Max-Planck-Gesellschaft

2018

Topological periodic homology

2018, Max Planck Institute for Mathematics

Thomas; Nikolaus

It is a classical algebraic problem to study solutions of polynomial equations. Since this can be very hard, people have invented certain algebraic invariants, called cohomology theories, to get a qualitative picture. Crystalline cohomology is one of these cohomology theories which is especially important in counting solutions over finite fields. It is an algebraic invariant, but it has recently been possible to express crystalline cohomology in terms of a topological invariant - called topological periodic homology. This leads to natural generalizations, extensions and computations.

2017

2017, Max Planck Institute for Mathematics

Kaiser, Christian

After introducing elliptic modular forms we consider Borcherds products as singular theta lifts on orthogonal groups. Finally we discuss a characterization of Borcherds products by symmetries.

2016

2016, Max Planck Institute for Mathematics

Mnev, Pavel

We discuss the problem of counting paths going along the edges of a graph as a toy model for Feynman’s path integral in quantum mechanics.

2015

2015, Max Planck Institute for Mathematics

von Känel, Rafael; Matschke, Benjamin

We discuss certain classical Diophantine equations and we explain how geometry can help to understand their solutions. In particular we consider some highlights in the field of Diophantine equations, including Faltings' resolution of the Mordell conjecture and Wiles' proof of Fermat's last theorem. We also discuss a project at the MPIM Bonn which combines the methods of Faltings and Taylor-Wiles in order to make some progress concerning the classical problem of finding all squares and cubes of given difference.

2014

2014, Max Planck Institute for Mathematics

Vogel, Thomas

We discuss an application of contact topology to the space of foliations by surfaces on 3-manifolds. Without further geometric restrictions on the foliation, the connected components of the space of foliations are well understood. In contrast to this, if one requires that all foliations under consideration are taut, then the space of these foliations is more complicated. So far, its connected components are classified only in few special situations.

2014, Max Planck Institute for Mathematics

Swoboda, Jan

We discuss the moduli space of solutions to Hitchin's selfduality equations and its connection with representation varieties. Afterwards some recent results concerning the degeneration profile of solutions for large Higgs fields will be presented.

2013

2013, Max Planck Institute for Mathematics

Göttsche, Lothar

The counting of algebraic curves on surfaces is a classical topic of algebraic geometry, which has gained new interest because of the advent of string theory from physics. After a short introduction into enumerative invariants of curves a refinement is considered. Instead of a number of curves one gets a polynomial, which can be related to invariants of real and tropical algebraic geometry.

2013, Max Planck Institute for Mathematics

Williamson, Geordie

We give a glimpse into the world of reflection groups, Coxeter groups, Lie groups and Kazhdan-Lusztig polynomials.

2012

2012, Max Planck Institute for Mathematics

Rossi, Carlo Antonio

After a brief comparison between the commutative algebra of classical observables in Hamiltonian mechanics and non-commutative quantum observables in quantum mechanics, an algebraic way of realizing the quantization process (deformation quantization) is introduced and in particular the universal formula of Kontsevich is explained in some details. Finally, some (open) problems related to deformation quantization are considered, some of which have been positively addressed at the Max Planck Institute for Mathematics in Bonn.

2012, Max Planck Institute for Mathematics

Raum, Martin; Raum, Sven

Periods are a special kind of numbers, which play an outstanding role in number theory. An important aspect to consider are relations between different representations of periods. These can be studied in great detail for L-values, which are more accessible than most other examples are.

2011

2011, Max Planck Institute for Mathematics

Schneider, Ansgar

C*-algebras are of central importance in Physics, Mathematics and in the border area between. Mathematicians at the MPIM in Bonn try to find new insights between C*-algebras and other mathematical structures.

2011, Max Planck Institute for Mathematics

Ilia Itenberg, Grigory Mikhalkin

Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this survey we look at a few simple examples of correspondence principle between classical and tropical geometries.

2010

2010, Max Planck Institute for Mathematics

Mellit, Anton

We explain the important idea of L-function in mathematics and Deligne-Beilinson conjectures about values of L-functions at integral points. As main examples we consider here the Riemann zeta function and L-functions of elliptic curves. In the end we explain relation between values of L-functions of elliptic curves at 2 and logarithmic Mahler measures.

2010, Max Planck Institute for Mathematics

Vogel, Thomas

Open books are particular decompositions of manifolds which are useful for the construction of foliations. There are also important applications to the theory of contact structures.

2009

2009, Max Planck Institute for Mathematics

Aizenbud, Avraham; Gourevitch, Dmitry

An important question of Representation Theory is "what happens to an irreducible representation of a group when one restricts it to a subgroup?". Usually it stops being irreducible and it is interesting to know whether it decomposes to distinct irreducible representations. This report contains an exposition on this question and a short description of a recent proof that in the case of the pair (GL(n+1),GL(n)) over local fields the answer is positive.

2009, Max Planck Institute for Mathematics

Möller, Martin

Sequences defined by recurrence relations are known since the work of Fibonacci. They occur in number theory as well as for the solutions of differential equations. This report explains why special billiard tables give rise to recurrence relations such that the corresponding sequences consists unexpectedly of integers only.

2008

2008, Max Planck Institute for Mathematics

Zagier, Don

Three sources of q-hypergeometric series will be discussed: Mock theta functions, "characters" of rational conformal field theories and the Witten-Reshitikhin-Turaev invariant of the icosahedral Poincaré sphere.

2008, Max Planck Institute for Mathematics

Durov, Nikolai

The aim of this note is to give a very informal introduction into the world of arithmetic geometry, and especially Arakelov geometry.

2007

2007, Max Planck Institute for Mathematics

Kapovich, Mikhail

The purpose of this article is to give an overview of generalized triangle inequalities in symmetric spaces and their application in representation theory.

2007, Max Planck Institute for Mathematics

Hamenstädt, U.

From a polygon in the plane with additional symmetries one can glue together closed surfaces. On the space of all such surfaces two flows act. Statistical properties of the thus obtained are closely related to statistical properties of the action of the transformation class group.

2006

2006, Max Planck Institute for Mathematics

Petridis, Yiannis

Geometry studies geodesics in various settings, in particular on hyperbolic surfaces. The distribution of geodesics on arithmetic hyperbolic surfaces gives information on the arithmetic of quadratic forms, an important branch of number theory.

2006, Max Planck Institute for Mathematics

Marcolli, Matilde

In the sixties and seventies of the last century physicists developed procedures to renormalize infinities appearing in quantum field theory. We explain recent work of Connes-Kreimer and Connes-Marcolli, giving a conceptual understanding of these procedures and indicating unexpected links to number theory.

2005

2005, Max Planck Institute for Mathematics

Moree, Pieter

The notion of multiplicative order is very old and has interested mathematicians since the days of Fermat. It plays a role in different branches of mathematics. We will give a short overview of the history of research in this area up to recent work concerning the distribution of the order over residue classes.

2005, Max Planck Institute for Mathematics

Ward, Tom

The theory of dynamical systems provides powerful methods for studying diverse problems from planetary orbits to number theory. Recent work by Einsiedler, Katok and Lindenstrauss used methods from dynamical systems to make significant progress on a classical conjecture in number theory.

2004

2004, Max Planck Institute for Mathematics

Marcolli, Prof. Matilde

We describe how noncommutative geometry, a mathematical formulation of geometry adapted to quantum phenomena, interacts with number theory through quantum statistical mechanical systems with phase transitions and spontaneous symmetry breaking. This provides a unified setting for many important arithmetic results including the spectral realization of zeros of the Riemann zeta function and the Galois theory of modular functions.

2004, Max Planck Institute for Mathematics

Baues, Hans-Joachim; Jibladze, Mamuka