Plenary Talks

Lev Birbrair, UFC 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, USP

Title: Structures on 1-, 2- and 3-dimensional manifolds coming from dynamical systems: an impressionistic overview 

Abstract: We will discuss several examples of geometric structures on manifolds which come from the action of a topological dynamical system on the manifold. Examples are the linear structure on an interval coming from a multimodal map, the quasi-Euclidean structure on surfaces coming from Thurston classification and some interesting structures on the 3-sphere coming from expanding Thurston maps, among others. The talk will (I hope) be accessible to non-experts.    

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:   Giants in graphs and digraphs

Abstract:  The theory of random graphs begin when Erdös and Rényi studied the threshold for a random graph to contain a "giant component", meaning a component which includes a positive proportion of the vertices.  Since then this problem has been studied in many random graph models.  In particular, Molloy and Reed proved major results for the case of random graphs with a given degree sequence, a model which is more versatile and general than the Erdös-Rényi model.  This has become an area of intense research.  We discuss recent progress on a related problem for directed graphs.

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. 

Yoshiharu Kohayakawa, USP

Title:   Arithmetic progressions in sumsets and in subsetsums of sparse random sets

Abstract: Given a set A, its sumset A + A is defined as the set of all sums of pairs of elements of A. Given p: N → [0, 1], we let Aₙ = [n]ₚ be the p-random subset of [n] = {1, . . . , n} and consider m* = m*(n), the largest m for which Aₙ + Aₙ contains an m-element arithmetic progression with high probability. We shall see that for certain choices of p⁻ = p⁻(n) and p⁺ = p⁺(n) with p⁻ ≤ n −1/2 ≤ p⁺ and p⁺/p⁻ = (log n)ˆω, where ω can be chosen to grow arbitrarily slowly, for p ≤ p⁻ we have m* ≪ (log n)/ log log n and for p ≥ p⁺ we have m* = Θ(n). Furthermore, if p ≥ n−1/2+ε for any fixed ε > 0, then long progressions exist in the sumset of any positive density subset of Aₙ: with high probability, for any subset S of Aₙ with a fixed proportion of elements of Aₙ, the sumset S + S contains arithmetic progressions with 2ˆ(Ω(√log n)) elements.

Based on joint with with Rafael K. Miyazaki and with Marcelo Campos and Gabriel Dahia.

Taisa Lopes Martins, UFF

Title: The size-Ramsey number of powers of bounded degree trees 

Abstract: Given a positive integer s, the s-colour size-Ramsey number of a graph H is the smallest integer m such that there exists a graph G with m edges where in any s-colouring of E(G) there is a monochromatic copy of H. We prove that, for any positive integers k and s, the s-colour size-Ramsey number of the kth power of any n-vertex tree is linear on n.

This is a joint work with S. Berger, Y. Kohayakawa, G. S. Maesaka, W. Mendonça, G. O. Mota and O. Parczyk. 

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, USP

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:   At the drawing board with Nicolau

Abstract: I will present some of my favourite moments of our collaboration.

Jeanine Van Order, PUC-Rio

Title:   Arithmetic heights of CM cycles and Fourier coefficients of half-integral weight forms 

Abstract: The refined conjecture of Birch and Swinnerton-Dyer predicts that the L-series of an elliptic curve has an analytic continuation given by a functional equation and that its Taylor series expansion around the central point is given by a certain constant. This constant contains a mysterious square term, which at least in some cases can be accounted for by the squares

of certain Fourier coefficients of half-integral weight forms. I will also explain this connection, as well as that to arithmetic heights of CM cycles on modular curves in the case of rank one, and some new research directions. 

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: Holomorphic correspondences 

Abstract: This talk is about holomorphic  correspondences, which are finite-to-finte multivalued maps, on the Riemann sphere with a.e locally defined holomorphic branches. The iteration of these generalize the iterations of rational maps and finitely generated Kleinian groups.  I will provide go through interesting examples of holomorphic correspondences and if time allows, mention some results based on work in progress joint with Xiaoran Li and Zhiqiang Li.

 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. We try to generalize this work to other surfaces. In this talk, I will describe one method of studying positive networks embedded on a cylinder and provide a new description for certain families of networks by matrices of rank 2 and 3. This is based on a current work in progress with M. Shapiro.