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Events
Applications of Tropical Geometry in Leipzig-Berlin

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

17/03/2023 17/03/2023

Applications of Tropical Geometry in Leipzig-Berlin

The purpose of this event is to bring together the two groups of researchers in Tropical Geometry at the Technische Universität Berlin and the MPI Leipzig through an afternoon of talks focused on applications. There will also be time for discussion about recent or ongoing research.

Speakers

Marie Brandenburg

Max Planck Institute for Mathematics in the Sciences

Andrei Comăneci

Technische Universität Berlin

Sylvain Spitz

Technische Universität Berlin

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Program

15:00 - 15:30	Bernd Sturmfels (Max Planck Institute for Mathematics in the Sciences) 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.
15:45 - 16:15	Marie Brandenburg (Max Planck Institute for Mathematics in the Sciences) 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.
16:15 - 16:45	Coffee Break
16:45 - 17:15	Sylvain Spitz (Technische Universität Berlin) 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.
17:30 - 18:00	Andrei Comăneci (Technische Universität Berlin) 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.
19:00 - 00:00	Dinner

Applications of Tropical Geometry in Leipzig-Berlin

Speakers

Marie Brandenburg

Andrei Comăneci

Sylvain Spitz

Bernd Sturmfels

Program

Participants

Carlos Améndola

Barbara Betti

Marie Brandenburg

Laura Casabella

Andrei Comăneci

Jane Coons

Karel Devriendt

Nicholas Early

Kamillo Ferry

Alheydis Geiger

Christian Haase

Benjamin Hollering

Henrik J. Munch

Marta Panizzut

Raul Penaguiao

Ángel Ríos

Kemal Rose

Lisa Seccia

Sylvain Spitz

Bernd Sturmfels

Roan Talbut

Marcel Wack

Scientific Organizers

Laura Casabella

Administrative Contact

Mirke Olschewski