## Topological Data Analysis - Troy Zeier (Queen's University)

### Tuesday, January 28th, 2020

**Time:** 2:30-4:00 p.m. **Place:** Goodes Hall 118

**Speaker:** Troy Zeier (Queen's University)

**Topics: **Applications of Topological Data Analysis.

All are welcome!

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**Time:** 2:30-4:00 p.m. **Place:** Goodes Hall 118

**Speaker:** Troy Zeier (Queen's University)

**Topics: **Applications of Topological Data Analysis.

All are welcome!

**Time:** 2:30 p.m. **Place:** Jeffery Hall 234

**Speaker:** Hok Kan Ling (Columbia University)

**Title:** Shape-constrained Estimation and Testing.

**Abstract: ** Shape-constrained inference has been gaining more attention recently. Such constraints are sometimes the direct consequence of the problem under investigation. In other times, they are used to replace parametric models while retaining qualitative shape properties that exist in problems from diverse disciplines. In this talk, I will first discuss the estimation of a monotone density in s-sample biased sampling models, which has been long missing in the literature due to certain non-standard nature of the problem. We established the asymptotic distribution of the maximum likelihood estimator (MLE) and the connection between this MLE and a Grenander-type estimator. In the second part of the talk, a nonparametric likelihood ratio test for the hypothesis testing problem on whether a random sample follows a distribution with a decreasing, k-monotone or log-concave density is proposed. The obtained test statistic has a surprisingly simple and universal asymptotic null distribution, which is Gaussian, instead of the well-known chi-square for generic likelihood ratio tests. We also established rates of convergence of the maximum likelihood estimator under weaker conditions than the existing literature that are of independent interest.

**Hok Kan (Brian) Ling** is a Ph.D. candidate in the Department of Statistics at Columbia University, working under the supervision of Dr. Zhiliang Yin. His research interests primarily lie in the areas of multivariate statistics, latent variable models, event history analysis, nonparametric estimation, semiparametric models and shape-restricted statistical inference.

**Time:** 10:30 a.m** Place:** Jeffery Hall 319

**Speaker:** Rylee Lyman (Tufts University)

**Title:** Train tracks and pseudo-Anosov braids in automorphisms of free products.

**Abstract:** The Nielsen–Thurston classification of surface homeomorphisms says that every homeomorphism of a surface either has a finite power isotopic to the identity, preserves the isotopy class of some essential multi-curve, or is isotopic to a pseudo-Anosov map, the most interesting kind. Bestvina and Handel introduced a similar classification for automorphisms of free groups. Here the analogue of a pseudo-Anosov homeomorphism is a train track map for an outer automorphism which is fully irreducible, a homotopy equivalence of a graph with extra structure. The analogy really is correct: pseudo-Anosov mapping classes of once-punctured surfaces induce fully irreducible outer automorphisms preserving a nontrivial conjugacy class and vice-versa. We discuss extensions of the train track theory to automorphisms of free products. Here the analogy is to mapping classes of punctured spheres. We show that fully irreducible automorphisms of free products of finite subgroups of SO(2) may be represented as pseudo-Anosov braids on orbifolds if and only if they preserve a non-peripheral conjugacy class.

**Time:** 2:30 p.m. **Place:** Jeffery Hall 234

**Speaker:** Siliang Gong (University of Pennsylvania)

**Title:** Per-family Error Rate Control for Gaussian Graphical Models via Knockoffs.

**Abstract: ** Driven by many real applications, the estimation of and inference for Gaussian Graphical Models (GGM) are fundamentally important and have attracted much research interest in the literature. However, it is still challenging to achieve overall error rate control when recovering the graph structures of GGM. In this paper, we propose a new multiple testing method for GGM using the knockoffs framework. Our method can control the overall finite-sample Per-Family Error Rate up to a probability error bound induced by the estimation errors of knockoff features. Numerical studies demonstrate that our method has competitive performance compared with existing methods. This is joint work with Qi Long and Weijie Su.

**Siliang Gong** is a postdoctoral fellow in the Department of Biostatistics at the University of Pennsylvania. She completed her Ph.D. in statistics at the University of North Carolina at Chapel Hill in 2018. She works on high-dimensional data analysis and statistical machine learning.

**Time:** 5:30 - 6:30 p.m ** Place:** Jeffery Hall 319

**Speaker:** Ivan Dimitrov (Queen's University)

**Title:** Proof of the Sensitivity Conjecture.

**Abstract:** Last year Hao Huang proved that, if $P$ is a set of $2^{n-1} +1$ vertices of an n-dimensional cube, it contains a vertex with at least $\sqrt{n}$ neighbours in $P$. This settled the nearly 30-year old Sensitivity Conjecture. I will present Huang’s proof and show that the estimate $\sqrt{n}$ is sharp.

**Time:** 4:00-5:30 p.m ** Place:** Jeffery Hall 319

**Speaker:** Gregory G. Smith (Queen's University)

**Title:** Green-Lazarsfeld nonvanishing.

**Abstract:** By exploiting vector bundle techniques for Koszul cohomology, we see how non-trivial geometry leads to non-trivial syzygies. In particular, this establishes one part of Green’s conjecture.

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**Time:** 2:30-4:00 p.m. **Place:** Goodes Hall 120

**Speaker:** Abdullah Zafar (University of Toronto)

**Topics: **Topological Data Analysis in Soccer.

All are welcome!

**Time:** 4:30-5:30 p.m. **Place:** Jeffery Hall 422

**Speaker:** Francesco Cellarosi (Queen's University)

**Title:** Smooth arithmetical sums over k-free integers.

**Abstract:** We use partial zeta functions to analyse the asymptotic behaviour of certain smooth arithmetical sums over smooth $k$-free integers. This is joint work with Ram Murty.

**Time:** 10:30 a.m** Place:** Jeffery Hall 319

**Speaker:** Kasun Fernando (University of Toronto)

**Title:** Edgeworth Expansions for (mostly) hyperbolic dynamical systems.

**Abstract:** Given a dynamical system which shows hyperbolicity on a large part of phase space, one would expect it to exhibit good statistical properties like rapid decay of correlations, the Central Limit Theorem (CLT), Large Deviation Principle and etc. In this talk, I will discuss sufficient conditions for such mostly hyperbolic dynamical systems to admit Edgeworth expansions in the CLT. Our focus is on systems that admit a Young tower with return times with an exponentially decaying tail. This is an on-going joint work with Françoise Pène.

**Time:** 2:30 p.m. **Place:** Jeffery Hall 234

**Speaker:** Kasun Fernando (University of Toronto)

**Title:** Error terms in the Central Limit Theorem.

**Abstract: ** Expressing the error terms in the Central Limit Theorem as an asymptotic expansion (commonly referred to as the Edgeworth expansion) goes back to Chebyshev. In the setting of sums of independent identically distributed (iid) random variables, sufficient conditions for the existence of such expansions have been extensively studied. However, there is almost no literature that describe this error when the expansions fail to exist. In this talk, I will discuss the case of sums of iid non-lattice random variables with $d+1$ atoms. It can shown that they never admit the Edgeworth expansion of order d. However, using tools from Homogeneous Dynamics, it can shown that for almost all parameters the Edgeworth expansion of order $d-1$ holds and the error of the order $d-1$ Edgeworth expansion is typically of order $n^{-d/2}$ but the order $n^{-d/2}$ terms have wild oscillations (to be made precise during the talk). This is a joint work with Dmitry Dolgopyat.

**Kasun Fernando** is a postdoctoral fellow in the Department of Mathematics at the University of Toronto. He completed his Ph.D. in 2018 at the University of Maryland, College Park. His research is primarily focused on possible extensions of this theory of asymptotic expansions to more general settings that are not included in the classical theory, including the case of random variables arising as observations of chaotic dynamical systems.