Plenary Talks
Lev Birbrair, Universidade Federal do Ceará and Jagiellonian University
Title: Germs of Real Surfaces
Abstract: I am going to describe old and new results related to Inner, Outer and Ambient Lipschitz geometry of germs of Real semi-algebraic and definable surfaces. The subject is closely related to non-archimedean geometry and Knot Theory.
André Salles de Carvalho, Universidade de São Paulo
Title: TBA
Abstract: TBA
Peter C. Gibson, York University
Title: Orthogonal polynomials and 1D scattering theory
Abstract: We present recent results concerning inverse scattering for both discrete and continuous one-dimensional models. We highlight the key role of orthogonal polynomials on both the circle and the line. Time permitting, related open questions involving the spectral theory of banded matrices will also be discussed.
Simon Griffiths, PUC-Rio
Title: TBA
Abstract: TBA
Umberto Leone Hryniewicz, RWTH Aachen University
Title: Generic existence of Birkhoff sections for Reeb flows in dimension three
Abstract: In this talk we will explain how pseudo-holomorphic curves can be used to show that C^\infty generically a Reeb flow on a closed 3-manifold has a Birkhoff section. As a consequence, we can bring arguments due to Le Calvez and Sambarino for surface maps to the realm of flows, and prove that C^\infty generically the Reeb flow has positive topological entropy. This is joint work with Colin, Dehornoy and Rechtman.
Caroline Klivans, Brown University*
Title: TBA
Abstract: TBA
Yoshiharu Kohayakawa, USP
Title: TBA
Abstract: TBA
Taisa Lopes Martins, UFF
Title: TBA
Abstract: TBA
Carlos Gustavo Tamm de Araujo Moreira, IMPA
Title: Fractal geometry of the Markov and Lagrange spectra and their set difference.
Abstract: We will discuss some recent results on the fractal geometry of the Markov and Lagrange spectra, which are classical objects from the theory of Diophantine approximations, and their set difference. In particular, we discuss recent results in collaboration with Erazo, Gutiérrez-Romo and Romaña, which give precise asymptotic estimates for the fractal dimensions of the Markov and Lagrange spectra near 3 (their smaller accumulation point) and other recent collaborations with Jeffreys, Matheus, Pollicott and Vytnova, in which we prove that the Hausdorff dimension of the complement of the Lagrange spectrum in the Markov spectrum has Hausdorff dimension between 0.593 and 0.796445.
We will relate these results to symbolic dynamics, continued fractions and to the study of the fractal geometry of arithmetic sums of regular Cantor sets, a subject also important for the study of homoclinic bifurcations in Dynamical Systems.
James Propp, University of Massachusetts Lowell
Title: Spaces of Tilings
Abstract: In 1995, Saldanha and coauthors Tomei, Casarin, and Romualdo published their paper "Spaces of Domino Tilings", generalizing Thurston's notion of height-functions via the idea of height-sections. In my talk I'll generalize height-functions in a different direction and apply multidimensional height functions to tilings in a hexagonal grid. Along the way I'll use Mathematica to show what spaces of tilings actually look like.
Sinai Robins, Universidade de São Paulo
Title: multi-tilings and the geometry of numbers, using the Siegel-Bombieri approach
Abstract: We extend the Bombieri-Siegel formula from the geometry of numbers, by studying a lattice sum of the cross-covariogram of two bounded sets A, B ⊂ R^d . Using a variation of Poisson summation, our extension also refines the summation index of the Bombieri-Siegel formula, allowing us to obtain some interesting applications. One of the consequences of these results is a new characterization of multi-tiling Euclidean space by translations of a compact set. Another consequence is a spectral formula for the volume of a compact set. Finally, we give an application to arithmetic combinatorics, namely an identity for finite sums of discrete covariograms over any set of integer points in Z d . As a consequence, we arrive at an equivalent condition for multi-tiling Z^d by any finite set of integer points. This is joint work with my Ph.D. student Michel Faleiros Martins.
Boris Shapiro, Stockholm University
Title: Mystery of points charges after Gauss-Maxwell-Morse
Abstract: In his 2 volume chef-d'oeuvre “Treatise of electricity and Magnetism” J.C.Maxwell (among thousands of much more important claims) formulated the following statement.
Given any configurations of N fixed point charges in R^3, the electrostatic field created by them has at most (N-1)^2 points of equilibrium.
Maxwell’s arguments are incomplete and this problem was considered much later by M.Morse and revitalised about two decades ago. However Maxwell’s original claim is still open already in case of N=3 charges. In my talk I will survey what is known in this direction and, in particular, formulate a calculus 1 problem which currently still remains unsolved.
Michael Shapiro, Michigan State University
Title: Grassmann convexity
Abstract: We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation. We show that this formula gives the lower bound for the required total number of real zeros for equations of arbitrary order, and we prove that the formula is correct for equations of orders 4 and 5.
The talk is based on the joint work with N. Saldanha and B. Shapiro.
Carlos Tomei, PUC-Rio
Title: TBA
Abstract: TBA
Jeanine Van Order, PUC-Rio
Title: TBA
Abstract: TBA
David Matínez Torres, Universidad Politécnica de Madrid
Title: Grassmannians and Poisson geometry.
Abstract: The Poincaré disk has a natural holomorphic embedding into the complex projective line. One verifies that the inverse of the Poicaré Kahler form extends to the projective line to a "pseudo-Kahler" Poisson structure. It turns out that this property is common to any symmetric domain. In the talk we shall discuss this interaction between Poisson geometry and Hermitian symmetric spaces.
This is (ongoing) joint work with P. Frejlich.
Short Talks
Harold Erazo, IMPA
Title: Fractal geometry of the Markov and Lagrange spectra near 3.
Abstract: We will talk about some recent results about the Lagrange and Markov spectra near 3. It is well known that they coincide before 3 (which is the first accumulation point) and exhibit fractal behavior immediately afterward. We will discuss a recent joint work with Moreira, Gutiérrez-Romo and Romaña were we obtained precise asymptotics for the Hausdorff dimension of the Markov and Lagrange spectra near 3 and report on work in progress about this region together with Moreira, Matheus, Lima, Vieira.
Nils Hemmingsson, Stockholm University
Title: TBA
Abstract: TBA
Giovanna Leal, PUC-Rio
Title: Homotopy type of intersections of real Bruhat cells in dimension 6
Abstract: We examine arbitrary intersections of real Bruhat cells. Arising in various contexts across several disciplines -- such as in the Kazhdan-Lusztig theory and the study of the spaces of locally convex curves -- these objects have attracted the attention of many authors. We present a stratification of an arbitrary pairwise intersection of big real Bruhat cells. Also, we show the dual CW-complex of such stratification is homotopically equivalent to the intersection under analysis. Finally, both classical and new topological results about such intersections stem from our methods. Our focus here is on such intersections for real 6 X 6 matrices. For at least two different permutations, there exist non-contractible connected components. This is joint work with E. Alves and N. Saldanha.
David Whiting, Michigan State University
Title: Positive networks on a cylinder
Abstract: Networks embedded into a disk with positive weights have been previously classified by A.Postikov, and efforts to generalize this work to networks embedded into a cylinder have been done by S. Arthamonov, N. Ovenhouse, and M. Shapiro. Currently, these networks have found use in the study of scattering amplitudes. In this talk, I will present a new description for certain families of positive networks embedded on a cylinder, particularly those given by certain matrices of rank 2 and 3. This is based on a current work in progress with M. Shapiro.