Oberseminar "Algebra und Theoretische Mathematik"
Zeit und Ort:
Montagstermine in 39 - 733 (Reichenhainer Straße 39), 11.30 Uhr - 13.00 Uhr,Donnerstagstermine in 39 - 733 (Reichenhainer Straße 39), 9.15 Uhr - 10.45 Uhr,
Die Vorträge dauern 60 Minuten, anschließend bleibt etwas Zeit für Diskussionen.
Aommersemester 2019:
Datum | Sprecher | Titel und Zusammenfassung |
31.01.2019 | Vladimir Lazić, Universität des Saarlandes | On Generalized Abundance Abstract |
11.04.2019 | Martin Schwald, Universität Duisburg-Essen |
Titel Abstract |
Vergangene Semester:
Datum | Sprecher | Titel und Zusammenfassung |
09.06.2016 | Damian Brotbek, Strasbourg | On the hyperbolicity of general hypersurfaces A complex manifold is said to be (Brody) hyperbolic if it doesn't contain any entire curves. Kobayashi conjectured in 1970 that a general hypersurface of sufficiently high degree in projective space is hyperbolic. This statement was only settled recently by a work of Siu. In this talk, after a gentle introduction to hyperbolic manifolds, we outline a new proof of this conjecture of Kobayashi. |
11.08.2016 | Diletta Martinelli, Imperial College | What are minimal models? How many are there? Algebraic geometry studies the structure of algebraic varieties, solutions of a system of polynomial equations in an affine or projective space. The final goal of the subject is to achieve a complete classification of algebraic varieties up to some kind of equivalence relations. I will explain in the talk why one of the most natural choice is the classification up to birational equivalence (two varieties are birational if they are isomorphic up to some subvarieties of smaller dimension). Then the first step of the classification is to find a representative inside the birational equivalence class that is in some sense simpler than the others, we call this variety a minimal model. The Minimal Model Program (MMP) is an algorithm that establishes a series of steps to find the minimal model. A very natural question is whether the minimal model is unique, and if not how many minimal models does a variety admit and how are they related. After describing the general ideas of the MMP I will focus on these last questions and explain that for a special class of varieties the number of minimal model is finite and that in some cases it is possible to bound this number using topological information. These results are part of my PhD thesis and of a recent joint work with Stefan Schreieder and Luca Tasin. |
24.10.2016 | Christian Rose, TU Chemnitz | The first Betti number and the Kato class on compact Riemannian manifolds The starting point of this talk will be the dependence of the first Betti number of a compact Riemannian manifold on the so-called Kato-condition on the negative part of the Ricci curvature. This kind of condition can be controlled as soon as one knows something about the short time behavior of the heat semigroup. We show that L^p-type curvature conditions lead to upper bounds on the heat kernel. Furthermore, we discuss that in fact the Kato-condition on the negative part of the curvature is enough to obtain such bounds. |
03.11.2016 | Fabrizio Catanese, Bayreuth (im CMC) | Configurations of lines and interesting algebraic surfaces. Many important questions in the theory of surfaces and in algebraic geometry have been solved thanks to explicit constructions of algebraic surfaces as abelian coverings branched over special configurations of lines. After recalling the classical configurations (Pappus, Desargues, Fano, Hesse) and some new ones, I shall describe simple equations for such surfaces, as the Fermat, and Hirzebruch-Kummer coverings. As the configuration of lines becomes special some interesting geometry shows up, as in the case of the six lines of a complete quadrangle, related to the Del Pezzo surface of degree 5 and its icosahedral symmetry. After mentioning many important such examples and applications, by several authors, I shall concentrate on a recent simple series of such surfaces, studied in my joint work with Ingrid Bauer and Michael Dettweiler, discussing new results and quite general open questions, concerning rigid manifolds, and projective classifying spaces. |
10.11.2016 | Bernd Sturmfels, UC Berkeley and MPI Leipzig time: 11.30 h, room: 2/B102 |
Nearest Point on Toric Varieties We determine the Euclidean distance degree of a projective toric variety. This is the intrinsic algebraic complexity of a ubiquitous optimization problem, namely to compute the point on a real toric variety that is closest to a given data point. Our results generalize the formula due to Matsui and Takeuchi for the degree of the A-discriminant in terms of Euler obstructions. This is joint work with Martin Helmer. |
17.11.2016 | Giovanni Mongardi, Bologna | Chow rings of holomorphic symplectic manifolds: known structure, conjectures and speculations In this talk I will present some results on holomorphic symplectic manifolds and their Chow rings, in particular i will outline possible approaches to solve a series of conjectures made by Voisin. |
17.11.2016 | Joe Kileel, UC Berkeley | Using Algebraic Geometry for Computer Vision In computer vision, 3D reconstruction is a fundamental task: starting from photographs of a world scene, taken by cameras with unknown positions and orientations, how can we best create a 3D model of that world scene? Algorithms that do this built Street View (Google) and are instrumental in autonomous robotics. In 2004, David Nister (Tesla) used Grobner bases to build a solver for robust reconstruction given just two photographs. This is a key routine in much larger-scale reconstructions today. In this talk, I will discuss reconstruction given three photographs, where efforts to replicate Nister have so far proven elusive. My approach relies on applied algebraic geometry. In particular, I shall introduce an algebraic variety whose points are 3x3x3 tensors in correspondence with configurations of three calibrated cameras. Special linear sections of this variety recover camera configurations from image data. The main result is the determination of the algebraic degree of minimal problems for this recovery. These comprise interesting enumerative geometry problems; the solution is by way of homotopy continuation calculations. |
16.01.2017 | Vladimir Shikhman, Chemnitz | On the local representation of piecewise smooth equations as a Lipschitz manifold We study systems of equations given by piecewise differentiable functions The focus is on the representability of their solution set locally as a Lipschitz manifold. For that, nonsmooth versions of inverse function theorems are applied. It turns out that their applicability depends on the choice of a particular basis. To overcome this obstacle we introduce a strong full-rank assumption (SFRA) in terms of Clarkeʼs generalized Jacobians. The SFRA claims the existence of a basis in which Clarkeʼs inverse function theorem can be applied. Aiming at a characterization of SFRA, we consider also a full-rank assumption (FRA). The FRA insures the full rank of all matrices from the Clarkeʼs generalized Jacobian. The talk is devoted to the conjectured equivalence of SFRA and FRA. |
19.01.2017 | Stefan Müller-Stach, Mainz (im CMC) | Von Dedekind zur modernen Arithmetik: Rekursionstheorie und Perioden Dedekind (1831-1916) hat neben seinen vielen anderen einflussreichen Werken die Rekursionstheorie begründet. Von dort kann man eine Verbindung zur modernen Logik und zur Arithmetischen Geometrie und insbesondere zu berechenbaren reellen Zahlen und Perioden herstellen. Diese historische Reise versuchen wir im Vortrag darzustellen. |
30.01.2017 | Paolo Stellari, Milano | A derived category approach to some moduli spaces on cubic threefolds and fourfolds. We exploit the homological properties and the geometric meaning of Kuznetsov's semiorthogonal decomposition of the derived categories of cubic fourfolds (and threefolds) to study the (birational) geometry of some interesting moduli spaces on such varieties. We will start working out the very instructive example of the moduli space of stable aCM bundles of a given rank on a cubic threefold. Then we will discuss our main result concerning the case of generalized twisted cubics on cubic fourfolds not containing a plane. We will show that we can recover the picture by Lehn-Lehn-Sorger-van Straten in terms of moduli spaces of (weakly) stable sheaves/complexes. This is joint work with M. Lahoz, M. Lehn, and E. Macri'. |
02.02.2017 | Simon Brandhorst, Leibniz Universität Hannover | Minimal Salem numbers on supersingular K3 surfaces The entropy of a surface automorphism is either zero or the logarithm of a Salem number, that is an algebraic integer $lambda>1$ which is conjugate to $1/lambda$ and all whose other conjugates lie on the unit circle. In the case of a complex K3 surface McMullen gave a strategy to decide whether a given Salem number arises in this way. To do this he combined methods from linear programming, number fields, lattice theory and the Torelli theorems. In this talk we extend these methods to automorphisms of supersingular K3 surfaces using the crystalline Torelli theorems and apply them in the case of characteristic $5$. This is joint work with Víctor González-Alonso. |
23.03.2017 | Ziyu Zhang, Leibniz Universität Hannover | Holomorphic symplectic manifolds among Bridgeland moduli spaces We consider moduli spaces of semistable complexes on a projective K3 surface with respect to generic Bridgeland stability conditions. Similar to the sheaf case, the smooth ones among them are holomorphic symplectic manifolds, and the 10-dimensional singular ones admit symplectic resolutions. I will explain why these examples of holomorphic symplectic manifolds are all deformation equivalent to the known ones. By generalizing the prominent work of Bayer and Macri, we can also study the birational geometry of the 10-dimensional singular moduli spaces via wall-crossing on the stability manifold. This is a joint work with C.Meachan. |
03.04.2017 | Carsten Liese, Leibniz Universität Hannover | The KSBA compactification of the moduli space of degree 2 K3 pairs: a toroidal interpretation Work of Gross, Hacking, Keel and Siebert shows that the Gross-Siebert reconstruction algorithm provides a partial toroidal compactification of the moduli space of polarized K3 surfaces for any genus. The construction comes with a family $mathfrak{X}to bar{mathbb{P}}^g$ over a partial toroidal compactification $bar{mathbb{P}}^g$ of a subset $mathbb{P}^g$ of the Kollár-Shepherd-Barron moduli space of stable K3 pairs $M_{SP}$. A conjecture of Keel says that $mathfrak{X}to bar{mathbb{P}}^g$ extends to a compactification of ${mathbb{P}^g}$ and in particular, all surfaces in the boundary of $M_{SP} appear as fibres of $mathfrak{X}to bar{mathbb{P}}^g$. In the genus $2$ case, $M_{SP}$ is known by work of Laza. In this talk, I check the prediction of Keel's conjecture and show that all degenerate $K3$ surfaces in the boundary of $M_{SP}$ appear as fibres of $mathfrak{X}to mathbb{P}^g$. I do not assume familiarity with any part of the subject. |
06.04.2017 | Maxim Smirnov, Augsburg | On quantum cohomology of isotropic Grassmannians Dubrovin’s conjecture (ICM 1998) predicts an intriguing relation between the quantum cohomology ring of a smooth projective variety X and its derived category of coherent sheaves. I will explain some aspects of this story taking symplectic isotropic Grassmannians IG(m,2n) as the main example and stress the importance of the big quantum cohomology in the formulation of the conjecture. If time permits I will exhibit a relation between the quantum cohomology of IG(m,2n) and unfoldings of isolated hypersurface singularities, and its counterpart for the derived category of coherent sheaves on IG(m,2n). The talk is based on joint works, some finished and some still in progress, with A. J. Cruz Morales, S. Galkin, A. Mellit, N.Perrin, and A. Kuznetsov. |
08.06.2017 | Emanuel Scheidegger, Freiburg | On the hemisphere partition function Starting with a variation of GIT quotients with potential we will review how to associate a function on the K-theory of the associated derived categories, called the hemisphere partition function, which conjecturally defines a Bridgeland stability condition. We will discuss various (conjectural) properties of this function and relate it to known functions and differential equations. |
29.06.2017 | Ferran Dachs-Cadefau, Halle | Computing jumping numbers in higher dimensions Multiplier ideals and jumping numbers are invariants that encode relevant information about the structure of the ideal to which they are associated. A first part of this talk will be devoted to introduce some basics about multiplier ideals in the case of 2-dimensional local rings. We will also present the relations with other invariants. In the second part, we present a formula to compute the multiplicity of jumping numbers of an m-primary ideal in a 2-dimensional local ring with rational singularities. This formula leads to a simple way to detect whether a given rational number is a jumping number. Another consequence of the formula is that it allows us to give an explicit rational expression for the Poincar é series of the multiplier ideals introduced by Galindo and Monserrat in 2010. This Poincar é series encodes in a unified way the jumping numbers and its corresponding multiplicities. In the third part of the talk, we will introduce some results for the multiplier ideals in the higher-dimensional case. For this, we introduce the notion of π-antieffective divisors, a generalization of antinef divisors to higher dimensions. Using these divisors, we present a way to find a small subset of the ‘classical’ candidate jumping numbers of an ideal, containing all the jumping numbers. Moreover, many of these numbers are automatically jumping numbers, and in many other cases, it can be easily checked. The second part of this talk is a joint work with Maria Alberich Carramiñana, Josep Àlvarez Montaner and Víctor González Alonso, while the third is a joint work with Hans Baumers. |
06.07.2017 | Andrés Reyes, Bogotá | Topology and the phase transition of the Ising model After briefly reviewing the Shale-Stinespring theorem, I will discuss the relevance of the choice of complex structure for a topological characterization of quantum phase transitions in systems that are described in terms of quasi-free states of the CAR (i.e. fermionic) algebra. Making use of the transfer matrix method, I will then show how this interpretation can be carried over to the case of the thermodynamic phase transition of the two dimensional classical Ising model. |
12.07.2017 | Martin Ulirsch, MPI Leipzig | Tropical and logarithmic moduli theory The moduli space of tropical curves (and its variants) are some of the most-studied objects in tropical geometry. So far this moduli space has only been considered as an essentially set-theoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not define a universal curve (at least in the positive genus case). The classical work of Deligne-Knudsen-Mumford has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces. In this talk I am going to give an introduction to these fascinating moduli spaces and report on joint work with Renzo Cavalieri, Melody Chan, and Jonathan Wise, where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this $2$-categorical framework one can give a natural interpretation of the forgetful morphism as a universal curve. Our theory naturally arises as a combinatorial shadow of the F. Kato's moduli stack of logarithmically smooth curves, which we can now think of as a hybrid of both the moduli stack of stable algebraic curves and our tropical moduli stack. The process of tropicalization connects these two worlds via a natural smooth and surjective tropicalization morphism. |
13.07.2017 | Patrick Graf, Bayreuth | Finite quotients of complex tori Let $X$ be a compact Kähler threefold with canonical singularities and vanishing first Chern class. I will show that if the second orbifold Chern class of $X$ intersects some Kähler form trivially, then $X$ admits a quasi-étale (i.e. étale in codimension one) cover by a complex torus. This result generalizes a theorem of Shepherd-Barron and Wilson for projective varieties. It should be seen as complementing the structure theory of Kähler threefolds (aka Minimal Model Program). Part of the talk is devoted to explaining the notion of second orbifold Chern class for complex spaces with canonical singularities, since this topic has not been treated in the literature up to now. If time permits, I will also discuss possible generalizations to klt singularities and to higher dimensions. Joint with Tim Kirschner (Essen). |
01.03.2018 | Carolin Peternell, Leibniz Universität Hannover | Birational Models for Moduli of Quartic Rational Curves Ch. Lehn, M. Lehn, Ch. Sorger and D. van Straten constructed a family of irreducible holomorphic symplectic manifolds via a contraction of the compactified moduli space $M_3(Y)$ of rational curves of degree 3 on a smooth cubic fourfold Y. This suggests that also the moduli space $M_4(Y)$ could be connected to a family of holomorphic symplectic manifolds. In order to understand $M_4(Y)$, we study the Hilbert scheme of curves on $P^4$ with Hilbert polynomial 4n+1, a moduli space of Kronecker modules, a moduli space of semi-stable sheaves and their relations. |
30.04.2018 | Mathew Dawes, Riemann Fellow, Leibniz Universität Hannover Note that this talk will exceptionally take place in W037 from 11-12! |
Modular forms and the birational geometry of modular varieties Orthogonal modular varieties are arithmetic quotients of a Hermitian symmetric domain of type IV by a subgroup of O(2,n). They often arise in connection with moduli; for example, the moduli of K3 surfaces or the moduli of irreducible symplectic manifolds. Many aspects of the geometry of orthogonal modular varieties can be understood by studying modular forms for O(2,n); in particular, their Kodaira dimension. This problem is particularly interesting in small dimensions as, in addition to encountering questions on modular forms, one also needs a detailed understanding of singularities in modular varieties. I shall discuss my work on this problem for modular varieties associated with the moduli of irreducible symplectic manifolds of generalised Kummer type. If time permits, I shall discuss current work to generalise these results and explain how my methods shed some light on automorphisms and certain families of lattice polarised varieties. |
24.05.2018 | Milena Wrobel, MPI MiS Leipzig | On Cox rings of T-varieties An important invariant of an algebraic variety is its Cox ring. We consider structural properties of these rings and give for varieties with torus action an explicit description via generators and relations. Based on this description we look at the question of iterability of Cox rings and give classification results for smooth Fano varieties. |
28.05.2018 | Thomas Jahn, TU Chemnitz Note that this talk will take place in 39-733 from 11-12! |
Uniqueness of circumcenters in generalized Minkowski spaces We study an extension of a familiar concept of Euclidean plane geometry: circumballs. After an introduction to our setting of generalized Minkowski spaces, a characterization of uniqueness of circumcenters in terms of the boundary structure of balls will be presented. Finally, we discuss an extension involving the dimension of the set of circumcenters. This is joint work with Bernardo González Merino and Christian Richter. |
31.05.2018 | Emre Sertöz, MPI MiS Leipzig | Computing periods of hypersurfaces The periods of a smooth complex projective variety X are complex numbers, typically expressed as integrals, which give an explicit representation of the Hodge structure on the cohomology of X. Although they provide great insight, periods are often very hard to compute. In the past 20 years, an algorithm for computing the periods existed only for plane curves. We will give a different algorithm which can compute the periods of any smooth projective hypersurface and can do so with much higher precision. As an application, we will demonstrate how to reliably guess the Picard rank of quartic K3 surfaces and the Hodge rank of cubic fourfolds from their periods. |
07.06.2018 | Thomas Krämer, Humboldt-Universität zu Berlin | A categorification of Kashiwara's index formula On any commutative group variety, the category of holonomic D-modules has a natural Tannakian description with respect to the convolution product. While for affine varieties this has been studied by Gabber, Loeser, Sabbah, Katz, Dettweiler and others from many perspectives, much less is known in the case of abelian varieties. The talk will discuss a microlocal categorification of Kashiwara's index formula that gives a dictionary between characteristic cycles of holonomic D-modules and Weyl group orbits of weights for the corresponding reductive Tannakian Galois groups. While most previous results required the characteristic cycles to be reduced, we will explain how to get rid of this assumption and what this means for the geometry and singularities of subvarieties in abelian varieties. |
11.06.2018 | Caterina Cejp, Leibniz Universität Hannover Note that this talk will exceptionally take place in 39-633 from 15.30-16.30! |
Die Methode der Borel Invarianten für das invariante Hilbertschema |
14.06.2018 | Ronan Terpereau, Université de Bourgogne, Dijon | Automorphism groups of P1-bundles over rational surfaces.
In this talk I will explain 1) how to construct a very simple moduli space (=a projective space) for the non-decomposable P1-bundles with no "jumping fibres" over Hirzebruch surfaces, and 2) how to deduce the classification of all the P1-bundles over rational surfaces whose automorphism group G is maximal in the sense that every G-equivariant birational map to another P1-bundle is necessarily a G-isomorphism. (This is a joint work with Jérémy Blanc and Andrea Fanelli.) |
21.06.2018 | Bernd Schober, Leibniz Universität Hannover | Resolution of Surface singularities
Nowadays, there are several quite accessible accounts for Hironaka's theorem of resolution of singularities for algebraic varieties over a field of characteristic zero. In contrast to this, there exist only results in small dimensions for the case of positive or mixed characteristic. Based on a result by Hironaka for excellent hypersurfaces of dimension two Cossart, Jannsen and Saito (CJS) gave a proof for resolution of singularities of two dimensional excellent schemes via blowing ups in regular centers. More precisely, they introduced a canonical strategy for the choice of the centers and showed by contradiction that the constructed sequence of blowing ups can not be infinite. In order to avoid technical details as good as possible, I will consider only the case of hypersurfaces (i.e., varieties defined by the vanishing of a single polynomial) in my talk. First, I will give an introduction to the problem of resolution of singularities and the strategy of CJS. After that I will explain how polyhedra can be used to obtain an invariant that captures the strict improvement of the singularity along the CJS process. Therefore, the invariant provides the basis for a direct proof of the result by CJS. The constructions involve Hironaka’s characteristic polyhedron which is a certain minimal projection of the Newton polyhedron. Hence, the ideas for the very technical proof of the improvement, can be explained by drawing rather simple pictures. This is joint work with Vincent Cossart. |
26.06.2018 | Henri Guenancia, Université Paul Sabatier Note that this talk will exceptionally take place in W065 from 13:45 -14:45! |
Bochner principle on singular varieties The well-known Bochner principle states that on any compact Kähler manifold with trivial first Chern class, any global holomorphic tensor is parallel with respect to any Ricci flat Kähler metric (provided by Yau's theorem), thus establishing a 1-1 correspondence between holomorphic tensors and invariant vectors under the so-called holonomy representation. I will explain a singular version of this statement for projective varieties with klt singularities and trivial first Chern class. This is joint work with Daniel Greb and Stefan Kebekus. |
28.06.2018 | Tim Kirschner, Universität Duisburg-Essen | Deformations of hyperkähler twistor spaces I present some novel results concerning the deformation theory of twistor spaces of hyperkähler type. First and foremost, I show that the (local) deformations of such twistor spaces are unobstructed—a result which is new even in the K3 surface case. Time permitting, I touch upon the (global) structure of the moduli of hyperkähler twistor spaces. |
05.07.2018 | Anna-Laura Sattelberger, Universität Augsburg | Topological Computation of Stokes Data of Weighted Projective Lines The quantum connection of (weighted) projective lines P(a, b) defines a system of linear differential equations, which have an irregular singularity at ∞. According to a conjecture of B. Dubrovin, the Stokes matrices can be obtained by the Gram matrix of an exceptional collection of the bounded derived category of coherent sheaves on P(a, b). By mirror symmetry, the quantum connection of P(a, b) is closely related to the Fourier transform of the Gauß–Manin connection of a Landau–Ginzburg model (X, f) of P(a, b). We apply the results of A. D’Agnolo, G. Morando, M. Hien and C. Sabbah from 2017 to the perverse sheaf Rf∗C[1] in order to obtain the Stokes matrices in a purely topological way and compare them to the Gram matrix that we obtain by Dubrovin’s conjecture. |
03.09.2018 | Makiko Mase, Tokyo Metropolitan University | Polytope/Lattice dualities among families of K3 surfaces associated to strange duality of singularities As a generalisation of Arnold’s strange duality, Ebeling and Takahashi found a strange duality of invertible polynomials. We consider the case of bimodal singularities, which is studied by Ebeling and Ploog, and study the families of K3 surfaces associated to them. Our question is whether or not the strange duality of bimodal singularities can be explained in terms of the families. As a conclusion, we can prove that the families are polytope dual, and that with some exceptions, they are lattice dual shown by combining toric geometry and lattice theory. |
24.01.2019 | Konstantin Jakob, Universität Duisburg-Essen |
Rigid Local Systems and the Geometric Langlands Correspondence Classically, rigid local systems arise as solution sheaves of certain regular singular differential equations in the complex domain. They have been studied in depth by N. Katz who devised a way of constructing these local systems from systems of rank one by means of a convolution operation. D. Arinkin has generalized this to allow for the construction of arbitrary rigid connections (with possibly irregular singularities). However, this construction has several disadvantages, for example it only works for GL_n-local systems. In this talk I will report on work in progress (joint with Z. Yun) on another approach to construct rigid local systems for more general reductive groups via the geometric Langlands correspondence based on ideas of Heinloth, Ngô & Yun. |
29.01.2019 | Robert Laterveer, IRMA, Strasbourg |
Algebraic cycles and Verra fourfolds A Verra fourfold is a smooth projective complex variety defined as double cover of P^2x P^2 branched along a divisor of bidegree (2,2). These varieties are similar to cubic fourfolds in several ways (Hodge theory, relation to hyperkaehler fourfolds, derived categories). Inspired by these multiple analogies, in this talk I will consider the Chow ring of a Verra fourfold. Among other things, I will show that the multiplicative structure of this Chow ring has a curious K3-like property. |
31.01.2019 | Vladimir Lazić, Universität des Saarlandes | On Generalized Abundance Abstract |