We establish central limit theorems for principal eigenvalues and eigenvectors under a large factor model setting, and develop two-sample tests for differences in either principal eigenvalues or principal eigenvectors. As an application, these tests can be used to detect structural breaks in large factor models. To the best of our knowledge, our tests are the first that can distinguish between individual eigenvalues and/or eigenvectors, and hence provide unique insights into the source of structural breaks. Based on joint work with Jianqing Fan, Yingying Li and Ningning Xia.

Homology 3-spheres, i.e. 3-dimensional manifolds with the same homology groups as the standard 3-sphere, play a central role in topology. Their study was initiated by Poincare in 1904, who constructed the first nontrivial example of a homology 3-sphere, and conjectured that the standard sphere is the only simply connected example. A century later, Poincare's conjecture was finally resolved by Perelman, but we are still far from understanding the general classification of homology 3-spheres. This classification problem can be packaged in terms of the homology cobordism group, which is an abelian group formed by the set of all homology 3-spheres modulo a cobordism relation. I will survey what is known about this group, as well as discuss a closely related group classifying knots in the 3-sphere, including recent results joint with Irving Dai, Jennifer Hom, and Matthew Stoffregen.

The collection of birational self-maps of an algebraic variety X forms a group Bir(X). When X is a surface, this group acts on an infinite hyperbolic space, constructed via divisors on blowups of the surface. In this talk, we introduce this construction and use it to understand Bir(X). Time permitting, we will discuss very recent work of Lonjou and Urech on Bir(X) for higher dimensional X. The main source for this talk is the paper "Sur les groupes de transformations birationnelles des surfaces" by Serge Cantat.

We will continue the introduction to quandles from the last meeting. This talk is self-contained and will review the previous talk quickly. An (involuntary) quandle is an algebraic structure Q with one binary operation satisfying a*a=a , (a*b)*b = a, (a*b)*c = (a*c)*(b*c) for all a,b,c in Q. Quandles can be used to make invariants of knots and links. Here is an algebraic example: Let G be a group with multiplication ab. Define a*b = ba^{-1}b and verify that this gives a quandle structure on G. This construction is related to the fundamental group of the double branched covering of a knot or link in the three sphere, and can be used to define a quandle invariant of knots that detects the unknot.

Artin groups emerged from the study of braid groups, complex hyperplane arrangements and Coxeter groups. Recently they also play an important role in the understanding of 3-manifolds. Despite the seemingly simple presentation of Artin groups, they have rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric properties of these groups. No previous knowledge on Artin groups or Garside groups is required. Based on work with D. Osajda.

The widespread adoption of electronic health records (EHR) and their subsequent linkage to specimen biorepositories has generated massive amounts of routinely collected medical data for use in translational research. These integrated data sets enable real-world predictive modeling of disease risk and progression. However, data heterogeneity and quality issues impose unique analytical challenges on the development of EHR-based prediction models. For example, ascertainment of validated outcome information, such as the presence of a disease or treatment response, is particularly challenging because it requires manual chart review. Outcome information is therefore only available for a small subset of patients in the cohort of interest, unlike the traditional setting where this information is available for all patients. In this talk, I will discuss semi-supervised and weakly-supervised learning methods for predictive modeling in such constrained settings where the proportion of labeled data is very small. I demonstrate that leveraging unlabeled examples can improve the efficiency of model estimation and evaluation procedures, which in turn substantially reduces the amount of labeled data required for developing prediction models.

The high dimensional graphical model, a powerful tool for studying conditional dependency relationship of random variables, has attracted great attention especially in biological network analysis with different types of omics data. While significant progress has been achieved recently in computing confidence intervals and p-values of each edge for Gaussian graphical model (GGM), the usage of GGM on important discrete-type omics data can be statistically and biologically inappropriate. On the other hand, discrete-type graphical models were proposed to tailor the network analysis of count-valued data, but the statistical inference of these models is not well studied. In this talk, we investigate statistical inference of each edge for large Ising and modified Poisson-type graphical models. The key role in most existing inferential methods is played by a linear projection method to de-bias an initial regularized estimator. Major drawback of this approach in those discrete-type graphical models is that an extra sparsity assumption on the linear projection coefficient is required, which cannot be checked in practice. In addition, efficiency often is compromised by the usage of sample splitting in these methods. To solve these challenges, we first propose a novel estimator of each edge for Ising model via quadratic programming and show that our estimator is asymptotically normal without the above mentioned extra sparsity condition. Our proof applies a novel low dimensional maximum likelihood method for the de-bias procedure and a data swap technique to avoid loss of efficiency. In addition, we further show that whenever the extra sparsity condition is satisfied, our estimator is adaptively efficient and achieves the Fisher information. Otherwise, we still provide a restricted Fisher information as a lower bound. We then extend our approach to modified Poisson-type graphical models for both edge-wise and global statistical inference. The practical merit of the proposed method is demonstrated by an application to a novel RNA-seq gene expression data set in childhood atopic asthma in Puerto Ricans. Compared to sole estimation and statistical inference of GGM, our method provides more biologically meaningful results.

It's been known since work of Harrington in the early 1970s that computable differential fields have computable differential closures. Recently Calvert, Frolov, Harizanov, Knight, McCoy, Soskova, and Vatev showed that the countable saturated differentially closed field is computable. Their proof involves first creating an effective listing of all types and then using a result of Morley's on existence of computable saturated models. I will give a significant simplification of the enumeration result and, for completeness, sketch Morley's priority construction of a saturated model. Pillay has also given an alternative enumeration argument though ours seems more robust and generalizes to quantifier free types in ACFA.

Variable selection has been well studied in the recent literatures due to the surge of enormous high dimensional data. Interaction between predictors is commonly expected to exist in all kinds of real applications. Recently some parametric interaction selection methods have been proposed. In this talk, we will present a new method to perform nonparametric interaction selection and screening, based on the measurement error selection likelihood approach. This method uses local constant smoothing and backfitting algorithm to perform main and interaction selection for additive model. Resulting solution path will exhibit the importance of predictors. Finite-sample simulation shows this method performs well.

After a brief introduction to classical hypergraph Ramsey numbers, I will focus on the following problem. What is the minimum t such that there exist arbitrarily large k-uniform hypergraphs whose independence number is at most polylogarithmic in the number of vertices and every s vertices span at most t edges? Erdos and Hajnal conjectured (1972) that this minimum can be calculated precisely using a recursive formula and Erdos offered a \$500 prize for a proof. For $k = 3$, this has been settled for many values of s, but it was not known for larger k. Here we settle the conjecture for all k at least 4. Our method also answers a question of Bhat and Rodl about the maximum upper density of quasirandom hypergraphs. This is joint work with Alexander Razborov.

In contemporary scientific research, it is often of great interest to predict a categorical response based on a high-dimensional tensor (i.e. multi-dimensional array). Motivated by applications in science and engineering, we propose two probabilistic methods for machine learning on tensor data in the supervised and the unsupervised context, respectively. For supervised problems, we develop a comprehensive discriminant analysis model, called the CATCH model. The CATCH model integrates the information from the tensor and additional covariates to predict the categorical outcome with high accuracy. We further consider unsupervised problems, where no categorical response is available even on the training data. A doubly-enhanced EM (DEEM) algorithm is proposed for model-based tensor clustering, in which both the E-step and the M-step are carefully tailored for tensor data. CATCH and DEEM are developed under explicit statistical models with clear interpretations. They aggressively take advantage of the tensor structure and sparsity to tackle the new computational and statistical challenges arising from the intimidating tensor dimensions. Efficient algorithms are developed to solve the related optimization problems. Under mild conditions, CATCH and DEEM are shown to be consistent even when the dimension of each mode grows at an exponential rate of the sample size. Numerical studies also strongly support the application of CATCH and DEEM. Finally, we discuss how these developments in tensor data advance vector data analysis, such as differential network analysis.

It is a classical result (combining results of Tits and Hall) that there is no sharply 4-transitive action of a group on an infinite set. For algebraic groups acting on (infinite) varieties, there is no 4-transitive group action. Things get more interesting when we loosen the requirements slightly and merely demand that the action has a "large" orbit for a suitable notion of large. In this talk we will discuss some conjectures in this area and the current prospects for solving them. We will also talk about the connection between conjectures in this area and some notions from geometric stability theory.

The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. In this talk, I will sketch a proof showing that every $n$-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least $e^{(\log n)^{1 - o(1)}}$. The dependence on the VC-dimension is hidden in the $o(1)$ term. This improves the general lower bound, $e^{c\sqrt{\log n}}$, due to Erdos and Hajnal. This result is almost optimal and nearly matches the celebrated Erdos-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least $e^{\Omega(\log n)}$ If time permits, I will also discuss a multicolor Ramsey theorem for graphs with bounded VC-dimension. This is joint work with Jacob Fox and Janos Pach.

Prediction with expert advice is an area of online machine learning, which aims to synthesize advice from different experts. We consider the case of a stock prediction problem with an investor who relies on history-dependent experts, and an adversarial market. This forms a two-person game, and we are interested in the optimal strategies of the market and the player when the game is played over a long time. We prove that the discrete value function converges to the unique solution of a nonlinear parabolic PDE, which determines asymptotically optimal strategies.

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We develop a novel shrinkage rule for prediction in a high-dimensional non-exchangeable hierarchical Gaussian model with an unknown spiked covariance structure. We propose a family of commutative priors for the mean parameter, governed by a power hyper-parameter, which encompasses from perfect independence to highly dependent scenarios. Corresponding to popular loss functions such as quadratic, generalized absolute, and linex losses, these prior models induce a wide class of shrinkage predictors that involve quadratic forms of smooth functions of the unknown covariance. By using uniformly consistent estimators of these quadratic forms, we propose an efficient procedure for evaluating these predictors which outperforms factor model based direct plug-in approaches. We further improve our predictors by introspecting possible reduction in their variability through a novel coordinate-wise shrinkage policy that only uses covariance level information and can be adaptively tuned using the sample eigen structure. We extend our methodology to aggregation based prescriptive analysis of generic multidimensional linear functionals of the predictors that arise in many contemporary applications involving forecasting decisions on portfolios or combined predictions from dis-aggregative level data. We propose an easy-to-implement functional substitution method for predicting linearly aggregative targets and establish asymptotic optimality of our proposed procedure. We present simulation experiments as well as real data examples illustrating the efficacy of the proposed method.

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Inference of causal relations between interacting units in a directed acyclic graph (DAG), such as a regulatory gene network, is common in practice, imposing challenges because of a lack of inferential tools. In this talk, I will present constrained likelihood ratio tests for inference of the connectivity as well as directionality subject to nonconvex acyclicity constraints in a Gaussian directed graphical model. Particularly, for testing of connectivity, the asymptotic distribution is either chi-squared or normal depending on if the number of testable links in a DAG model is small; for testing of directionality, the asymptotic distribution is the minimum of d independent chi-squared variables with one-degree of freedom or a generalized Gamma distribution depending on if d is small, where d is the number of breakpoints in a hypothesized pathway. Computational methods will be discussed, in addition to some numerical examples to infer a directed pathway in a gene network. This work is joint with Chunlin Li and Wei Pan of the University of Minnesota.

When considering N-player differential games, making the approximation that there are instead infinitely many agents leads to the mean field games system of PDEs. This system has two unknowns, the probability distribution of the players, and the value function being optimized by a representative agent. One of these satisfies a forward parabolic equation and the other satisfies a backward parabolic equation. The forward parabolic equations comes with initial data while terminal data (at a fixed time T>0) is specified for the backward parabolic equation. We will describe some existence results for this coupled forward-backward system, including for a specific system which has been given as a model of household wealth.

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