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Stefan Müller-Stach

Publications and source records attributed to Stefan Müller-Stach.

At least 19 recordsLinked to original sources
arXiv2026

The Code of Mathematics

This text summarizes and expands the content of a general audience talk given in 2018 at the University of Mainz. Motivated by recent developments in dependent type theory and infinity category theory, it presents a history of ideas around the concepts of truth, proof, equality, and equivalence as well as their relation to human thought. We describe a few selected ideas of Platon, Aristoteles, Leibniz, Kant, Frege and others and then pass to the results of Gödel and Tarski about incompleteness, undecidability and truth in deductive systems and their semantic models. The main focus of this text, however, is the development of dependent type theory through the work of Per Martin--Löf and recent developments in homotopy type theory, i.e., the univalent foundations program of Vladimir Voevodsky and others. These theories allow the notion of identity types, which gives new possibilities for handling equality, symmetry, equivalence and isomorphisms in a conceptual way. Martin--Löf type theories have semantic models in (infinity,1)-categories, which are related to simplicial localizations of Quillen model categories. The interaction of type theory with infinity category theory is a new paradigm for a structural view on mathematics which is superior to set theory. It also supports the recent emerging trend for computer assisted proofs in mathematics and verification of algorithms and software in computer science.

math.HO↗
arXiv2023

Max Dehn, Axel Thue, and the Undecidable

This is a short essay on the roles of Max Dehn and Axel Thue in the formulation of the word problem for (semi)groups, and the story of the proofs showing that the word problem is undecidable.

math.HO↗
arXiv2020

The unequal mass sunrise integral expressed through iterated integrals on $\overline{\mathcal M}_{1,3}$

We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter $\varepsilon$. In order to do so, we transform the system of differential equations for the master integrals to an $\varepsilon$-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space ${\mathcal M}_{1,3}$ of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on $\overline{\mathcal M}_{1,3}$. On the hypersurface $τ=\mbox{const}$ our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.

hep-th↗
arXiv2019

Special subvarieties in Mumford-Tate varieties

Let X be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain D by a discrete subgroup. Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety Y in X (of Shimura type), and formulate necessary criteria for Y to be special. Our method consists in looking at finitely many compactified special curves C_i in Y, and testing whether the inclusion of the union of all C_i in Y satisfies certain properties. One of them is the so-called relative proportionality condition. In this paper, we give a new formulation of this numerical criterion in the case of Mumford-Tate varieties X. In this way, we give necessary and sufficient criteria for a subvariety Y of X to be a special subvariety in the sense of the Andre-Oort conjecture. We discuss in detail the important case where X=A_g, the moduli space of principally polarized abelian varieties.

math.AG↗
arXiv2018

Specialization of cycles and the K-theory elevator

A general specialization map is constructed for higher Chow groups and used to prove a "going-up" theorem for algebraic cycles and their regulators. The results are applied to study the degeneration of the modified diagonal cycle of Gross and Schoen, and of the coordinate symbol on a genus-2 curve.

math.AG↗
arXiv2017

Richard Dedekind: Style and Influence

This text is based on an invited talk at the Dedekind Symposium at Braunschweig in October 2016. It summarizes views from my recent commented edition of Dedekinds two books on the foundations of mathematics.

math.HO↗
arXiv2014

Motives of graph hypersurfaces with torus operations

We investigate graph hypersurfaces and study conditions under which graph hypersurfaces admit algebraic torus operations. This leads in principle to a computation of graph motives using the theorem of Bialynicki-Birula, provided one knows the fixed point loci in a resolution of singularities.

math.AG↗
arXiv2014

Degeneration of the modified diagonal cycle

In this note, we revisit the modified diagonal cycle of Gross and Schoen. We look at degenerations of this cycle, induced by a degeneration of the curve C, and explain how the specialization map with respect to the central fiber produces a higher Chow cycle. When C is non-hyperelliptic of genus three, and degenerates to a nodal curve, the degeneration of the cycle corresponds to an indecomposable higher Chow cycle on a curve of genus two.

math.AG↗
arXiv2014

On the cohomology groups of local systems over Hilbert modular varieties via Higgs bundles

Let $X$ be a Hilbert modular variety and $\mathbb{V}$ a non-trivial local system over $X$ with infinite monodromy. In this paper we study Saito's mixed Hodge structure (MHS) on the cohomology group $H^k(X,\mathbb{V})$ using the method of Higgs bundles. Among other results we prove the Eichler-Shimura isomorphism, give a dimension formula for the Hodge numbers and show that the mixed Hodge structure is split over $\mathbb{R}$. These results are analogous to Matsushima-Shimura [Annals of Mathematics 78, 1963] in the cocompact case and complement the results in Freitag [Book: Hilbert modular forms, Springer-Verlag, Berlin, 1990] for constant coefficients.

math.AG↗
arXiv2014

What is a period ?

This is an essay about Kontsevich-Zagier periods and their relation to mixed motives a la Madhav Nori.

math.NT↗
arXiv2014

On the relation between Nori Motives and Kontsevich Periods

We show that the spectrum of Kontsevich's algebra of formal periods is a torsor under the motivic Galois group for mixed motives over the rational numbers. This assertion is stated without proof by Kontsevich and originally due to Nori. In a series of appendices, we also provide the necessary details on Nori's category of motives.

math.AG↗
arXiv2013

Hodge numbers for the cohomology of Calabi-Yau type local systems

We use Higgs cohomology to determine the Hodge numbers of the first intersection cohomology group of a local system V arising from the third direct image of a family of Calabi-Yau 3-folds over a smooth, quasi-projective curve. We give applications to Rhode's families of Calabi-Yau 3-folds without MUM.

math.AG↗
arXiv2013

A characterization of special subvarieties in orthogonal Shimura varieties

Let $Y$ be a subvariety contained in a smooth Mumford compactification of an orthogonal Shimura variety $M \subset A_g$, where $A_g$ is the moduli space of principally polarized abelian varieties of dimension $g$ with some level structure, such that $Y$ intersects the boundary of $A_g$ transversally. Then we give necessary and sufficient conditions of André-Oort type for $Y$ itself being the compactification of a special subvariety $Y^0 \subset M$

math.AG↗
arXiv2013

Picard-Fuchs equations for Feynman integrals

We present a systematic method to derive an ordinary differential equation for any Feynman integral, where the differentiation is with respect to an external variable. The resulting differential equation is of Fuchsian type. The method can be used within fixed integer space-time dimensions as well as within dimensional regularisation. We show that finding the differential equation is equivalent to solving a linear system of equations. We observe interesting factorisation properties of the D-dimensional Picard-Fuchs operator when D is specialised to integer dimensions.

hep-ph↗
arXiv2012

A second-order differential equation for the two-loop sunrise graph with arbitrary masses

We derive a second-order differential equation for the two-loop sunrise graph in two dimensions with arbitrary masses. The differential equation is obtained by viewing the Feynman integral as a period of a variation of a mixed Hodge structure, where the variation is with respect to the external momentum squared. The fibre is the complement of an elliptic curve. From the fact that the first cohomology group of this elliptic curve is two-dimensional we obtain a second-order differential equation. This is an improvement compared to the usual way of deriving differential equations: Integration-by-parts identities lead only to a coupled system of four first-order differential equations.

hep-ph↗
arXiv2012

Abelian varieties and theta functions associated to compact Riemannian manifolds; constructions inspired by superstring theory

We look into a construction of principal abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to place it in a broader framework. This is related to Weil intermediate Jacobians but it also suggests to associate abelian varieties to polarized even weight Hodge structures. The latter construction can also be explained in terms of algebraic groups which might be useful from the point of view of Tannakian categories. The constructions depend on moduli much as in Teichmüller theory although the period maps in general are only real analytic. One of the nice features is how the index for certain differential operators canonically associated to the geometry of the situation (spin structure, complex structure etc.) leads to integrality of skew pairings on the topological K-group (coming from the Index Theorem) which then serves as a polarization for the jacobian.

math.AG↗