**Speaker:**

Stefan Hollands

**Coordinator:**

Jörg Lehnert

**Address:**

Inselstraße 22

04103 Leipzig

Germany

**Contact:**

+49(0)341-9959-641

+49(0)341-9959-658* ***IMPRS**

# Algebra and Geometry (AG)

## Scientific Members working in this area

- Judith Brinkschulte (Mathematical Institute of Leipzig University)
- James Reed Farre (MPI for Mathematics in the Sciences)
- Tobias Finis (Mathematical Institute of Leipzig University)
- Lukasz Grabowski (Mathematical Institute of Leipzig University)
- Jürgen Jost (MPI for Mathematics in the Sciences)
- Michael Joswig (MPI for Mathematics in the Sciences)
- Guido Montúfar (MPI for Mathematics in the Sciences)
- Sayan Mukherjee (MPI for Mathematics in the Sciences and ScaDS.AI)
- Marta Panizzut (MPI for Mathematics in the Sciences)
- İrem Portakal (MPI for Mathematics in the Sciences)
- Hans-Bert Rademacher (Mathematical Institute of Leipzig University)
- Érika Roldan (MPI for Mathematics in the Sciences)
- Matthias Schwarz (Mathematical Institute of Leipzig University)
- Rainer Sinn (Mathematical Institute of Leipzig University)
- Peter Smillie (MPI for Mathematics in the Sciences)
- Peter F. Stadler (Institute for Computer Science of Leipzig University)
- Bernd Sturmfels (MPI for Mathematics in the Sciences)
- Simon Telen (MPI for Mathematics in the Sciences)
- Anna Wienhard (MPI for Mathematics in the Sciences)

## Current PhD Students

Surname, first name | Phone | Office | Homepage | |

Barth, Lukas | Lukas.Barth | 986 | E1 04 | |

Calvo Cortes, Veronica | Veronica.Calvo | 763 | F3 06 | external |

Camacho Cadena, Fernando | Fernando.Camacho | 830 | G2 02 | external |

Freund, Valerie | Valerie.Freund | 666 | ---- | |

Hsiao, Enya | Enya.Hsiao | 830 | G2 02 | |

Kendal-Freedman, Nat Michael | Nat.Kendal | 834 | G2 07 | |

Koefler, Joris | Joris.Koefler | 768 | F3 07 | external |

Niemeyer, Merik Juljan | Merik.Niemeyer | 831 | G2 03 | |

Paul, Niharika Chakrabarty | Niharika.Paul | 812 | G3 04 | external |

Shi, Jiajun | Jiajun.Shi | 831 | G2 03 | external |

## Description of Research

### Algebraic Geometry.

This is the study of solution sets to polynomial equations in several variables. If the variables range over the real numbers, then one speaks of real algebraic geometry, which is a subject with many applications, e.g., in optimization and statistics.

### Lie Groups and Number Theory.

Number theory is a central field in mathematics, with ramifications in logic, algebra, and algebraic geometry. In this context, we study automorphic forms and their relations to representations of Lie groups over fields of an arithmetic nature. This stands in contrast to the appearance of Lie groups in the study of homogeneous spaces, particularly in higher Teichmüller theory, which is also studied in this working group.

### Differential and Symplectic Geometry.

The strong geometry division at the IMPRS covers Riemannian geometry, complex geometry, and symplectic geometry. Riemannian geometry enters the classification problems for black holes, rigidity results, or the geometric analysis of initial data sets. Symplectic geometry underpins all of classical mechanics. In some cases, symplectic automorphisms describe the evolution of physical systems. Such systems exhibit both a generic tendency for chaotic behavior as well as rigidity in the Hamiltonian case. At the intersection of complex geometry and complex dynamics, one studies holomorphic foliations and rigidity results for manifolds admitting a Levi foliation.

### Combinatorics.

It has been said that combinatorics is the nanotechnology of mathematics. The term is sometimes used synonymously with "discrete mathematics". This field is closely connected with computer science. Local expertise includes geometric combinatorics and polyhedral algorithms, graphs, hypergraphs and their applications, and tropical combinatorics and their connection to computer algebra.

### Data and Computation.

Recent years have seen explosive development in data science. Mathematics plays an important double role: it provides method for analyzing data, but it also generates interesting data in its own right. Mathematical software packages play a significant role here. Data analysis that uses various innovative approaches from geometry and topology to make sense of high-dimensional data is studied in our IMPRS.

## Alumni

Name | Advisor | Graduation | Topic |

Zachary Adams | Jürgen Jost | 2023 | Quasi-ergodicity of SPDE : spectral theory and phase reduction (see more) |

Renan Assimos Martins | Jürgen Jost | 2019 | The geometry of maximum principles and a Bernstein theorem in codimension 2 (see more) |

David Bauer | Matthias Schwarz | 2013 | Towards discretization by piecewise pseudoholomorphic curves (see more) |

Philipp Benner | Jürgen Jost | 2016 | Combining prior information for the prediction of transcription factor binding sites (see more) |

Sarah Berkemer | Peter F. Stadler | 2020 | Towards dynamic programming on generalized data structures and applications of dynamic programming in bioinformatics (see more) |

Guy Buss | Jürgen Jost | 2009 | On the construction and the behaviour of automorphic forms on completions of Teichmüller spaces and higher Bers maps |

Brian Clarke | Jürgen Jost | 2009 | The completion of the manifold of Riemannian metrics with respect to its L2 metric (see more) |

Masud Ehsani | Jürgen Jost | 2021 | Self-organized criticality in neural networks by inhibitory and excitatory synaptic plasticity (see more) |

Marzieh Eidi | Jürgen Jost | 2022 | Topological and geometric methods with a view towards data analysis (see more) |

Christoph Eikemeier | Tobias Finis | 2022 | Asymptotics of Hecke operators for quasi-split simple groups (see more) |

Vishal Gupta | Tobias Finis | 2018 | Limit multiplicity problem (see more) |

Julian Hofrichter | Jürgen Jost | 2014 | On the diffusion approximation of Wright-Fisher models with several Alleles and Loci and its geometry (see more) |

Danijela Horak | Jürgen Jost | 2012 | Combinatorial Laplace operators on simplicial complexes |

Leonhard Horstmeyer | Jürgen Jost | 2017 | Towards a geometric theory of exact lumpability (see more) |

Shuhan Jiang | Jürgen Jost | 2023 | Mathematical structures of cohomological field theories (see more) |

Nidhi Kaihnsa | Bernd Sturmfels | 2019 | Algebraic methods for dynamical systems and optimisation (see more) |

Martin Kell | Jürgen Jost | 2014 | On curvature conditions using Wasserstein spaces : with an application to the q-Laplace heat equation (see more) |

Enno Keßler | Jürgen Jost | 2017 | The super conformal action functional on super Riemann surfaces (see more) |

Ilona Kosiuk | Jürgen Jost | 2012 | Relaxation oscillations in slow-fast systems beyond the standard form (see more) |

Alexei Lebedev | Jürgen Jost | 2008 | Simple modular Lie superalgebras |

Shiping Liu | Jürgen Jost | 2012 | Synthetic notions of curvature and applications in graph theory (see more) |

Arun Maiti | Hans-Bert Rademacher | 2017 | On the Goresky-Hingston product (see more) |

Chiara Meroni | Rainer Sinn | 2022 | Semialgebraic convex bodies (see more) |

Raffaella Mulas | Jürgen Jost | 2020 | Spectra of normalized Laplace operators for graphs and hypergraphs (see more) |

Johannes Müller | Guido Montufar, Jürgen Jost | 2024 | Geometry of optimization in Markov decision processes and neural network based PDE solvers (see more) |

Wiktor Młynarski | Jürgen Jost | 2015 | Functional sensory representations of natural stimuli : the case of spatial hearing (see more) |

Wolfgang Otto | Peter Stadler | 2011 | Transcriptional regulatory elements : detection and evolutionary analysis (see more) |

Paolo Perrone | Jürgen Jost | 2018 | Categorical probability and stochastic dominance in metric spaces (see more) |

Stephan Poppe | Jürgen Jost | 2017 | Foundational aspects of coherent predictive and statistical inference for exchangeable categorical sequences |

Sharwin Rezagholi | Jürgen Jost | 2020 | Hyperspace-type monads on top and the entropy theory of cynamical systems |

Angel Eduardo Rodríguez Fernández | Peter F. Stadler | 2021 | On the study of fitness landscapes and the Max-Cut problem (see more) |

Christoph Sachse | Jürgen Jost | 2007 | Global analytic approach to super Teichmüller spaces (see PDF) |

Murat Sağlam | Matthias Schwarz | 2017 | A search for finer topological information by holomorphic curves in symplectizations : the case of lens spaces and their unit cotangent bundles |

Camilo Eduardo Sarmiento Cortés | Jürgen Jost | 2014 | Concerning triangulations of products of simplices (see more) |

Mahsa Sayyary Namin | Bernd Sturmfels | 2021 | Real algebraic geometry of the sextic curves (see more) |

Matti Schneider | Matthias Schwarz | 2013 | The Leray-Serre spectral sequence in Morse homology on Hilbert manifolds and in Floer homology on cotangent bundles (see more) |

Gerardo Sosa Garciamarín | Jürgen Jost | 2017 | On symmetric transformations in metric measure geometry (see more) |

Maximilian Stegemeyer | Hans-Bert Rademacher | 2023 | On global properties of geodesics : the string topology coproduct and geodesic complexity (see more) |

Linlin Sun | Jürgen Jost | 2015 | Dirac equations, Dirac-harmonic maps and their heat flows |

Tat Dat Tran | Jürgen Jost | 2012 | Information geometry and the Wright-Fisher model of mathematical population genetics (see more) |

Hanna Tseran | Bernd Sturmfels, Guido Montúfar | 2023 | Expected complexity and gradients of deep maxout neural networks and implications to parameter initialization (see more) |

José Javier Vasquez | Hans-Bert Rademacher | 2017 | Isospectral nearly Kähler manifolds (see more) |

Ying Wang | Jürgen Jost | 2010 | Branching processes : optimization, variational characterization, and continuous approximation (see more) |

Anatol Wegner | Jürgen Jost | 2015 | Subgraph covers : an information theoretic approach to motif analysis in networks (see more) |

Ruijun Wu | Jürgen Jost | 2017 | Analysis of a two-dimensional nonlinear sigma model with gravitino (see more) |

Marius Emar Yamakou | Jürgen Jost | 2018 | Weak-noise-induced phenomena in a slow-fast dynamical system |

Odette Sylvia Yaptieu Djeungue | Jürgen Jost | 2018 | Generalizations of discrete Morse theory (see more) |

Tinggui Zhang | Jürgen Jost | 2014 | Separability and local unitary equivalence of quantum states |

Miaomiao Zhu | Jürgen Jost | 2008 | Harmonic maps and Dirac-Harmonic maps from degenerating surfaces |