Achieving High Accuracy in Neural Network Training for PDEs with Energy Natural Gradient Descent.
Dienstag, 18.4.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We will talk about energy natural gradient descent, a natural gradient method with respect to a Hessian-induced Riemannian metric as an optimization algorithm for physics-informed neural networks (PINNs) and the deep Ritz method. As a main motivation we show that the update direction in function space resulting from the energy natural gradient corresponds to the Newton direction modulo an orthogonal projection onto the model's tangent space. We present numerical results illustrating that energy natural gradient descent yields highly accurate solutions with errors several orders of magnitude smaller than what is obtained when training PINNs with standard optimizers like gradient descent, Adam or BFGS, even when those are allowed significantly more computation time.
Informalizing formalized mathematics using the Lean theorem prover
Freitag, 21.4.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
One of the applications of interactive theorem provers in\npure mathematics is being able to produce machine-verified formal proofs. I will talk about a less-obvious application, which is using formalized mathematics to author interactive informal expositions. I will demonstrate a prototype of an "auto-informalization" system written in Lean that presents the reader with an interface to view proofs at a desired level of detail. I will also discuss thoughts on the impact of such tools in mathematics. This is joint work with Patrick Massot.\n
Automorphisms of the Boutet de Monvel algebra
Montag, 24.4.23, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
In a remarkable work, Duistermaat and Singer in 1976 studied the algebras of all classical pseudodifferential operators on smooth (boundaryless) manifolds. They gave a description of order preserving algebra isomorphism between the algebras of classical pseudodifferential operators of two manifolds. The subject of this talk is the generalisation of their results to manifolds with boundary. The role of the algebra of pseudodifferential operators that we are interested in is the Boutet de Monvel algebra.\n\nThe main fact of life about manifold with boundary is that vector fields do not define global flows and the "boundary conditions" are a way of dealing with this problem. The Boutet de Monvel algebra corresponds to the choice of local boundary conditions and is, effectively, a non-commutative completion of the manifold. One can think of it as a parametrised version of the classical Toeplitz algebra as a completion of the half-space.\n\nWhat appears in the study of automorphisms are Fourier integral operators and we will try to explain their appearance - both in boundaryless and boundary case. as it turns out, the non-trivial boundary case introduces both some complications but also some simplifications of the analysis involved, Once this is done, the analysis that we need reduces to a high degree to relatively classical results about automorphisms and homology of the Toeplitz algebra and some basic facts from K-theory.\n\nThis is a joint work in progress with Elmar Schrohe.
The fundamental gap conjecture
Dienstag, 25.4.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
At the end of his study [J. Statist. Phys., 83] on thermodynamic functions\nof a free boson gas, van den Berg conjectured that the difference between the\ntwo smallest eigenvalues\n\nΓ\nV\n(Ω) := λ\nV\n2\n(Ω) − λ\nV\n1\n(Ω);\n\nof the Schr¨odinger operator −∆ + V on a convex domain Ω in R\nd\n, d ≥ 1,\nequipped with homogeneous Dirichlet boundary conditions satisfies\n\nΓ\nV\n(Ω) ≥ Γ (ID) = 3π\n2\nD2\n, (1)\n\nwhere ID is the interval (−D/2, D/2) of length D = diameter(Ω) . The term\nΓ\nV\n(Ω) is called the fundamental gap and describes an important physical quantity: for example, in statistical mechanics, ΓV\n(Ω) measures the energy needed to\njump from the ground state to the next excited eigenstate, or computationally,\nit can control the rate of convergence of numerical methods to compute large\neigenvalue problems [SIAM, 2011]. Thus, one is interested in (optimal) lower\nbounds on ΓV\n(Ω). Since the late 80s, the fundamental conjecture (1) attracts\nconsistently the attention of many researcher including M. S. Ashbaugh & R.\nBenguria [Proc. Amer. Math. Soc., 89], R. Schoen and S.-T. Yau Camb. Press,\n94., B. Andrews and J. Clutterbuck [J. Amer. Math.\nSoc., 11].\n\nIn this talk, I present new results on the fundamental gap conjecture (1)\nfor the Schr¨odinger operator −∆ + V on a convex domain Ω equipped with\nRobin boundary conditions. In particular, we present a proof of this conjecture\nin dimension one, and mention results for the p-Laplacian.\n\nThe talk is based on the joint works [1, 2] with B. Andrews and J. Clutterbuck.\nReferences\n[1] Ben Andrews, Julie Clutterbuck, and Daniel Hauer. Non-concavity of the\nRobin ground state. Camb. J. Math., 8(2):243–310, 2020.\n[2] Ben Andrews, Julie Clutterbuck, and Daniel Hauer. The fundamental gap\nfor a one-dimensional Schr¨odinger operator with Robin boundary conditions.\nProc. Amer. Math. Soc., 149(4): 1481–1493, 2021.
Strong distributivity and games on posets
Dienstag, 25.4.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
A forcing order is said to be \(<\bkappa\)-distributive iff it does not add new sequences of length \(<\bkappa\). A sufficient but not necessary condition for this is that the forcing is \(<\bkappa\)-closed, i.e. any \(<\bkappa\)-sequence of conditions has a lower bound. We introduce a strenghtening of \(<\bkappa\)-distributivity called strong \(<\bkappa\)-distributivity which can replace \(<\bkappa\)-closure in many applications. A main benefit of this property is that a \(<\bkappa\)-closed forcing remains strongly \(<\bkappa\)-distributive in any extension by a \(\bkappa\)-cc. order, even though it no longer necessarily is \(<\bkappa\)-closed.
Syzygies of the cotangent complex
Freitag, 28.4.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
The cotangent complex is an important but difficult to understand object associated to a map of commutative rings (or schemes). It is connected with some easier to compute invariants: the module of differential forms, the conormal module, and Koszul homology can all be seen as syzygies inside the cotangent complex. Quillen conjectured that, for maps of finite flat dimension, the cotangent complex can only be bounded for locally complete intersection homomorphisms. This was proven by Avramov in 1999. I will explain how to get a new proof by paying attention to these syzygies, and how to simultaneously prove a conjecture of Vasconcelos on the conormal module.
Combinatorics of toric bundles
Freitag, 28.4.23, 14:00-15:00, SR 119
Toric bundles are fibre bundles which have a toric variety as a fiber. One particularly studied class of toric bundles are horospherical varieties which are toric bundles over generalized flag varieties. Similar to toric varieties, toric bundles admit a combinatorial description via polyhedral geometry. In my talk, I will explain such a combinatorial description, and describe a couple of results which rely on it. In particular, I will present a generalization of the BKK theorem and the Fano criterion for toric bundles.
Noetherianity and equationality
Dienstag, 2.5.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
A theory is noetherian if there is a family of definable sets with the descending chain condition such that every definable set is a boolean combination of those in the family. Noetherianity captures some of the desired properties of algebraically closed fields in any characteristic or differentially closed fields in characteristic 0. Noetherian theories are in particular omega-stable and equational. In recent work with M. Ziegler, we have shown that the theory of proper pairs of algebraically closed fields in any characteristic is noetherian.
Shifted Lagrangian structures in Poisson geometry
Dienstag, 2.5.23, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
It is well known that BG carries a 2-shifted symplectic structure. In this talk, I will study the shifted lagrangian groupoids of BG. I will show how many constructions on Poisson geometry unify using the language of shifted symplectic groupoids. This is work in progress with Daniel Alvarez and Henrique Bursztyn.
Entwicklung algebraischen Denkens in der Schule
Dienstag, 2.5.23, 19:30-20:30, Hörsaal II, Albertstr. 23b
Das Verstehen und der flexible Gebrauch von Variablen, Termen und Gleichungen ist ein wichtiges Ziel im Mathematikunterricht. Der souveräne Umgang mit diesen algebraischen Objekten ist ein wichtiges Instrumentarium bei inner- und außermathematischen Problemstellungen, insbesondere im Rahmen der Funktionenlehre. Im Vortrag wird erörtert, welche Aspekte verstanden werden müssen und welche Fehlvorstellungen hierbei hinderlich sind. Es werden Erkenntnisse aus verschiedenen Forschungsprojekten vorgestellt, die den Blick für Lehrkräfte schärfen können, algebraisches Denken erfolgreich zu entwickeln.
Heights on commutative algebraic groups
Freitag, 5.5.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
In diophantine geometry a height function is measure of the complexity of an algebraic number. This talk is a presentation of my master thesis and will look at height estimates on connected commutative algebraic groups for taking integral multiples of points. Such estimates are used in the Analytic Subgroup Theorem by Gisbert Wüstholz.
Workshop: Numerik in BaWü
Montag, 8.5.23, 10:00-11:00, Raum 226, Hermann-Herder-Str. 10
Dreitägiger Workshop mit Sprecher:innen verschiedener Universitäten in Baden-Württemberg
Paths towards Open World Generalization
Montag, 8.5.23, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
ODD Riemannian metrics
Montag, 8.5.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We describe a generalization of Riemannian metrics motivated from\nKähler geometry of singular complex varieties. These generalizations\nare semipositive symmetric 2-tensors, but degenerate in such way, that\ne.g. they still induce a metric space structure on the underlying\nmanifold.\n\nIn this talk, we will mostly use instructive examples to sketch how far\nRiemannian Geometry can (hopefully) be pursued for these ODD metrics.\n
Acceleration of quantum mechanical systems by exploiting similarity
Dienstag, 9.5.23, 09:00-10:00, Raum 226, Hermann-Herder-Str. 10
Total (generalized) variation for images and shapes
Dienstag, 9.5.23, 10:00-11:00, Raum 226, Hermann-Herder-Str. 10
A categorical perspective on scattering amplitudes
Freitag, 12.5.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
Scattering amplitudes are physical observables which play a central role in interpreting scattering experiments at particle colliders. In recent years, a new perspective on scattering amplitudes, known as amplituhedron programme, has revealed a fascinating link to various mathematical structures of a combinatorial nature, such as positive Grassmannians and cluster algebras. In this talk I will explain this connection from the point of view of derived and cluster categories of type A quivers, from which the formulae for scattering amplitudes can be obtained from projectives of hearts of intermediate t-structures. This talk is based on arXiv:2101.02884 joint with K. Ray and on arXiv:2112.14288 joint with P. Oak, A. Pal, K. Ray and H. Treffinger.
Shifted convolution sums
Montag, 15.5.23, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
In talk I will evaluate shifted convolution sums of divisor functions of the form \(\bdisplaystyle\bsum_{n_1,n_2\bin\bmathbb{Z}\bsetminus\b{0\b}, n_1+n_2=n}\nQ_{d}^{(r_1,r_2)}\bBig(\bfrac{n_2-n_1}{n_1+n_2}\bBig)\bsigma_{r_1}(n_1)\bsigma_{r_2}(n_2)\) where \(\bsigma_{r}(n) = \bsum_{d \bmid n} d^ r\) and \(Q_{d}^{(r_1,r_2)}(x)\) is the Jacobi function of the second kind. These sums can be considered as a shifted version of the Ramanujan sum \(\bsum_{n_1 \bin \bmathbb{Z}} \bsigma_{r_1}(n_1) \bsigma_{r_2}(n_1) n_1^s\). \n\nKey words that appear in the proof and the final result: non-holomorphic Eisenstein series, cusp forms, values of \(L\)-functions, Mellin transform and Whittaker's functions.
Nonlinear bending-torsion theory for inextensible tapered rods by \(\bGamma\)-convergence
Dienstag, 16.5.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The upcoming presentation will focus on a modified version of the main theorem initially stated and proven by Mora and Mueller in their\nwork titled "Derivation of the nonlinear bending-torsion theory for inextensible rods by Γ-Convergence", published in 2003. Specifically, we will direct our attention towards studying a rod with a non-constant cross-section, known as a taper, and answer the question of how the \(\bGamma\)-limit of the elastic energy changes in this case. Additionally, we will combine this new result with previous findings, using banana plants (Musa sp.) as an example.\n
Statistical Learning for Structured Models: Tree Based Methods and Neural Networks
Freitag, 19.5.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
Estimation in regression and classification problems which include low dimensional structures are considered. The underlying question is the following. How well do statistical learning methods perform for models with low dimensional structures? We approach this question using various algorithms in various settings. First, we introduce a new tree based algorithm we named random planted forest. It adapts particularly well to models which consist of low dimensional structures. We examine its performance in simulation studies and include some theoretical backing by proving optimal convergence rates in certain settings for a modification of the algorithm. A generalized version of the algorithm is included, which can be used in classification settings. Furthermore, we prove optimal convergence rates in a classification setting using neural networks. While non-optimal rates existed for this problem, we are the first to prove optimal ones.
The exterior Dirichlet problem for the homogeneous k-Hessian equation
Freitag, 19.5.23, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
We studied the exterior Dirichlet problem for the homogeneous k-Hessian equation in \(R^n\). The main idea are the uniform gradient estimates and second derivative estimates. Then we use these estimates to study the Green function in a \(k-1\) convex domain in \(R^n\). This is joint works with Zhang dekai and Gao zhenghuan.
Metric inequalities with positive scalar curvature
Montag, 22.5.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We will discuss various situations where a certain perturbation of the Dirac operator on spin manifolds can be used to obtain distance estimates from lower scalar curvature bounds. \n\nA first situation consists in an area non-decreasing map from a Riemannian spin manifold with boundary \(X\) into the round sphere under the condition that the map is locally constant near the boundary and has nonzero degree. Here a positive lower bound of the scalar curvature is quantitatively related to the distance from the support of the differential of f and the boundary of \(X\). \n\nA second situation consists in estimating the distance between the boundary components of Riemannian “bands” \(M×[−1,1]\) where \(M\) is a closed manifold that does not carry positive scalar curvature. Both situations originated from questions asked by Gromov. \n\nIn the final part, I will compare the Dirac method with the minimal hypersurface method and show that if \(N\) is a closed manifold such that the cylinder \(N \btimes \bmathbb{R}\) carries a complete metric of positive scalar curvature, then \(N\) also carries a metric of positive scalar curvature. This answers a question asked by Rosenberg and Stolz. Based on joint work with Daniel Raede and Rudolf Zeidler.
"Can you escape?" Escape-Aktivitäten im Mathematikunterricht
Dienstag, 23.5.23, 19:30-20:30, Hörsaal II, Albertstr. 23b
Escape Rooms haben sich seit einiger Zeit als ein sehr erfolgreiches Unterhaltungsformat etabliert. In der Regel werden Besucher in kleinen Gruppen in einen speziell präparierten Raum "eingesperrt" und müssen eine Vielzahl von Rätseln lösen, um zu entkommen. Aufgrund ihrer interaktiven und herausfordernden Natur sind Escape Rooms besonders motivierend für die Teilnehmer*innen. Ein erfolgreicher Fluchtversuch erfordert jedoch die Anwendung von Problemlösungsstrategien, Frustrationstoleranz und die Zusammenarbeit im Team. In diesem Vortrag werden verschiedene Escape Rooms vorgestellt, die sich auf mathematische Rätsel konzentrieren. Es werden theoretische Grundlagen zur Konzeption und Gestaltung eines Escape Rooms behandelt und die Möglichkeiten und Grenzen eines sinnvollen Einsatzes im Bildungsbereich diskutiert.
Deformations of Lagrangian Q-submanifolds
Montag, 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
Dienstag, 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
Montag, 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
Dienstag, 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
Dienstag, 13.6.23, 19:30-20:30, Hörsaal II, Albertstr. 23b
Laver Trees in the Generalized Baire Space
Mittwoch, 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
Donnerstag, 15.6.23, 11:00-12:00, Raum 226, Hermann-Herder-Str. 10
Hodge numbers of moduli spaces of principal bundles on curves
Freitag, 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
Freitag, 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
Dienstag, 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
Freitag, 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
Freitag, 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
Freitag, 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
Montag, 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
Dienstag, 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
Dienstag, 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
Freitag, 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.
Unendliche Körper mit Quantorenelimination
Dienstag, 4.7.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Heintze-Karcher inequality and Alexandrov’s theorem for capillary hypersurfaces
Dienstag, 4.7.23, 16:00-17:00, Raum 127, Ernst-Zermelo-Str. 1
Heintze-Karcher’s inequality is an optimal geometric inequality for embedded closed hypersurfaces, which can be used to prove Alexandrov’s soap bubble theorem on embedded closed CMC hypersurfaces in the Euclidean space. In this talk, we introduce a Heintze-Karcher-type inequality for hypersurfaces with boundary in the half-space. As application, we give a new proof of Wente’s Alexandrov-type theorem for embedded CMC capillary hypersurfaces. Moreover, the proof can be adapted to the anisotropic case, which enable us to prove an Alexandrov-type theorem for embedded anisotropic capillary hypersurfaces.\nThis is based on joint works with Xiaohan Jia, Guofang Wang and Xuwen Zhang.\n
On a Mystery in Machine Learning
Freitag, 7.7.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
In classical regression modelling, the complexity of the model, e.g. measured by the number of parameters, is smaller than the amount of training data. The prediction error exhibits a U-shaped behaviour. The (first) descent is due to decreasing bias, the ascent due to increasing variance. In modern machine learning, often the number of parameters by far exceeds the number of training data points. Intuitively, one could expect that the prediction error explodes with increasing model complexity due to overfitting. Belkin et al. (2019) observed that this is not the case. Instead, the prediction error decreases again when surpassing a certain threshold in model complexity, in some case even below the minimum of the classical, U-shaped regime. A phenomenon the authors denominated as double descent. To understand double descent, we study the simplest setting of linear regression and show that it can be explained by investigating the singular values of the design matrix. Finally, we give an outlook for the non-linear model setting.\n\nBelkin, M.; Hsu, D.; Ma, S.; Mandal, S.: Reconciling modern machine-learning practice and the classical bias–variance trade-off. In: Proceedings of the National Academy of Sciences 116 (2019), jul, Nr. 32, 15849–15854.
Fakultätsfest und Abschlussfeier
Freitag, 7.7.23, 14:00-15:00, Großer Hörsaal Physik, Hermann-Herder-Straße 3a
Stability properties of the \(L^2\)-projection mapping to finite element spaces on adaptively generated meshes
Dienstag, 11.7.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\n\nThe \(L^2\)-projection mapping to Lagrange finite element spaces is a crucial tool in numerical analysis. Its Sobolev stability is known to be the key to discrete stability and quasi-optimality estimates for parabolic problems. For adaptively generated meshes the proof of Sobolev stability is challenging and requires conditions on how strongly the mesh size varies.\n\nHence, for the newest vertex bisection and its generalisation to higher dimensions by Maubach and Traxler we present optimal estimates on the mesh grading. Previously, grading estimates have been available only for 2D mesh refinement strategies. For such adaptively generated meshes we discuss Sobolev stability of the \(L^2\)-projection mapping to Lagrange finite element spaces under certain conditions on the polynomial degree and on the space dimension. In particular, the \(L^2\)-projection is \(W^{1,2}\)-stable for any polynomial degree, for any space dimension smaller than \(7\).\n\nThis is joint work with Lars Diening and Johannes Storn (Bielefeld University).\n
On the Six Functor Formalism for Nori Motivic Sheaves
Freitag, 14.7.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
A Principal-Agent Framework for Optimal Incentives in Renewable Investments
Freitag, 14.7.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
We investigate the optimal regulation of energy production reflecting the long-term goals of the Paris climate agreement.\n\nWe analyze the optimal regulatory incentives to foster the development of non-emissive electricity generation when the demand for power is served either by a monopoly or by two competing agents. The regulator wishes to encourage green investments to limit carbon emissions, while simultaneously reducing intermittency of the total energy production. We find that the regulation of a competitive market is more efficient than the one of the monopoly as measured with the certainty equivalent of the Principal’s value function. This higher efficiency is achieved thanks to a higher degree of freedom of the incentive mechanisms which involves cross-subsidies between firms. A numerical study quantifies the impact of the designed second-best contract in both market structures compared to the business-as-usual scenario. In addition, we expand the monopolistic and competitive setup to a more general class of tractable Principal-Multi-Agent incentives problems when both the drift and the volatility of a multi-dimensional diffusion process can be controlled by the Agents. We follow the resolution methodology of Cvitanić et al. (2018) in an extended linear quadratic setting with exponential utilities and a multi-dimensional state process of Ornstein-Uhlenbeck type. We provide closed-form expression of the second-best contracts. In particular, we show that they are in rebate form involving time-dependent prices of each state variable.
Spinoren, kalibrierte Untermannigfaltigkeit und Instantonen
Montag, 17.7.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
In der Modulraumtheorie gibt es eine tiefe, weitgehend unverstandene Dualität zwischen den Instanton-Zusammenhängen auf Hauptfaserbündeln über einer Mannigfaltigkeit und den kalibrierten Untermannigfaltigkeiten, welche als Modelle für „singuläre“ Instantonen auftreten. Während des Vortrags werde ich mithilfe von Dirac-Operatoren und Spinoren eine entsprechende Dualität (im adiabatischen Limes) der linearisierten Deformationstheorien dieser Modulräume herstellen. Das Hauptergebnis hat Anwendungen auf die Konstruktion von Orientierungsdaten in der Donaldson-Thomas Theorie.
Freitag, 21.7.23, 13:30-14:30, Raum 226, Hermann-Herder-Str. 10
On the role of discrete Green’s operator preconditioning in FFT-based computational homogenization methods
Dienstag, 25.7.23, 15:00-16:00, Raum 226, Hermann-Herder-Str. 10
Solving computational homogenization problems on fine grids leads to systems of linear equations with millions to billions of unknowns, which favour iterative solvers over direct solvers.\nHowever, the number of iterations of iterative solvers can grow with increasing system size. To\novercome this issue, the so-called FFT-based solvers use the discrete Green’s operator preconditioning, which makes the condition number of the resulting linear system independent of the\nsystem/grid size [1, 2]. We studied the discrete Green’s operator preconditioning from a linear\nalgebra viewpoint and showed that all individual eigenvalues of such preconditioned systems can\nbe bounded purely from the knowledge of the material data of the problems, both original and\nreference. We developed a simple algorithm to compute these bounds [3, 4]. In my talk, I will\ndiscuss the theoretical aspects of these results and practical applications of the discrete Green’s\noperator preconditioning to periodic homogenisation problems discretised on regular grids [5].\n\n\nReferences\n\n[1] Moulinec, H. and Suquet, P., A fast numerical method for computing the linear and nonlinear\nmechanical properties of composites, Comptes Rendus de l’Acad´emie des sciences. S´erie II.\nM´ecanique, physique, chimie, astronomie, 318 (1994) 1417–1423\n\n[2] Schneider, M., A review of nonlinear FFT-based computational homogenization methods,\nActa Mechanica, 29 (2021) DOI 10.1007/s00707-021-02962-1,\n\n[3] Ladeck´y, M. and Pultarov´a, I. and Zeman, J., Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method,\nApplications of Mathematics, 66, 21–42 (2020) DOI 10.21136/AM.2020.0217-19\n\n[4] Pultarov´a, I., Ladeck´y, M., Two-sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems, Numerical Linear Algebra Applications\n28 (2021) e2382. DOI 10.1002/nla.2382.\n\n[5] Ladeck´y, M., Leute, J.R., Falsafi, A., Pultarov´a, I., Pastewka, L., Junge, T., and Zeman,\nJ. An Optimal Preconditioned FFT-accelerated Finite Element Solver for Homogenization.\nApplied Mathematics and Computation 446 (2023) 127835 DOI 10.1016/j.amc.2023.127835
A regularity result for a class of free boundary problems
Dienstag, 15.8.23, 15:00-16:00, Raum 125, Ernst-Zermelo-Str. 1
The talk focuses on the study of a class of free boundary problems involving both bulk and interface energies. The bulk energy is of Dirichlet type, albeit of very general form, allowing the dependence from the unknown variable u and the position x. We employ the regularity theory of Λ-minimizers to study the regularity of the free interface. Mild assumptions concerning the dependence of the coefficients on x and u are made and are of Hölder type.
Intermediate curvature and a generalization of Geroch's conjecture
Dienstag, 15.8.23, 16:30-17:30, Raum 125, Ernst-Zermelo-Str. 1
In this talk we explain a non-existence result for metrics of positive m-intermediate curvature (a notion of curvature reducing to positive Ricci curvature for m = 1, and positive scalar curvature for m = n-1) on closed orientable manifolds with topology \(N^n = M^{n-m} x \bmathbb{T}^m\) for \(n \bleq 7\).\nOur proof uses a slicing constructed by minimization of weighted areas, the associated stability inequality, and estimates on the gradients of the weights and the second fundamental form of the slices. This is joint work with Simon Brendle and Sven Hirsch.
Segmentation and Estimation of Change-Points
Donnerstag, 14.9.23, 12:00-13:00, Raum 404, Ernst-Zermelo-Str. 1
We consider the problem of segmentation of (usually normal) observations according to changes in their mean. Changes can occur continuously, e.g., a change in the slope of a regression line, or discontinuously, e.g., a jump in the level of a process.Theoretical results will be illustrated by applications to copyn umber changes, historical weather records, and COVID-19--daily incidence, wastewater analysis, and excess deaths. Sequential detection, confidence regions for the change-points, and difficulties associated with dependent observations will also be discussed.\n\nAspects of this research involve collaboration with Fang Xiao, Li Jian, Liu Yi, Nancy Zhang, Benjamin Yakir, Keith Worsley, and Li (Charlie) Xia.
Freitag, 29.9.23, 00:00-01:00, Hörsaal II, Albertstr. 23b