MATH+ Incubator Projects were short-term 12-month projects that aimed to generate the potential for innovation. These projects were expected to have a high potential impact within the MATH+ Topic Development Lab by advancing fundamental mathematics or building new bridges within mathematics or with closely related disciplines. Incubator projects were expected to have a lasting impact in terms of new research directions and fields. Incubator Projects could only be applied for from 2019-2022.

In 2022, the following Incubator Projects were still active:

- IN-5: Efficient Minimization of Parametric Submodular Functions for Quickest Transshipments

*Martin Skutella* - IN-6: Homotopy Methods for Dynamic Flows

*Max Klimm* - IN-7: Electronic Properties of Gate-Confined Quantum Dots in Si-Ge Heterostructures for Qubit Generation

*Thomas Koprucki (WIAS), Alexander Mielke (WIAS), Torsten Boeck (IKZ), Oliver Brandt (PDI)* - IN-8: Infinite-Dimensional Supervised Least Squares Learning as a Noncompact Regularized Inverse Problem

*Péter Koltai*

- IN-9: Holonomic Functions and Koszul Modules

*Gavril Farkas*

- IN-10: Towards a Framework for Decision Theatre Models

*Sarah Wolf, Stefanie Winkelmann, Thomas Steinke* - IN-11: Identifying and Efficiently Computing Band-Edge Energies for Charge Transport Simulations in Strained Materials

*Costanza Lucia Manganelli, Christian Merdon, Patricio Farrell*

MATH+ Incubator Projects were short-term projects that aim to generate the potential for innovation. In a first round in 2020, Incubator Projects ran in two tracks:

Track A: Projects that progress theory by advancing fundamental mathematics or build new bridges within mathematics or with closely related disciplines. These projects have a high potential of impact within the MATH+ Topic Development Lab, preferably in conjunction with one of the past or future Thematic Einstein Semesters.

Track B: Interdisciplinary projects together with a strong partner from other scientific disciplines. These projects have a high potential of opening up novel perspectives on the fruitful interplay of mathematics with other disciplines.

Incubator projects were expected to have lasting impact in terms of new research directions and fields.

**Track A
**

- IN-A1: Algebraic and Tropical Methods for Periodic Timetabling

*Ralf Borndörfer* - IN-A2: Applications of Algebra in the Geometry of Materials

*Myfanwy Evans, Frank Lutz, Bernd Sturmfels* - IN-A3: The Fukaya Algebra of Secondary Polytopes in Landau-Ginzburg Models

*Chris Wendl* - IN-A4: Learning Hypergraphs

*Tibor Szabó*

**Track B
**

- IN-B1: Biological Validation of Mathematically Predicted Algebraic Conditional Expectation Structures Arising in Gene Regulatory Networks

*Vikram Sunkara, Mir-Farzin Mashreghi, Gitta Anne Heinz, Heike Siebert, Tim Conrad* - IN-B2: Determining the Probability Distribution Function of Molecular Overlap Times in Single Molecule Microscopy: Random Collisions and Constructive Interactions

*Paolo Annibale, Martin Lohse, Christof Schütte* - IN-B3: Understanding Doping Variations in Silicon Crystals

*Nella Rotundo, Patricio Farrel, Natascha Dropka*

The success of the format of Incubator Projects is reflected by new research projects originating from them:

- AA1-11*: Receptor Dynamics and Interactions in Complex Geometries: An Inverse Problem in Particle-Based Reaction-Diffusion Theory

*Originating from IN-B2* - AA3-8: The Tropical Geometry of Periodic Timetables

*Originating from IN-A1* - EF1-12: Learning Extremal Structures in Combinatorics

*Originating from IN-A4*

The following former ECMath research projects were being continued until the end of 2019 with the aim to re-bundle research activities and possibly integrate them into the MATH+ research agenda:

**Clinical Research and Health Care (CH)
**

- CH12: Advanced Magnetic Resonance Imaging: Fingerprinting and Geometric Quantification

*Michael Hintermüller* - CH14: Understanding Cell Trajectories with Sparse Similarity Learning

*Tim Conrad, Gitta Kutyniok, Christof Schütte* - CH15: Analysis of Empirical Shape Trajectories

*Hans-Christian Hege, Tim J. Sullivan, Christian von Tycowicz* - CH17: Hybrid Reaction-Diffusion / Markov-State Model of Systems with Many Interacting Molecules

*Frank Noé, Christof Schütte* - CH18: Boundary-Sensitive Hodge Decompositions

*Konrad Polthier* - CH20: Stochasticity Driving Robust Pattern Formation in Brain Wiring

*Max von Kleist, Martin Weiser* - CH21: Data-Driven Modelling of Cellular Processes and beyond

*Tim Conrad, Stefan Klus, Christof Schütte*

**Metropolitan Infrasctructure (MI)
**

- MI-CH1: Robust Optimization of Load Balancing in the Operating Theatre

*Guillaume Sagnol* - MI7: Routing Structures and Periodic Timetabling

*Ralf Borndörfer* - MI8: Understanding and Improving Traffic with Unknown Demands

*Max Klimm* - MI10: Acyclic Network Flows

*Benjamin Hiller, Thorsten Koch, Martin Skutella* - MI12: Dynamic Models and Algorithms for Equilibria in Traffic Networks

*Martin Skutella*

**Optical Technologies (OT)
**

- OT9: From Single Photon Sources to Tailored Multi-Photon States

*Sven Burger, Frank Schmidt* - OT10: Model Reduction for Nonlinear Parameter-Dependent Eigenvalue Problems in Photonic Crystals

*Volker Mehrmann*

**Sustainable Energies (SE)
**

- SE17: Stochastic Methods for the Analysis of Lithium-Ion Batteries

*Jean-Dominique Deuschel, Peter Friz, Clemens. Guhlke, Manuel Landstorfer* - SE18: Models for Heat and Charge-Carrier Flow in Organic Electronics

*Annegret Glitzky, Matthias Liero* - SE23: Multilevel Adaptive Sparse Grids for Parametric Stochastic Simulation Models of Charge Transport

*Sebastian Matera*

**Education and Outreach (EO)
**

- EO2: Fostering Multipliers’ Noticing of Teachers’ Mathematics Learning by Means of Video Examples (NoTe)

*Bettina Rösken-Winter, Jürg Kramer*