. A classical result in model theory is that all abelian groups are stable. Nilpotent groups are, in some sense, the simplest class of groups that properly contains the abelian groups. This led naturally to the question of investigating the degree of complexity of nilpotent groups. \n\nIn this talk, we will give some insight into the complexity of some generic theories of nilpotent groups. We will explain how those questions relate to more algebraic questions related to Lie algebras and we will illustrate an intriguing discontinuity of complexity when passing from generic 2-nilpotent groups to 3-nilpotent: the former are NSOP1 whereas the latter have SOP3.
On Line Bundle Twists for Unitary Bordisms
Monday, 26.6.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
Classical theorems of Conner-Floyd and Hopkins-Hovey say that complex \(K\)-theory is completely determined by unitary bordism and \(\bmathrm{Spin}^c\) bordism respectively. The isomorphisms appearing in these theorems are induced by the maps that send a bordism class to its orientation-class in complex \(K\)-theory. Despite this geometric description, the proofs that they are indeed isomorphism are rather abstract and homotpy-theoretical.\n\nMotivated by theoretical physics, Baum, Joachim, Khorami and Schick extend Hopkins and Hovey’s result in a forthcoming paper to twisted \(\bmathrm{Spin}^c\) bordism and twisted \(K\)-theory. Here, the twists are given by (representatives of) elements in third integral cohomology.\n\nSince every almost complex structure induces a \(\bmathrm{Spin}^c\) structure and since the classical Conner-Floyd orientation factors through the Hopkins-Hovey orientation, one may wonder whether there is a twisted unitary bordism theory and a twisted Conner-Floyd orientation that extends the result of Baum, Joachim, Khorami and Schick ‘to the left’.\nIn this talk, I answer this question in the negative.
The Łoś-Tarski Theorem and Forbidden Induced Substructures
Tuesday, 27.6.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
The well-known Łoś-Tarski Theorem from classical model theory implies that a class of structures that is closed under induced substructures is axiomatizable in first-order\nlogic by a sentence if and only if it has a finite set of forbidden induced finite substructures. Furthermore, by the Completeness Theorem, we can compute from the axiomatization of the class the corresponding forbidden induced substructures. This machinery fails on finite graphs as shown by our results.\n
Grundvorstellungen zu Produkten in der Analytischen Geometrie
Tuesday, 27.6.23, 19:30-20:30, Hörsaal II, Albertstr. 23b
Grundvorstellungen sind anschauliche Deutungen eines mathematischen Begriffs, die diesem Sinn geben und Verständnis ermöglichen. In der Analytischen Geometrie sind für den Aufbau von Grundvorstellungen Übersetzungsprozesse zwischen abstrakten algebraischen Konzepten und der geometrischen Anschauung relevant. Eine besondere Bedeutung kommt im Rahmen der Vektorrechnung den verschiedenen Produkten zu (Produkt reeller Zahlen, Skalare Multiplikation, Skalarprodukt und Vektorprodukt), für die im Vortrag normativ formulierte Grundvorstellungen als didaktische Leitlinien vorgestellt werden. Dabei werden mit einem Schwerpunkt auf das Skalar- und Vektorprodukt anhand konkreter Aufgabenstellungen Wege zu einem grundvorstellungsorientierten Mathematikunterricht in der Analytischen Geometrie aufgezeigt.
Evaluation of risks under dependence uncertainty
Friday, 30.6.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
Generalized Hoeffding-Fréchet functionals aiming at describing the possible influence of dependence on functionals of a statistical experiment, where the marginal distributions are fixed, have a long history. The problem of mass transportation can be seen as a particular case of this problem with two marginals and a linear functional induced by a distance. In the first part we give a review of some basic developments of this topic. We also describe several results for the solution for nonlinear functionals in the context of the analysis of worst case risk distributions. We show that these problems can be reduced to a variational problem and the solution of a finite class of (linear) masstransportation problems. \n\nIn the second part we review several approaches to improve risk bounds for aggregated portfolios of risks based on marginal information only. This endevour is motivated by the fact that the dependence uncertainty on the aggregated risks based on marginal information only is typically too wide to be acceptable in applications. Several methods have been developed in recent years to include structural and partial dependence information in order to reduce the model uncertainty. These include higher order marginals (method of reduced bounds), global variance or higher order moment bounds, partial positive or negative dependence restrictions and structural information given by common risk factors (partially specified risk factor models) or given by models with subgroup structures. Also an effective two-sided variant of the method of improved standard bounds has been developed.\n\nThe third part is devoted to some recent more detailed ordering results w.r.t. dependence orderings of relevant risk models making essential use of structural properties (like subgroup structure, graph structure or factor models) and on dependence properties of the models. The dependence structure of these models is given by *-products of copulas. Comparison results for *-products then allow to derive (sharp) risk bounds invarious subclasses of risk models induced by additional restrictions.\n\nSeveral applications show that these improved risk bounds may lead to results acceptable in praxis.