Fall 2020 Titles and Abstracts

Date: 18 September 2020
Speaker: Emmanuel Breuillard
Title: A subspace theorem for manifolds
Abstract: Schmidt’s subspace theorem is a fundamental result in diophantine approximation and a natural generalization of Roth’s celebrated theorem. In this talk I will discuss a geometric understanding of this theorem that blends homogeneous dynamics and geometric invariant theory. Combined with the Kleinbock-Margulis quantitative non-divergence estimates this yields a natural generalization of the subspace theorem to systems of linear forms that depend nicely on a parameter. I will also present several applications and consequences of the main result. Joint work with Nicolas de Saxcé. 

Date: 25 September 2020
Speaker: Yotam Smilansky
Title: Multiscale substitution tilings
Abstract: Multiscale substitution tilings are a new family of tilings of Euclidean space that are generated by multiscale substitution rules. Unlike the standard setup of substitution tilings, which is a basic object of study within the aperiodic order community and includes examples such as the Penrose and the pinwheel tilings, multiple distinct scaling constants are allowed, and the defining process of inflation and subdivision is a continuous one. Under a certain irrationality assumption on the scaling constants, this construction gives rise to a new class of tilings, tiling spaces and tiling dynamical systems, which are intrinsically different from those that arise in the standard setup. In the talk I will describe these new objects and discuss various structural, geometrical, statistical and dynamical results. Based on joint work with Yaar Solomon.

Date: 2 October 2020
Speaker: Samantha Fairchild
Title: Counting social interactions for discrete subsets of the plane
Abstract: Given a discrete subset V in the plane, how many points would you expect there to be in a ball of radius 100? What if the radius is 10,000? Due to the results of Fairchild and forthcoming work with Burrin, when V arises as orbits of non-uniform lattice subgroups of SL(2,R), we can understand asymptotic growth rate with error terms of the number of points in V for a broad family of sets. A crucial aspect of these arguments and similar arguments is understanding how to count pairs of saddle connections with certain properties determining the interactions between them, like having a fixed determinant or having another point in V nearby. We will focus on a concrete case used to state the theorem and highlight the proof strategy. We will also discuss some ongoing work and ideas which advertise the generality and strength of this argument.

Date: 9 October 2020
Speaker: Nattalie Tamam
Title: Effective equidistribution of horospherical flows in infinite volume
Abstract: We want to provide effective information about averages of orbits of the horospherical subgroup acting on a hyperbolic manifold of infinite volume. We start by presenting the setting and results for manifolds with finite volume. Then, discuss the difficulties that arise when studying the infinite volume setting, and the measures that play a crucial role in it. This is joint work with Jacqueline Warren.

Date: 16 October 2020
Speaker: Douglas Lind
Title: Decimation Limits of Algebraic Actions
Abstract: This is intended to be an expository talk using simple examples to illustrate what’s going on, and so will (hopefully) be a gentle introduction to these topics. Given a polynomial in d commuting variables we can define an algebraic action of ℤ^d by commuting automorphisms of a compact subgroup of 𝕋^(ℤ^d). Restricting the coordinates of points in this group to finite-index subgroups of ℤ^d gives other algebraic actions, defined by polynomials whose support grows polynomially and whose coefficients grow exponentially. But by “renormalizing” we can obtain a limiting object that is a concave function on ℝ^d with interesting properties, e.g. its maximum value is the entropy of the action. For some polynomials this function also arises in statistical mechanics models as the “surface tension” of a random surface via a variational principle. In joint work with Arzhakova, Schmidt, and Verbitskiy, we establish this limiting behavior, and identify the limit in terms of the Legendre transform of the Ronkin function of the polynomial. The proof is based on Mahler’s estimates on polynomial coefficients using Mahler measure, and an idea used by Boyd to prove that Mahler measure is continuous in the coefficients of the polynomial. Refinements of convergence questions involve diophantine issues that I will discuss, together with some open problems. 

Date: 23 October 2020
Speaker: Mishel Skenderi
Title: Small values at integer points of generic subhomogeneous functions
Abstract: This talk will be based on joint work with Dmitry Kleinbock that has been motivated by several recent papers (among them, those of Athreya-Margulis, Bourgain, Ghosh-Gorodnik-Nevo, Kelmer-Yu). Given a certain sort of group $G$ and certain sorts of functions $f: \mathbb{R}^n \to \mathbb{R}$ and $\psi : \mathbb{R}^n \to \mathbb{R}_{>0},$ we obtain necessary and sufficient conditions so that for Haar-almost every $g \in G,$ there exist infinitely many (respectively, finitely many) $v \in \mathbb{Z}^n$ for which $|(f \circ g)(v)| \leq \psi(\|v\|),$ where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^n.$ We also give a sufficient condition in the setting of uniform approximation. As a consequence of our methods, we obtain generalizations to the case of vector-valued (simultaneous) approximation with no additional effort. In our work, we use probabilistic results in the geometry of numbers that go back several decades to the work of Siegel, Rogers, and W. Schmidt; these results have recently found new life thanks to a 2009 paper of Athreya-Margulis.

Date: 6 November 2020 
Speaker: Byungchul Cha
Title: Intrinsic Diophantine Approximation of circles
Abstract: Let $S^1$ be the unit circle in $\mathbb{R}^2$ centered at the origin and let $Z$ be a countable dense subset of $S^1$, for instance, the set $Z = S^1(\mathbb{Q})$ of all rational points in $S^1$. We give a complete description of an initial discrete part of the Lagrange spectrum of $S^1$ in the sense of intrinsic Diophantine approximation. This is an analogue of the classical result of Markoff in 1879, where he characterized the most badly approximable real numbers via the periods of their continued fraction expansions. Additionally, we present similar results for a few different subsets $Z$ of $S^1$. This is joint work with Dong Han Kim.

Date: 13 November 2020
Speaker: Jacqueline Warren
Title: Joining classification and factor rigidity in infinite volume
Abstract: For a group acting on two spaces, a joining of these systems is a measure on the product space that is invariant under the diagonal action and projects to the original measures on each space. Joinings are a powerful tool in ergodic theory, and joinings for the horocycle flow were classified by Ratner in the finite volume setting, with many interesting applications. In this talk, I will discuss some of these applications and present joining classification for horospherical flows in the infinite volume setting, as well as a key factor rigidity theorem that is used in the proof. This talk is intended to be accessible to graduate students.

Date: 20 November 2020
Speaker: Shahriar Mirzadeh
Title: On the dimension drop conjecture for diagonal flows on the space of lattices
Abstract: Consider the set of points in a homogeneous space X=G/Gamma whose g_t orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when it has real rank 1. In this talk we will prove the conjecture for probably the most important example of the higher rank case, namely: G=SL(m+n, R), Gamma=SL(m+n,Z), and g_t = diag(exp(t/m), …, exp(t/m), exp(-t/n), …, exp(-t/n)). We can also use our main result to produce new applications to Diophantine approximation. This project is joint work with Dmitry Kleinbock.

Date: 4 December 2020
Speaker: Osama Khalil
Title: Large centralizers and counting integral points on affine varieties
Abstract: Duke-Rudnick-Sarnak and Eskin-McMullen initiated the use of ergodic methods to count integral points on affine homogeneous varieties. They reduced the problem to one of studying limiting distributions of translates of periods of reductive groups on homogeneous spaces. The breakthrough of Eskin, Mozes and Shah provided a rather complete understanding of this question in the case the reductive group has a “small centralizer” inside the ambient group. In this talk, we describe work in progress giving new results on the equidistribution of generic translates of closed orbits of semisimple groups with “large centralizers”. The key new ingredient is an algebraic description of a partial compactification (for lack of a better word) of the set of intermediate groups which act as obstructions to equidistribution. This allows us to employ tools from geometric invariant theory to study the avoidance problem.

Date: 11 December 2020 
Speaker: Anthony Sanchez
Title: Gaps of saddle connection directions for some branched covers of tori
Abstract: Holonomy vectors of translation surfaces provide a geometric generalization for higher genus surfaces of (primitive) integer lattice points. The counting and distribution properties of holonomy vectors on translation surfaces have been studied extensively. In this talk, we consider the following question: How random are the holonomy vectors of a translation surface? We motivate the gap distribution of slopes of holonomy vectors as a measure of randomness and compute the gap distribution for the class of translation surfaces given by gluing two identical tori along a slit. No prior background on translation surfaces or gap distributions will be assumed.