Schnupperstudium 2020 - abgesagt!
Montag, 6.4.20, 09:00-10:00, Veranstaltung ist abgesagt!
Das Schnupperstudium wird seit über zehn Jahren angeboten und bietet Schülerinnen und Schüler ab Klasse 10 einen intensiven Einblick in die Universität und eine Auswahl ihrer Studiengänge.
Einstellung des Lehrbetriebs an der Universität Freiburg bis 19.04.2020
Sonntag, 19.4.20, 00:00-01:00, .
Donnerstag, 23.4.20, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 14.5.20, 17:00-18:00, Hörsaal II, Albertstr. 23b
TBA
Montag, 25.5.20, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Steenrod-Squares in differential cohomology
Dienstag, 2.6.20, 16:15-17:15, virtuellen SR 125
In my master's thesis I tried to construct the Steenrod squaring operations in a\ndifferential cohomology theory.\nOne construction method for singular cohomology is based on the graded\ncommutativity of the cup product. For odd Squares this method can be imitated in\nthe Cheeger-Simons-Model of differential cohomology and produces\nthe same differential operations which were already given by D. Grady and H.\nSati. It also gives another explanation\nwhy the even differential Squares cannot exist in general.\nIf time permits, I can discuss an approach to some even Squares that makes use of\nWu classes.\n\n
Twists of quaternionic Kähler manifolds
Montag, 8.6.20, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces
Freitag, 12.6.20, 10:30-11:30, virtueller Raum 404
K3 surfaces have been extensively studied over the past decades for\nseveral reasons. For once, they have a rich and yet tractable geometry\nand they are the playground for several open arithmetic questions.\nMoreover, they form the only class which might admit more than one\nelliptic fibration with section. A natural question is to ask if one\ncan classify such fibrations, and indeed that has been done by several\nauthors, among them Nishiyama, Garbagnati and Salgado.\nIn this joint work with A. Garbagnati, C. Salgado, A. Trbović and R.\nWinter we study K3 surfaces defined over a number field k which are\ndouble covers of extremal rational elliptic surfaces. We provide a list\nof all elliptic fibrations on certain K3 surfaces together with the\ndegree of the field extension over which each genus one fibration is\ndefined and admits a section. We show that the latter depends, in\ngeneral, on the action of the cover involution on the fibers of the\ngenus one fibration.
ODE methods in non-local equations
Dienstag, 16.6.20, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk, I will introduce some ODE-type methods to find radially symmetric solutions for non-local PDEs. I will provide an overview and some improvements of the main ideas behind these methods. The main underlying idea is understanding a non-local PDE problem as an infinite dimensional ODE system. These methods were first developed in the non-local gluing scheme to find solutions for the fractional Yamabe problem, which are singular along a non-zero dimensional submanifold. I will also show some new applications.\n\nIn particular, I will present a variation of constants formula for fractional Hardy operators, a suitable extension in the spirit of Caffarelli–Silvestre and an equivalent formulation as an infinite system of second order constant coefficient ODEs. Moreover, I will show that, like in classical ODEs, quantities such as the Hamiltonian and Wronskian may then be used. As an example of applications, we obtain a Frobenius theorem and a new proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane-Emden equation. We also establish new Pohozaev identities.\n\nThis is a joint work with Weiwei Ao, Hardy Chan, Marco Fontelos, María del Mar González and Juncheng Wei.
TBA
Montag, 22.6.20, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Donnerstag, 25.6.20, 17:00-18:00, Hörsaal II, Albertstr. 23b
"(Ir)rationality of L-values"
Freitag, 26.6.20, 10:30-11:30, virtueller Raum 404
Euler’s beautiful formula\nζ(2n) = −\n(2πi) 2n\nB 2n .\n2(2n)!\ncan be seen as the starting point of the investigation of special values of L-\nfunctions. In particular, Euler’s result shows that all critical zeta values are ra-\ntional up to multiplication with a particular period, here the period is a power of\n(2πi). Conjecturally this is expected to hold for all critical L-values of motives.\nIn this talk, we will focus on L-functions of number fields. In the first part of the\ntalk, we will discuss the ’critical’ and ’non-critical’ L-values exemplary for the\nRiemann zeta function. Afterwards, we will head on to more general number\nfields and explain our recent joint result with Guido Kings on the algebraicity\nof critical Hecke L-values for totally imaginary fields up to explicit periods.
"(Ir)rationality of L-values"
Freitag, 26.6.20, 10:30-11:30, v404
Euler’s beautiful formula\nζ(2n) = −\n(2πi) 2n\nB 2n .\n2(2n)!\ncan be seen as the starting point of the investigation of special values of L-\nfunctions. In particular, Euler’s result shows that all critical zeta values are ra-\ntional up to multiplication with a particular period, here the period is a power of\n(2πi). Conjecturally this is expected to hold for all critical L-values of motives.\nIn this talk, we will focus on L-functions of number fields. In the first part of the\ntalk, we will discuss the ’critical’ and ’non-critical’ L-values exemplary for the\nRiemann zeta function. Afterwards, we will head on to more general number\nfields and explain our recent joint result with Guido Kings on the algebraicity\nof critical Hecke L-values for totally imaginary fields up to explicit periods.
Shape optimization of convex, rotational symmetric domains for an eigenvalue problem arising in optimal insulation
Dienstag, 30.6.20, 14:15-15:15, virtueller SR 226, HH-10
I will present the main results from my master thesis about shape optimization for an eigenvalue problem arising in optimal insulation. Under the assumption of convexity, the existence of an optimal domain can be proven, but due to the difficulties of approximating convex domains in \(\bmathbb{R}^3\), the constraint to rotational symmetric domains is used to reduce the problem to a two-dimensional setting, allowing the numerical approximation of optimal domains.
Multiscale simulation of bone tissue regeneration
Dienstag, 7.7.20, 14:15-15:15, Raum 226, virtuell Euwe
The development of multiscale methods and technologies is of vital importance not only in the field of mechanobiology, but also in Mechanical and Biological Engineering in general. Therefore, the desire of the M2BE group is to extend these multiscale methodologies to different application areas of engineering, for example, bone tissue engineering.\n\nLarge bone defects represent a clinical challenge for which regenerative therapies and tissue engineering strategies aim at offering treatment alternative to conventional replacement approaches by metallic implants (Pobloth et al., 2018). Additionally, 3D printing technologies provide directly printed porous scaffolds with designed shape, controlled chemistry and interconnected porosity, where bone will regenerate (Valainis et al., 2019). In the last decades a strong effort has been made to develop computer tools to optimize scaffold designs. In fact, computer techniques and mathematically based models are becoming very useful tools for material engineers and biologists to advance the understanding of the scaffold behaviour under different environments (Metz et al., 2020). Bone tissue regeneration using scaffolds in vivo inherently works on two well-differentiated spatio-temporal scales: the tissue level and the pore scaffold.\n\nMathematical modelling of tissue regeneration within scaffolds has been traditionally restricted to one of these scales. I will present a micro-macro mathematical approach for bone tissue engineering regeneration within scaffolds (Sanz-Herrera et al., 2009). At the tissue level, the macroscopic mechanical, diffusive and flow properties are derived by means of the asymptotic homogenization theory (Sanz-Herrera et al., 2008). At the microscopic scale, bone tissue regeneration at the scaffold microsurface is simulated using a certain bone growth model based on a bone remodeling theory (Beaupre et al., 1990). \n\nAdditionally, a new approach will be presented modelling tissue regeneration but restricted to one of these scales (tissue scale). Our latest mechano-driven computational model of bone ingrowth where the local mechanical environment and the regenerative potential of an individual host regulate the mineralization process within a porous scaffold.
Computing discrete invariants of varieties in positive characteristic
Freitag, 10.7.20, 10:30-11:30, virtueller Raum 404
For varieties (smooth projective, say) over fields of\npositive characteristic, we can define discrete invariants that have no\nnatural analogue in characteristic 0. A well-known example is that an\nelliptic curve in characteristic p is either ordinary or supersingular.\nI will first review in general terms how this can be generalized to\narbitrary varieties - there is in fact more than one natural\ngeneralization!\n\nAfter this, I will focus on one particular type of discrete invariant;\nfor abelian varieties this is known under the name 'Ekedahl-Oort type'.\nI will address the question how such discrete invariants can be\nconcretely computed. In particular, I will explain a new method that\nallows to explicitly compute the Ekedahl-Oort type of (the Jacobian of)\na complete intersection curve. For plane curves, a magma implementation\nof this method is now available, so if you have a favourite curve of\nwhich you want to know the E-O type, you can ask me and we can let\nmagma calculate the answer.\n\nAt the end of the talk I will try to say a few words about\ngeneralizations for higher-dimensional projective hypersurfaces. There\nis a simple pattern that emerges, but so far I can only prove that it's\ncorrect for varieties of low dimension.\n
On fluctuations in particle systems and their links to partial differential equations
Dienstag, 14.7.20, 14:15-15:15, virtual SR226 (Euwe)
Partial differential equations often arise as a scaling limit of particle models. We present a link that allows to determine the evolution operator of a class of parabolic equations directly from particle data. As a preliminary step, we will explain how transport coefficients in these equations can be computed from particle simulations. The argument relies on a suitable representation of the governing PDE as gradient flow of the entropy. \n \nSpecifically, we study particle systems and analyse their fluctuations. Fluctuations can often provide useful information on underlying processes, for example in form of fluctuation-dissipation relations. These fluctuations can be described by stochastic differential equations or variational formulations related to large deviations. The link between finite systems and their many-particle limit will be analysed in this formulation.\nThe talk will focus on a class of stochastic equations which are often called equations of fluctuating hydrodynamic and study some relatively simple incarnations of these equations.\n \nThe talk will close with a discussion of analytic results. As will be explained, the (non-)existence theory of the underlying stochastic differential equations is delicate and surprising. Existence for a regularised model will be sketched. \n \nThis is joint work with X. Li, P. Embacher, N. Dirr and C. Reina (numerical work) and Federico Cornalba and Tony Shardlow (analytic work).
Motives of moduli spaces of bundles over a curve
Freitag, 17.7.20, 10:30-11:30, virtueller Raum 404
Following Grothendieck’s vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some properties of this category, I will explain how to define motives of certain algebraic stacks. I will then state and prove a formula the motive of the moduli stack of vector bundles on a smooth projective curve; this formula is compatible with classical computations of invariants of this stack due to Harder, Atiyah--Bott and Behrend--Dhillon. The proof involves rigidifying this stack using Quot and Flag-Quot schemes parametrising Hecke modifications as well as a motivic version of an argument of Laumon and Heinloth on the cohomology of small maps. If there is time, I will discuss how this result can be used to also study motives of moduli space of Higgs bundles. This is joint work with Simon Pepin Lehalleur.
On a stochastic version of transfer operators
Montag, 20.7.20, 16:15-17:15, virtual SR404
For hyperbolic manifolds, there exists a straightforward connection between the spectral and the geometric data. More precisely, the lengths of its closed geodesics and the spectrum of its Laplace operator acting are connected by the Selberg trace formula, that can be considered a sibling of the Poisson summation formula. Selberg trace formula provides the information on the eigenvalues of the Laplace operator, however, completely ignoring its eigenfunctions.\n\nThere exists a method, originated from the classical statistical mechanics, that allows to obtain more information on the eigenfunctions. The method, called the transfer operator approach, involves a construction of a so-called transfer operator from a certain discretisation of the geodesic flow on the manifold. For a modular surface, this transfer operator is ultimately connected to a Gauss map. One can show that the 1-eigenfunctions of this operator correspond via a certain integral transform to the eigenfunctions of the Laplace operator. The integral transform mirrors the Eichler-Shimura-Manin isomorphism.\n\nIn this talk, we try to construct an analogue of the transfer operator, using the Brownian paths on the manifold instead of the geodesics. We obtain an operator, whose 1-eigenfunctions turn out to be the boundary forms of eigenfunctions of the Laplace operator.
Discretizations of the total variation for singular functions
Dienstag, 21.7.20, 14:15-15:15, virtual SR226 (Euwe)
This talk will be about works in collaboration with Corentin Caillaud (CMAP) and Thomas Pock (TU Graz) on the numerical analysis of discretizations of the total variation functional, when solutions are discontinuous. We will discuss error bounds and (an)isotropy properties.
Donnerstag, 23.7.20, 17:00-18:00, Hörsaal II, Albertstr. 23b
K-Motives and Koszul duality
Freitag, 24.7.20, 10:30-11:30, virtueller Raum 404
Koszul duality, as first conceived by Beilinson-Ginzburg-Soergel, is a remarkable symmetry in the representation theory of Langlands dual reductive groups. This talks argues that Koszul duality - in it's most natural form - stems from a duality between equivariant K-motives and monodromic sheaves. I will give a short guide to K-motives and monodromic sheaves and then discuss examples of Koszul duality in increasing difficulty: (1) Tori (2) Toric varieties (3) Reductive groups.
Phase field approaches for tumour growth
Dienstag, 28.7.20, 14:15-15:15, virtual SR226 (Euwe)
Modelling of tumour growth is one of the challenging frontiers of applied mathematics. In the last years, phase field models for tumour growth have been studied intensively. Alike classical free boundary models they use a continuum approach to describe the growth of tumours. However, an advantage to free boundary models is that phase field models allow for topology changes like break up and coalescence. In addition, phase field methods can be used numerically without an explicit tracking of the interface which is necessary for free boundary models. In my talk I will introduce a macroscopic model for tumour growth in which cell-cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. The resulting evolution equation is a Cahn-Hilliard equation taking source and sink terms into account. In addition, nutrient diffusion is incorporated by a coupling to a reaction-diffusion equation. I will show existence, uniqueness and regularity results.\n\nFinally, using optimisation theory and reduced order modelling I will describe how parameter in the system can be estimated in a patient specific way. Properties of solutions will be illustrated with the help of numerical simulations.
Limit spaces of Riemannian manifolds under integral curvature bounds
Donnerstag, 30.7.20, 10:00-11:00, Hörsaal II, Albertstr. 23b
I will discuss some recent work concerning limit spaces of Riemannian manifolds under integral curvature bounds. In particular, I will show how to adapt convergence theory when no non-collapsing assumptions are made. A focus will be the contrast between uniformly bounded Riemann tensor and uniform bounds on the Riemann tensor in \(L^p\), \(p > n/2\). After explaining how to identify a regular subset of a limit space, I will turn to surfaces, where a much richer structure theory is possible.
Bernstein and Spherical Bernstein theorems in codimension 2
Donnerstag, 30.7.20, 11:00-12:00, Hörsaal II, Albertstr. 23b
In this talk we will present some recent progress of the author in joint works with Juergen Jost on the codimension 2 Bernstein theorem for graphs with bounded gradient. We will use the opportunity to describe some geometrical properties of the maximum principle, which is the fundamental tool in the proof of the above theorem. Based on the geometry of the first result and some examples we will give, we will present a very geometrical proof of Solomon's spherical Bernstein theorem in codimension one, and explain how it can be extended to codimension 2. If time allows, we will describe why the techniques cannot work in codimension 3 or higher, where things are more delicate due to the existence of the so called Lawson-Osserman cones, and discuss some open problems and related questions.
Prescribed Curvature Problems in Two Dimensions
Donnerstag, 30.7.20, 14:00-15:00, Raum 404, Ernst-Zermelo-Str. 1
The Willmore Flow of Tori of Revolution
Donnerstag, 30.7.20, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
This is joint work with Anna Dall'Acqua, Adrian Spener and Reiner Schätzle.\n\nIn this talk we study the L^2 gradient flow of the Willmore functional. The initial data we consider are tori of revolution, i.e. closed curves revolved around a fixed axis.\nOur main result is that each such initial datum of energy below 8pi generates a convergent evolution. Its limit is always the torus with least possible Willmore energy - the Clifford torus.\nThe result is sharp in the sense that singular evolutions of tori can be constructed just slightly above the energy level of 8pi.\nThose results are reminiscent of existing results on the Willmore evolution of spheres, where the same energy threshold could be identified, even without revolution symmetry.\nOur proof uses a connection between the Willmore energy of surfaces of revolution and Euler's elastica energy of curves in the hyperbolic plane. We will discuss this connection and see how it can be applied to understand the behavior of the Willmore flow.
Symmetries of deformed c-map metrics and the HK/QK correspondence
Donnerstag, 17.9.20, 16:15-17:15, vSR 404