This workshop will be offered virtually. The in-person meeting has been cancelled due to the COVID-19 outbreak. A schedule for virtual talks will be posted soon. Accepted participants will be notified how to access the virtual presentations.
Organizing Committee

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.

Image for "VIRTUAL ONLY: Symmetry, Randomness, and Computations in Real Algebraic Geometry"

Confirmed Speakers & Participants

Talks will be presented virtually or in-person as indicated in the schedule below.

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

Monday, August 24, 2020
8:00 - 8:10am EDTWelcome - ICERM Director  
8:10 - 8:55am EDTRigid Continuation Paths II- Structured Polynomial Systems - Peter Burgisser, TU Berlin 
9:10 - 9:55am EDTInflation of poorly conditioned zeros of systems of analytic functions - Anton Leykin, Georgia Tech 
10:00 - 10:45am EDTCoffee/ Tea Break  
10:45 - 11:30am EDTA Survey of random Real Algebraic Geometry - Erik Lundberg, Florida Atlantic University 
11:45 - 1:00pm EDTBreak for Lunch/ Free Time  
1:00 - 1:45pm EDTSparse Interpolation in Terms of Multivariate Chebyshev Polynomials Associated to a Weyl group - Evelyne Hubert, Inria, Méditerranée 
2:00 - 2:45pm EDTAlgebraic Methods for Tensor Data - Harm Derksen, Northeastern University 
Tuesday, August 25, 2020
8:00 - 8:45am EDTTropicalization in Real Algebraic Geometry and Extremal Combinatorics - Greg Blekherman, Georgia Tech 
9:00 - 9:45am EDTSpectral sets and derivatives of the psd cone - Mario Kummer, TU Berlin 
10:00 - 11:00am EDTPanel discussion on racism and diversity issues  
11:00 - 11:45am EDTQuantitative Singularity theory for Random Polynomials - Paul Breiding, Technische Universität Berlin 
12:00 - 1:00pm EDTBreak for Lunch/ Free Time  
1:00 - 2:00pm EDTShelling face posets for stratified spaces of electrical networks and of phylogenetic trees - Patricia Hersh, University of Oregon  
2:00 - 3:00pm EDTDiscussion Time  
Wednesday, August 26, 2020
9:00 - 9:45am EDTMLE for tensor normal models - Visu Makam, Institute for Advanced Study 
10:00 - 11:00am EDTGrad Students' Presentations - Margaret Regan, Duke University; Alison Rosenblum, Purdue University; Isabelle Shankar, UC, Berkeley; Kevin Shu, Georgia Institute of Technology; & Miruna-Stefana Sorea, Max Planck Institute for Mathematics in the Sciences, Leipzig  
11:00 - 11:45am EDTMaximal and Typical Topology of Real Polynomial Singularities - Michele Stecconi, SISSA 
12:00 - 1:00pm EDTBreak for Lunch/ Free Time  
1:00 - 1:45pm EDTSequences of Symmetric Ideals - Uwe Nagel, University of Kentucky 
2:00 - 3:00pm EDTReal Smooth Points - Agnes Szanto, North Carolina State University 
Thursday, August 27, 2020
9:00 - 9:45am EDTOn algebraic and geometric properties of spectral convex bodies - Raman Sanyal, Goethe University Frankfurt 
10:00 - 11:00am EDTCoffee/ Tea Break  
11:00 - 11:45am EDTBetti Numbers of Random Hypersurface Arrangements - Abhiram Natarajan, University of Colorado- Boulder 
12:00 - 1:00pm EDTBreak for Lunch/ Free Time  
1:00 - 1:45pm EDTThe p-adic integral geometry formula - Avinash Kulkarni, Dartmouth College 
2:00 - 2:45pm EDTLimit cycles in random polynomial vector fields - Erik Lundberg, Florida Atlantic University 
Friday, August 28, 2020
9:00 - 9:45am EDTThe Sum-of-Squares Approach to Disentangling Non-Spherical Gaussian Mixtures - Pravesh K. Kothari, Carnegie Mellon University 
10:00 - 11:00am EDTMentoring Session  
11:00 - 11:45am EDTReal lines on random cubic surfaces - Chiara Meroni, Sissa 
12:00 - 1:00pm EDTBreak for Lunch/ Free Time  
1:00 - 1:45pm EDTThe Fourier transform of a polytope, and its zero set - Sinai Robins, University of São Paulo 
2:00 - 2:45pm EDTExpected distances on special flag manifolds and lens spaces - Chris Peterson, Colorado State University 

Lecture Videos

Real Smooth Points

Agnes Szanto
North Carolina State University
August 26, 2020