Buildings are combinatorial objects that model geometric features of groups and their representations. They have numerous connections to algebraic and arithmetic geometry. Algebraic varieties associated to buildings, such as affine Grassmannians, Mustafin varieties and Shimura varieties, offer plenty of untapped potential for computational research and novel applications.
The aim of the meeting is to bring together researchers working in fields related to buildings and algebraic geometry, including those who are relatively unfamiliar with applications of buildings to their work. The program will consist of a number of invited lectures, contributed talks by participants, and unstructured time to build collaborations.
Limited funding is available for participants who will contribute a talk. Further instructions on the talk application follow in the confirmation of registration. The deadline for submission is July 31, 2019.
In this talk, I will give a gentle introduction to recent research to find efficient algorithms for natural problems in geometric invariant theory by drawing a connection to geodesically convex optimization. These algorithms minimize the norm of the moment map and test membership in moment polytopes for non-commutative group actions. As we will see, this setting captures a diverse set of problems in different areas of mathematics, computer science, and physics. Several of these were solved efficiently for the first time using optimization methods; the corresponding algorithms also lead to solutions of purely structural problems and to new connections between disparate fields.
In the spirit of standard convex optimization, we develop two general methods in the geodesic setting, a first and a second order method, which receive first and second order information, respectively, on the derivatives of the function to be optimized. We will discuss the key parameters of the underlying group actions which control the efficiency of these methods, and see how to bound these parameters in several general cases. We will end by discussing some intriguing open problems and further research directions.
Peter Bürgisser will give a follow-up talk on Tuesday on the same line of work (arXiv:1910.12375).
For a number field K with ring of integers O, the order zeta function of K is a Dirichlet generating series enumerating orders, i.e. unital subrings of O of finite index. In comparison with the Dedekind zeta function of K, the order zeta function of K is poorly understood: for number fields of degree larger than 5, next to nothing general is known. Encoding this Dirichlet series as a p-adic integral, we develop computational tools to repeatedly resolve singularities until it is distilled to enumerating points on polyhedra and p-rational points of algebraic varieties. This is joint work with Anne Fruehbis-Krueger, Bernd Schober, and Christopher Voll.
In 1997 Kuperberg gave a presentation for the spider categories of the rank 2 Lie algebras and showed that the non-elliptic webs form a basis in each invariant space. Fontaine, Kamnitzer, Kuperberg then showed that in type $A_2$ the non-elliptic webs are dual to CAT(0) diskoids in the affine building. Each such CAT(0) diskoid, aka dual non-elliptic web, is a triangulation of a generic polygon $P$ in the building. In this talk, we discuss work in progress for characterizing the dual non-elliptic webs as the intersection of two convex hulls of $P$ in the affine building. The convex sets are related to tropical convexity as discussed in work by Joswig, Sturmfels, Yu, and by Leon Zhang. We hope that these convexity ideas can be applied to higher rank $A_n$, where a good basis of webs is not known.
Maximal and Hitchin representations are discrete subgroups of semisimple Lie groups isomorphic to the fundamental group of a surface. Their parameter spaces are often referred to as Higher rank Teichmüller spaces: not only they form entire connected components of the character variety, but the subgroups they parametrize also share many common geometric properties with holonomies of hyperbolizations. After introducing these representations and motivating their study, I will discuss a project joint with Burger, Iozzi and Parreau aimed at compactifying these spaces by studying associated actions on affine buildings. The techniques we use range from geometric group theory to real algebraic geometry.
In his talk Peter Bürgisser introduced the "weight margin" of a representation. Moreover, he presented a second order algorithm for non-commutative optimization and the weight margin is the crucial complexity measure for this algorithm. Namely, the running time depends inversely on the weight margin.
In this talk we present bounds on the weight margin for concrete examples, including actions on quivers and on 3-order tensors. Thereby we will encounter inverse polynomial lower bounds for some examples as well as inverse (sub)exponential upper bounds for others. Thus, we exhibit situations in which the second order algorithm has polynomial respectively at best (sub)exponential running time.
This is joint work with Cole Franks.
For a Riemann surface (and really all “nice” topological spaces), all abelian covering spaces are classified by the singular cohomology groups. Similarly, for an algebraic curve, all finite abelian coverings spaces are classified by the étale cohomology groups, and for number fields the abelian extension are classified in terms of class field theory. We consider the analogous question for tropical curves: Given a tropical curve, how can you classify all of its (finite) abelian covers?
This question is less trivial than it first appears, since, due to the presence of dilation, there are many unramified tropical covers that are not covering spaces in a topological sense. To resolve this probelm, we propose a so-called dilated cohomology group that classifies all abelian covers with a certain fixed dilation profile. In this talk I will give an elementary introduction to this cohomology theory and highlight its connections to the Bass-Serre theory of graphs of groups. I will also present a framework to study the realizability problem for abelian tropical covers, which, in the case of cyclic covers, is surprisingly closely related to the nowhere zero flow problem on a finite graph. This is based on joint work with Yoav Len and Dmitry Zakharov.
Skew polynomials, which have a noncommutative multiplication rule between coefficients and an indeterminate, are the most general polynomial concept that admits the degree function with desirable properties. This talk presents the first algorithms to compute the maximum degree of the Dieudonné determinant of a $k \times k$ submatrix in a matrix $A$ whose entries are skew polynomials over a skew field $F$. Our algorithms make use of the discrete Legendre conjugacy between the sequences of the maximum degrees and the ranks of block matrices over $F$ obtained from coefficient matrices of $A$. Three applications of our algorithms are provided: (i) computing the dimension of the solution spaces of linear differential and difference equations, (ii) determining the Smith--McMillan form of transfer function matrices of linear time-varying systems and (iii) solving the "weighted" version of noncommutative Edmonds' problem with polynomial bit complexity. We also show that the deg-det computation for matrices over sparse polynomials is at least as hard as solving commutative Edmonds' problem.
During the talk I will describe a striking application of the non-commutative optimisation to a problem in geometric group theory, namely the Kazhdan property (T).It is known that property (T) is equivalent to positivity of the element $\Delta^2 - \lambda\Delta$ in the full group $*$-algebra, where $\Delta$ is group Laplacian associated to a generating set. It turns out that its positivity is equivalent to the existence of a sum of (hermitian) squares decomposition of $\Delta^2 - \lambda\Delta$ in the real group algebra, which might be understood as an algebra in transition between the classical polynomial algebra and Heltons free algebra.I will mention the algorithm encoding the optimisation problem, and how an (imprecise) numerical solution can be turned into a mathematical proof. We applied the method to answer the question of property (T) for $\operatorname{Aut}(F_{n})$, the automorphism group of the free group on $n$ generators. Since (even for $n=4$) the size of the problem is out of reach of the method, I will show how to exploit the inherent symmetry of the problem to obtain a smaller, equivalent problem. This leads to constructive, computer-assisted proof that $\operatorname{Aut}(F_{n})$ has Kazhdan\'s property (T).Property (T) for $\operatorname{Aut}(F_{n})$ has been a long-standing open problem and, as observed by Lubotzky and Pak, the positive resolution leads to better understanding of the effectiveness of the product replacement algorithm commonly used in computational group theory.
In this talk we shall see examples and recent progress on how buildings show up in the quest to determine certain cohomological finiteness conditions for groups of matrices.
