Genus numbers in families of number fields
Titchmarsh divisor problem for almost primes
In 1976, Fujii obtained an asymptotic formula for an analogue of the Titchmarsh divisor problem for the product of two primes. Recently, Drappeau and Topacogullari studied an asymptotic formula for the Titchmarsh divisor problem with a small shift for almost primes. In this talk, we will discuss an asymptotic formula for the Titchmarsh divisor problem for the product of primes, which is uniform in the shift parameter.
Minimum covolume of quaternionic lattices
We calculate the minimum volume of quaternionic hyperbolic orbifolds and give some explanations how it is related to number theory.
Cohomology of ($\phi$, $\Gamma$)-modules and duality
Cohomology of ($\phi$, $\Gamma$)-modules was studied by Herr, Liu, and Kedlaya-Pottharst-Xiao. Kedlaya-Pottharst-Xiao proved finiteness, duality, and Euler-characteristic formula for cohomology of families of ($\phi$, $\Gamma$)-modules.
In this talk, we will present an alternative proof of finiteness and duality by using analytic geometry introduced by Clausen-Scholze and 6-functor formalism refined by Heyer-Mann. One advantage of this proof is that it can handle families over Banach Qp-algebras that are not topologically of finite type over Qp. If time permits, we will also discuss potential future applications to the representability of the analytic Emerton-Gee stack.
(ARGOS Seminar, Endenicher Allee 60) Localization of D-modules on partial flag varieties
LOCATION: Seminarraum 1.007, Endenicher Allee 60, Center for Mathematics, University of Bonn
The duality map on symplectic moduli spaces of sheaves
Duality defines an involution on the moduli space of slope-stable bundles with trivial determinant on a projective surface X. It extends to a birational involution on the moduli space of Gieseker semistable sheaves. When X is a K3 surface of Picard rank one, we characterize when the duality map is biregular and non-trivial in terms of the Mukai vector. Further analysis of the quotient by this involution yields new (singular) irreducible holomorphic symplectic varieties, with simply connected smooth locus and second Betti number 24. (Joint work in progress with R. Yamagishi.)
(IMPRS seminar) Basics on D-modules
IMPRS seminar on various topics: D-modules
(Oberseminar Arithm. Geom. and Repres. Th.) A survey on Selmer schemes
Uniform bounds for Artin's density (online)
Assuming GRH, with Järviniemi and Sgobba we have obtained very general results on Artin-type problems, and with Shparlinski we have discovered uniform bounds on Artin's density. We will address this and also the computational problem of determining a primitive root modulo p by
testing candidates.
WIQI topology seminar
Seminar webpage: https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html
WIQI topology seminar
Seminar webpage: https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html
(Vorlesung) Habiro Cohomology
(IMPRS Seminar) The Hopf Invariant One Problem and the Adams Spectral Sequence
Twisted Milnor torsion for finite group actions, part 2
The Milnor torsion is an invariant of unitary flat vector bundles on closed odd-dimensional manifolds. Its analytic counterpart was introduced by Ray and Singer and the equality of these torsions is the celebrated Cheeger-Müller theorem. In this talk, I will discuss how to extend the Milnor torsion to certain equivariant flat superconnections (or representations up to homotopy) for finite group actions and its relation to analytic torsion of the twisted de Rham complex.
WIQI topology seminar
Seminar webpage: https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html
(Number theory seminar) Generalized Fermat Equations
The proof of Fermat’s Last Theorem pioneered a new approach to resolving families of ternary Diophantine equations using modularity of residual Galois representations attached to Frey curves. In the case of differing exponents, Darmon gave a framework for resolving generalized Fermat equations in one varying exponent using Frey varieties. In this talk, I will survey the methods and techniques of recent progress on Darmon’s program as well as the challenges that remain in the study of generalized Fermat equations.
(LDT seminar) Kontsevich integrals and configuration spaces
Arithmetic properties of generalised polynomials
We study generalised polynomials, that is, functions that can be expressed using standard algebraic operations and the floor function. Generalised polynomials have long been studied (both explicitly and implicitly) in number theory and dynamics, often using dynamical and ergodic-theoretic methods (especially dynamics on nilmanifolds). These methods enable one to deduce precise information about the average behaviour of generalised polynomials, but allow for complicated behaviour on special sets of density zero. We will present some of the classical results and give examples of various interesting arithmetic and combinatorial behaviour occurring along sets of density zero as well as restrictions to what is possible. We will also state an analogue of Hadamard's quotient theorem for generalised polynomials (the classical version concerns linear recurrences). We will also formulate several open problems. The talk is based on joint work with Jakub Konieczny (Oxford).
The p-adic monodromy theorem
In this talk I will discuss a new proof of the p-adic monodromy theorem in p-adic Hodge theory using the theory of diamonds and Analytic Geometry à la Clausen and Scholze. This new perspective does not use explicitly the classical theory of p-adic differential equations but the geometric aspects of Fargues-Fontaine curves and properties of the Fargues-Fontaine de Rham stack. If time permits I will mention the relation of this theory with Hyodo-Kato cohomology.
This is based on works in progress with Johannes Anschütz, Guido Bosco, Arthur-Cesar Le Bras and Peter Scholze.
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