Скачать или смотреть Optimization and Tropical Combinatorics (Lecture 4) 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: 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.

ICTS is committed to building an environment that is inclusive, non-discriminatory and welcoming of diverse individuals. We especially encourage the participation of women and other under-represented groups.

CONTACT US: cagtr@ icts.res.in

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-4)
0:00:13 4. Product-mix auctions
0:01:48 Product-mix auctions
0:06:09 Demand sets and aggregate demand
0:10:53 Competitive equilibrium
0:12:54 Tropical hypersurfaces and their union
0:16:48 Example
0:23:58 [Code Walkthrough]
0:36:36 Proposition
0:38:38 When does competitive equilibrium exist?
0:43:42 Lemma
0:45:54 The unimodularity theorem
0:49:04 [Demo]
0:54:18 References
0:55:04 Summary
0:58:04 Q&A
1:09:20 Wrap Up