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Organizing Committee
Purdue University
SISSA (International School for Advanced Studies)
University of Massachusetts, Amherst
UiT The Arctic University of Norway
Abstract
Real algebraic (and semi-algebraic) geometry studies subsets of R^n defined by a finite number of polynomial equalities and inequalities. Such sets occur ubiquitously in practice both inside and outside of mathematics. While being easy to define, semi-algebraic sets can be complicated topologically, which restricts the application of many algorithms. In recent years, there has been progress in proving much stronger results – both quantitative and algorithmic -- when the problem under consideration involves the invariance under some group action.
In this workshop, we plan to focus on two situations where this phenomenon happens.
The first one is the statistical study of the topology of random real algebraic varieties as well as semi-algebraic sets, where the polynomials defining these objects are picked from a distribution invariant under the action of a certain group (usually the orthogonal group) acting on the space of variables. The behavior of the set of zeros (or more generally level sets) of such random polynomial systems is an extremely active topic with deep connections with classical enumerative problems in algebraic geometry (over real as well as complex fields), theoretical computer science, statistical physics, and machine learning.
The second one is the deterministic behavior of the topology of semi-algebraic sets admitting a group action (say a finite reflection group or a Lie group). The cohomology modules of such symmetric semi-algebraic sets then acquire the structure of a finite-dimensional representation of the corresponding group. This added structure is extremely useful in studying the topology as well as developing algorithms for computing topological invariants of such sets. The study of symmetric semi-algebraic sets has promising connections with several other areas, including representational and homological stability, sums-of-squares proofs and lower-bound questions in theoretical computer science.
In this workshop, we plan to focus on two situations where this phenomenon happens.
The first one is the statistical study of the topology of random real algebraic varieties as well as semi-algebraic sets, where the polynomials defining these objects are picked from a distribution invariant under the action of a certain group (usually the orthogonal group) acting on the space of variables. The behavior of the set of zeros (or more generally level sets) of such random polynomial systems is an extremely active topic with deep connections with classical enumerative problems in algebraic geometry (over real as well as complex fields), theoretical computer science, statistical physics, and machine learning.
The second one is the deterministic behavior of the topology of semi-algebraic sets admitting a group action (say a finite reflection group or a Lie group). The cohomology modules of such symmetric semi-algebraic sets then acquire the structure of a finite-dimensional representation of the corresponding group. This added structure is extremely useful in studying the topology as well as developing algorithms for computing topological invariants of such sets. The study of symmetric semi-algebraic sets has promising connections with several other areas, including representational and homological stability, sums-of-squares proofs and lower-bound questions in theoretical computer science.
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Lecture Videos
Peter Bürgisser
Technical University Berlin
August 24, 2020 • 12:00 AM
Anton Leykin
Georgia Tech
August 24, 2020 • 12:00 AM
Erik Lundberg
Florida Atlantic University
August 24, 2020 • 12:00 AM
Evelyne Hubert
Inria
August 24, 2020 • 12:00 AM
Harm Derksen
University of Michigan
August 24, 2020 • 12:00 AM
Greg Blekherman
Georgia Tech
August 25, 2020 • 12:00 AM
Mario Kummer
TU Berlin
August 25, 2020 • 12:00 AM
Patricia Hersh
University of Oregon
August 25, 2020 • 12:00 AM
Viswambhara Makam
Institute for Advanced Study
August 26, 2020 • 12:00 AM
Uwe Nagel
University of Kentucky
August 26, 2020 • 12:00 AM
Agnes Szanto
North Carolina State University
August 26, 2020 • 12:00 AM
Raman Sanyal
Goethe-Universität Frankfurt
August 27, 2020 • 12:00 AM
Abhiram Natarajan
Purdue University
August 27, 2020 • 12:00 AM
Avinash Kulkarni
Dartmouth College
August 27, 2020 • 12:00 AM
Erik Lundberg
Florida Atlantic University
August 27, 2020 • 12:00 AM
Pravesh Kothari
Institute for Advanced Study and Princeton University
August 28, 2020 • 12:00 AM
Chiara Meroni
SISSA
August 28, 2020 • 12:00 AM
Sinai Robins
University of Sao Paulo
August 28, 2020 • 12:00 AM
Chris Peterson
Colorado State University
August 28, 2020 • 12:00 AM