Optimization and Tropical Combinatorics (Lecture 1) by Michael Joswig
PROGRAM
COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID)
ORGANIZERS: Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India)
DATE & TIME: 27 June 2022 to 08 July 2022
VENUE: Madhava Lecture Hall and Online
Algebraic geometry is the study of solutions to systems of polynomial equations. Such sets of solutions (often with additional structure) are usually referred to as algebraic varieties. Combinatorial algebraic geometry is an aspect of algebraic geometry where either combinatorial techniques are used to study algebraic varieties or methods (and analogies) from algebraic geometry are used to study combinatorial objects. Tropical geometry is a branch of algebraic geometry that is based on transforming an algebraic variety into a “polyhedral subset” called its tropicalisation. Tropicalisation has proven to be an efficient technique for dealing with limits of algebraic varieties called degenerations. This is a thriving area with connections to several other areas such as number theory and topics in physics. Real algebraic geometry is a related active area of mathematics that is inspired by Hilbert’s sixteenth and seventeenth problems, and is a fertile ground for rich interactions between algebraic and polyhedral geometry.
An important goal of this meeting is to initiate new dialogues between researchers working in disparate areas of combinatorics, geometry and physics that enjoy deep connections that have not yet been sufficiently explored. The interface between algebraic surfaces and their tropical variants is an example of an area with potential for such interactions. Another such area is the theory of scattering amplitudes in quantum field theory, where methods from combinatorial geometry have come to play an important role in recent years. We envision fruitful and lasting collaborations arising out of conversations between researchers working in these and other areas.
This workshop aims to serve as a platform for junior researchers working in algebraic geometry, combinatorics and related areas to interact with leading experts in tropical and real algebraic geometry, and with each other. The program will include pedagogical lectures with a view toward making some of the more advanced topics accessible to a broader audience.
CONTACT US: [email protected]
PROGRAM LINK: https://www.icts.res.in/program/cagtr...
Table of Contents (powered by https://videoken.com)
0:00:00 Optimization and Tropical Combinatorics (Lecture 1)
0:00:16 1. Shortest paths and the Hungarian method
0:02:15 Shortest path problems
0:05:11 Tropical arithmetic
0:07:21 Powers of tropical matrices
0:10:44 Keene stars
0:12:36 Floyd-Warshall algorithm (1962)
0:18:46 Floyd-Warshall algorithm (1962), continued
0:22:49 Tropical polynomials
0:27:01 A univariate tropical polynomial
0:29:09 Regions of linearity of tropical polynomials
0:30:46 Parameterized all-pairs shortest paths
0:35:15 Example
0:46:31 Parameterized all-pairs shortest paths, continued
0:51:28 Sketch of proof
0:53:30 The linear assignment problem
0:56:51 Hungarian method (Kuhn 1955; Munkres 1957)
1:02:02 References
1:02:43 Q&A
1:12:19 Wrap Up
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