*Humboldt-Universität zu Berlin *

*Mathem.-Naturwissenschaftliche Fakultät*

*Institut für Mathematik*

Wintersemester 2018/19

Das Forschungsseminar findet mittwochs in der Zeit von 13:00 - 15:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 3.007, statt.

Seminar: Algebraic Geometry an der FU

17.10.2018 | Daniele Agostini (HU Berlin) | |

Title: Algebraic statistics and abelian varieties |
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Abstract: Perhaps surprisingly, it turns out that many techniques
and questions in algebraic geometry have natural applications
in statistics. This is the subject of the relatively new field of Algebraic
Statistics. In my talk, I will give an introduction to Algebraic Statistics for geometers
trying to give an overview of some main ideas in the area. Then, I will
present some recent results (joint with Carlos Amendola) that show how abelian
varieties come up naturally when studying the Gaussian distribution on the
integers.
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24.10.2018 | Ania Otwinowska (HU Berlin) | |

Title: On the Zariski closure of the Hodge locus |
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Abstract: Given a variation of Hodge structures V on a smooth quasi projective complex manifold S, the Hodge locus is the subset of points of S where exceptional Hodge tensors do occur. A famous result of Cattani, Deligne and Kaplan states that this Hodge locus is a countable union of algebraic subvarieties of S. In this talk we study the Zariski closure in S of the union of positive dimensional components of the Hodge locus. This is joint work with B. Klingler.
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31.10.2018 | Alessandro Verra (Uni Roma Tre) | |

Title: Coble cubics, genus 10 Fano threefolds and the theta map |
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Abstract: The talk deals with the relations between two different moduli spaces.
From one side the branch divisor B is considered for the theta map of the moduli of semistable rank r vector bundles with trivial determinant on
a genus 2 curve C. Special attention is payed to the case r = 3. Then B is the sextic dual to the Coble cubic,
the unique cubic hypersurface singular along the Jacobian JC embedded by its 3-theta linear system.
From the other side the moduli space of Fano threefolds X of genus 10 is considered. Since the intermediate
Jacobian of X is JC, for a given genus 2 curve C, the assignement X ---> C defines a rational map f: F ---> M,
M being the moduli space of genus 2 curves.
Relying on a suitable description of the ramification divisor of the theta map, a description of f and of its fibres
is outlined. The main result is that the fibres of f are naturally birational to the Coble cubic defined by JC.
This is a joint work in progress with Daniele Faenzi.
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07.11.2018 | Sam Payne (University of Texas, Austin) | |

Title: Tropical methods for the Strong Maximal Rank Conjecture |
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Abstract: I will present joint work with Dave Jensen using tropical methods
on a chain of loops to prove new cases of the Strong Maximal Rank Conjecture of Aprodu and Farkas.
As time permits, I will also discuss relations to an analogous approach via limit linear series on chains
of genus 1 curves.
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14.11.2018 |
kein Seminar | |

21.11.2018 | Frank-Olaf Schreyer (Universität des Saarlandes) | |

Title: Godeaux surfaces via homological algebra |
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Abstract: In this talk I report on joint work with my student Isabel Stenger
about the construction of numerical Godeaux surfaces via homological algebra. Numerical Godeaux surfaces are
minimal surfaces of general type with K^{2}=1 and p_{g}=q=0. So they are the surfaces of general type with smallest
possible numerical invariants. It is conjectured that there are precisely 5 families of these surfaces, which are
distinguished by their fundamental groups, which is conjectured to be Z/n, for n=1,...,5. Our homological algebra
approach aims for a complete classification of these surfaces.
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28.11.2018 |
kein Seminar | |

05.12.2018 | Ana-Maria Castravet (Université de Versailles) | |

Title: Exceptional collections on moduli spaces of stable rational curves |
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Abstract: A question of Orlov is whether the derived category of the Grothendieck-Knudsen
moduli space of stable, rational curves with n markings admits a full, strong, exceptional collection that is invariant
under the action of the symmetric group S_{n}. I will present an approach towards answering this question.
This is joint work with Jenia Tevelev.
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12.12.2018 | Ziyang Gao (IMJ-PRG Paris) | |

Title: Application of mixed Ax-Schanuel to bounding the number of rational points on curves |
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Abstract: With Philipp Habegger we recently proved a height inequality, using which one can bound the number of rational points on 1-parameter families of curves in terms of the genus, the degree of the number field and the Mordell-Weil rank (but no dependence on the Faltings height). In this talk I will give a blueprint to generalize this method to arbitrary curves. In particular I will focus on: (1) how establishing a criterion for the Betti map to be immersive leads to the desired bound; (2) how to apply mixed Ax-Schanuel to establish such a criterion. This is work in progress, partly joint with Vesselin Dimitrov and Philipp Habegger.
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19.12.2018 | Hanieh Keneshlou (MPI MiS Leipzig) | |

Title: Unirational components of moduli of genus 11 curves with several pencils of degree 6 |
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Abstract: Considering a smooth d-gonal curve C of genus g, one may naturally ask about the existing possible number of pencils of degree d on C.
Motivated by some questions of Michael Kemeny, in this talk we will focus on this question for hexagonal curves of genus 11.
Inside the moduli space of genus 11 curves, we describe a unirational irreducible component of the locus of curves possessing
k mutually independent and type I pencils of degree 6, for the values k=5,...,10.
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09.01.2019 !RAUM 3.011! | 13:15 - 14:15 Frederic Campana (Université de Lorraine) | |

Title: Criterion for algebraicity of foliations, applications |
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Abstract (joint with M. Paun) A foliation F on X, complex projective smooth, is showed to have algebraic leaves if its dual is not pseudo-effective. In the particular case where F has positive minimal slope with respect to some movable class on X, the closures of the leaves are rationally connected. Combined with the existence of Viehweg-Zuo sheaves, this permits to show several versions of the Shafarevich-Viehweg 'hyperbolicity conjecture'.
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14:45 - 15:45 Jochen Heinloth (Uni Essen) | ||

Title: Existence of good moduli spaces for algebraic stacks |
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Abstract: Recently Alper, Hall and Rydh gave general criteria when a moduli problem can locally be described as a quotient and thereby clarified the local structure of algebraic stacks. We report on a joint project with Jarod Alper and Daniel Halpern-Leistner in which we use these results to show general existence and completeness results for good coarse moduli spaces.
In the talk we will focus on two aspects that illustrate how the geometry of algebraic stacks gives a new point of view on classical methods for the construction of moduli spaces. Namely we explain how one-parameter subgroups in automorphism groups allow to formulate a version of Hilbert-Mumford stability in stacks that are not global quotients and sketch how one can reformulate Langton's proof of semistable reduction for coherent sheaves in geometric terms. This allows to apply the method to an interesting class of moduli problems.
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16.01.2019 | Michael Dettweiler (U Bayreuth) | |

Title: Elliptic curve convolution and sheaves with Tannaka group G_{2} |
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Abstract: The talk is about a joint work with Collas, Reiter and Sawin. We study the convolution of a certain subcategory of perverse sheaves on elliptic curves. By the Tannaka formalism, to each such sheaf one has attached an algebraic group: the Tannaka group of the sheaf. Using explicit monodromy calculations, we prove a general theorem under which circumstances sheaves of rank two with 7 singularities have a Tannaka group equal to the simple algebraic group G_{2}.
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23.01.2019 !RAUM 3.011! | 13:15 - 14:15 Rahul Pandharipande (ETH Zürich) | |

Title: Double ramification cycles for target varieties |
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Abstract: A basic question in the theory of algebraic curves is whether
a divisor represents the zeros and poles of a rational function.
An explicit solution in terms of periods was given by the work of Abel
and Jacobi in the 19th century. In the past few years, a different
approach to the question has been pursued: what is the class
in the moduli of pointed curves of the locus of such divisors? The
answer in Gromov-Witten theory is given by Pixton's formula
for the double ramification cycle. I will discuss recent work
with F. Janda, A. Pixton, and D. Zvonkine which considers
double ramification cycles for target varieties X (where Pixton's
original question is viewed as the X=point case). I will also
discuss the associated relations studied by Y. Bae.
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14:45 - 15:45 Vadim Vologodsky (HSE Moscow) | ||

Title: On the periodic topological cyclic homology of (DG) algebras in characteristic p |
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Abstract: We prove that the periodic topological cyclic homology of a smooth proper DG
algebra over F_{p} is isomorphic to the periodic (algebraic) cyclic homology of a lifting of the algebra over Z_{p}.
This is a joint work with Alexander Petrov.
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24.01.2019 !RAUM 3.006! | 15:00 - 17:00 Carel Faber (Utrecht University) | |

Title: Ternary quartics and modular forms of genus three |
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Abstract: I will discuss the precise relation between concomitants
for plane quartics and (Teichmüller and Siegel) modular forms of genus three.
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30.01.2019 | Philipp Habegger (Uni Basel) | |

Title: Equidistribution of roots of unity and the Mahler measure |
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Abstract: It is a classical fact that roots of unity of given order
n equidistribute around the unit circle as n\rightarrow\infty. In other words, the average of a
continuous, real-valued function f on C taken over roots of unity of order n converges to the
normalized integral of f over the unit circle as n\rightarrow\infty. Baker, Ih, and Rumely extended
this to the test functions \log|P(\cdot)| where P is a univariate polynomial with algebraic coefficients.
I will discuss work in progress with Vesselin Dimitrov where we allow a certain class of multivariate polynomials.
In particular, f may have logarithmic singularities along a divisor. The resulting integral is the Mahler
measure of P. If time permits I will also discuss connections to work of Lind, Schmidt, and Verbitskiy on
dynamical systems of algebraic origin.
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06.02.2019 |
kein Seminar | |

13.02.2019 !RAUM 3.011! | 13:15 - 14:15 Lynn Chua (UC Berkeley) | |

Title: Schottky Algorithms: Classical meets Tropical |
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Abstract: We present a new perspective on the Schottky problem that links
numerical computing with tropical geometry. The task is to decide
whether a symmetric matrix defines a Jacobian, and, if so, to compute
the curve and its canonical embedding. We offer solutions and their
implementations in genus four, both classically and tropically. The
locus of cographic matroids arises from tropicalizing the
Schottky-Igusa modular form. This is joint work with Mario Kummer and
Bernd Sturmfels.
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14:30 - 15:30 Francesco Galuppi (MPI Leipzig) | ||

Title: The rough path signature variety |
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Abstract: Signature tensors are a useful tool in the study of paths X. When X runs among a given class of path (e.g. polynomial, piecewise linear, etc), the signature of X parametrizes an algebraic variety. The geometry of this variety reflects some of the properties of the chosen class of path. The most important example is the class of rough paths, that are widely studied in stochastic analysis. Their signature variety presents many similarities to the Veronese variety, and we'll illustrate the first nice results, as well as some open questions.
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