Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Clone of Curves Seminar - Mike Roth (Queen's University)

Wednesday, January 29th, 2020

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

Speaker: Mike Roth (Queen's University)

Title: An overview of Brill-Noether theory.

Abstract: The object of Brill-Noether theory is to answer the question : which $g^{r}_{d}$’s can a generic curve of genus $g$ have?

The relevance to Green’s conjecture is that this then allows us to compute the Clifford index of a generic curve of genus $g$.

The lecture will be an overview of Brill-Noether theory, including a sketch of the main lines of argument.

Number Theory Seminar - Seoyoung Kim (Queen's University)

Monday, January 27th, 2020

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

Speaker: Seoyoung Kim (Queen's University)

Title: On the density of irreducible polynomials which generate $k$-free polynomials over function fields.

Abstract: Let $M \in \FF_{q}[t]$ be a polynomial, and let $k \geq 2$ be an integer. In this talk, we will compute the asymptotic density of irreducible monic polynomials $P\in\FF_{q}[t]$ for which $P+M$ is not divisible by the $k$-th power of any irreducible polynomial.

Department Colloquium - Hok Kan Ling (Columbia University)

Hok Kan Ling (Columbia University)

Monday, January 27th, 2020

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 Ying. 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.

Dynamics, Geometry, & Groups - Rylee Lyman (Tufts University)

Friday, January 24th, 2020

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.

Department Colloquium - Siliang Gong (University of Pennsylvania)

Siliang Gong (University of Pennsylvania)

Thursday, January 23rd, 2020

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.

Math Club - Ivan Dimitrov (Queen's University)

Thursday, January 23rd, 2020

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.

Curves Seminar - Gregory G. Smith (Queen's University)

Wednesday, January 22nd, 2020

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.

Number Theory Seminar - Francesco Cellarosi (Queen's University)

Monday, January 20th, 2020

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.