• Universality and Homogeneity
• Special Seminar on Universality of moduli spaces and geometry
• Schedule

• Schedule
• Participants

The polytopes with the smallest possible realization space are the projectively unique polytopes; they are, by definition, the polytopes for which the projective linear group acts transitively on the realization space.
We present two new methods for the construction of polytopes, that in particular lead to the resolution of old conjectures concerning projectively unique polytopes:

• A construction based on solving Cauchy problems for discrete differential equations, and
• A construction that extends on the important Lawrence lifting technique.

We give an outline of the proof of the following universality theorem for realization spaces of polyhedral maps. Let be a semialgebraic set (real algebraic coefficients). Then there exists a map with vertices (on some orientable manifold) which contains only triangles and  quadrangles such that the image of the realization space of under the canonical projection is equal to , where . As a corollary we get that for each strict subfield of the field of real algebraic numbers there is a map which can be realized as a polyhedron in but not in .

A stratification of the set of critical points of a map is universal in the class of stratifications satisfying the classical Thom and Whitney-a conditions if it is the coarsest among all such stratifications. We show that a universal stratification exists if and only if the 'canonical subbundle' of the cotangent bundle of the source of the map (constructed via operations introduced by Glaeser) is Lagrangian. The proof relies on a new Bertini-type theorem for singular varieties proved via an intriguing use of resolution of singularities. Many examples are provided, including those of maps without universal stratifications. (joint work with P. Milman)

I will review several universality theorems on configuration/moduli spaces of geometric objects, going back to the 19th century. The main idea of  universality in this context is to encode arithmetic into some basic geometric and algebraic objects, like mechanical linkages, rod/gear mechanisms, projective arrangements, convex polytopes, Coxeter groups, etc.

I will talk about a recent universality theorem for moduli spaces of representations of 3-manifold groups to SL(2) that I proved with John Millson. This is an application of our old universality theorem for character schemes of Coxeter groups and a recent theorem of Panov and Petrunin on fundamental groups of 3-dimensional orbifolds. 

What is more universal than a free object? Of course, it may be not universal enough but often it is the best we can do. In research of free commutative associative algebras (polynomials) algebraically closed fields play very important role. Historically idea of an algebraically closed field started with complex numbers and it takes some effort to prove that the complex numbers indeed form an algebraically closed field. On the other hand if just an existence of an algebraically closed field is of interest to us and we are ready to accept the transfinite induction, construction is easy and even not very tedious. If we replace free commutative associative algebras with free associative algebras and try to built an algebraically closed skew field situation is com- pletely different: transfinite induction will not deliver. In my talk I'll outline a construction of a non-commutative “complex numbers” and discuss difficulties inherent in such a construction. (For example, it is still not know whether such a field exists in the finite characteristic case.)

I will present the following universality theorem for projectively unique polytopes: every polytope described by algebraic coordinates is the face of a projectively unique polytope. This result can be used to construct a combinatorial type of polytope that is not realizable as a subpolytope of any stacked polytope. This disproves a classical conjecture in polytope theory, first formulated by Shephard in the seventies.

This is joint work with Karim Adiprasito.

In a beautiful paper from 1996, R. Lang, a well known origamist, described an algorithm for designing folded origami shapes with an underlying metric tree structure. We show that, very often, Lang's crease patterns are rigid, and his flat-folded shapes (although highly flexible) belong to isolate components in the origami configuration space from which they cannot "unfold". This is joint work with my PhD student John Bowers, who also implemented a very nice program
which visually illustrates Lang's method.

The space of all metric measure spaces (X,d,m) plays an important role in image analysis, in the investigation of limits of Riemannnian manifolds and metric graphs as well as in the study of geometric flows that develop singularities. We show that this "space of spaces" -- equipped with the L^{^2}-distortion distance -- is a challenging object of geometric interest in its own. In particular, we show that it has nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on the space of spaces are presented. 

Low-rank matrix completion is the task to reconstruct (or “complete”) a large m x n matrix A of known rank r << n from a subset of its entries. The most successful algorithms with theoretical guarantees are based on convex relaxations and require assumptions on: (a) the sampling of the “true matrix” A; (b) the sampling of the observed entries. I’ll talk about how to formulate matrix completion in algebraic-combinatorial terms. This will lead to new algorithms and give an indication about which sampling models are computationally tractable.