# Applications of Tropical Geometry in Leipzig-Berlin

## Abstracts for the talks

**Marie Brandenburg **:**Tropical Geometry and Neural Networks**

Tropical Geometry is the geometry that is underlying ReLU neural networks, a class of networks which are heavily used in machine learning today. In the first part of this talk I will give an idea about the relation of these networks to tropical geometry, and how tropical geometry can contribute in understanding the geometry of neural networks.

In the second part I will present first results of ongoing work regarding classication problems within this framework.

**Andrei Comăneci **:**Tropical medians by transportation**

In this talk, we present the Fermat–Weber problem under an asymmetric tropical distance. We describe the location of the optimum in terms of tropical geometry, which gives a combinatorial interpretation of the set of solutions. Moreover, it turns out that this location problem is equivalent to a transportation problem, allowing for fast computation.

Finally, we show how we can exploit the connection to tropical

convexity for an application to the consensus problem from

computational biology. The geometric interpretation also gives desirable properties for the resulting consensus method.

**Sylvain Spitz **:**The Polyhedral Geometry of Truthful Auctions**

In an auction mechanism, the objective is to allocate m items to n players by asking the players for their valuations of the items. The difference set of an allocation is the set of valuation vectors that the auction mechanism maps to the given allocation. Using tropical geometry, we give a complete characterization of the geometry of the difference sets that can appear for an incentive compatible multi-unit auction showing that they correspond to regular subdivisions of the unit cube.

This observation is then used to construct mechanisms that are robust in the sense that the set of items allocated to a player does change only slightly when the player's reported valuation is changed slightly.

**Bernd Sturmfels **:**Tropical Geometry of Statistical Models**

This talk revisits the manifesto which Lior Pachter and I published in 2004. My aim is to explain its third thesis, which proclaims that “tropicalized statistical models are fundamental for parametric inference, and to initiate a discussion on how this could be relevant for present-day machine learning.

## Date and Location

**March 17, 2023**

Max Planck Institute for Mathematics in the Sciences

E1 05 (Leibniz-Saal)

Inselstr. 22

04103 Leipzig

## Scientific Organizers

**Laura Casabella**

MPI for Mathematics in the Sciences

## Administrative Contact

**Mirke Olschewski**

MPI for Mathematics in the Sciences

Contact by Email