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In this paper, we prove a transchromatic phenomenon for Hill--Hopkins--Ravenel and Lubin--Tate theories. This establishes a direct relationship between the equivariant slice spectral sequences of height-$h$ and height-$(h/2)$ theories. As applications of this transchromatic phenomenon, we prove periodicity and vanishing line results for these theories., Comment: 47 pages, 7 figures. Comments welcome!

In this paper, we construct a stratification tower for the equivariant slice filtration. This tower stratifies the slice spectral sequence of a $G$-spectrum $X$ into distinct regions. Within each of these regions, the differentials are determined by the localized slice spectral sequences, which compute the geometric fixed points along with their associated residue group actions. Consequently, the stratification tower offers an inductive method of understanding the entirety of the equivariant slice spectral sequence of $X$ by examining each of its distinct stratification regions., Comment: 26 pages, 10 figures. Comments welcome!

With the modern software and online platforms to collect massive amount of data, there is an increasing demand of applying causal inference methods at large scale when randomized experimentation is not viable. Weighting methods that directly incorporate covariate balancing have recently gained popularity for estimating causal effects in observational studies. These methods reduce the manual efforts required by researchers to iterate between propensity score modeling and balance checking until a satisfied covariate balance result. However, conventional solvers for determining weights lack the scalability to apply such methods on large scale datasets in companies like Snap Inc. To address the limitations and improve computational efficiency, in this paper we present scalable algorithms, DistEB and DistMS, for two balancing approaches: entropy balancing and MicroSynth. The solvers have linear time complexity and can be conveniently implemented in distributed computing frameworks such as Spark, Hive, etc. We study the properties of balancing approaches at different scales up to 1 million treated units and 487 covariates. We find that with larger sample size, both bias and variance in the causal effect estimation are significantly reduced. The results emphasize the importance of applying balancing approaches on large scale datasets. We combine the balancing approach with a synthetic control framework and deploy an end-to-end system for causal impact estimation at Snap Inc., Comment: KDD '23: Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining

Culture shapes people's behavior, both online and offline. Surprisingly, there is sparse research on how cultural context affects network formation and content consumption on social media. We analyzed the friendship networks and dyadic relations between content producers and consumers across 73 countries through a cultural lens in a closed-network setting. Closed networks allow for intimate bonds and self-expression, providing a natural setting to study cultural differences in behavior. We studied three theoretical frameworks of culture - individualism, relational mobility, and tightness. We found that friendship networks formed across different cultures differ in egocentricity, meaning the connectedness between a user's friends. Individualism, mobility, and looseness also significantly negatively impact how tie strength affects content consumption. Our findings show how culture affects social media behavior, and we outline how researchers can incorporate this in their work. Our work has implications for content recommendations and can improve content engagement., Comment: 9 pages, 4 pages of supplementary information. Accepted to be published in the Proceedings of the 2023 CHI Conference on Human Factors in Computing Systems

We show that at the prime 2, for any height $h$ and any finite subgroup $G \subset \mathbb{G}_h$ of the Morava stabilizer group, the $RO(G)$-graded homotopy fixed point spectral sequence for the Lubin--Tate spectrum $E_h$ has a strong horizontal vanishing line of filtration $N_{h, G}$, a specific number depending on $h$ and $G$. It is a consequence of the nilpotence theorem that such homotopy fixed point spectral sequences all admit strong horizontal vanishing lines at some finite filtration. Here, we establish specific bounds for them. Our bounds are sharp for all the known computations of $E_h^{hG}$. Our approach involves investigating the effect of the Hill--Hopkins--Ravenel norm functor on the slice differentials. As a result, we also show that the $RO(G)$-graded slice spectral sequence for $(N_{C_2}^{G}\bar{v}_h)^{-1}BP^{(\!(G)\!)}$ shares the same horizontal vanishing line at filtration $N_{h, G}$. As an application, we utilize this vanishing line to establish a bound on the orientation order $\Theta(h, G)$, the smallest number such that the $\Theta(h, G)$-fold direct sum of any real vector bundle is $E_h^{hG}$-orientable., Comment: 23 pages, accepted version. To appear in Geometry & Topology

In this paper, we investigate equivariant quotients of the Real bordism spectrum's multiplicative norm $MU^{((C_{2^n}))}$ by permutation summands. These quotients are of interest because of their close relationship with higher real $K$-theories. We introduce new techniques for computing the equivariant homotopy groups of such quotients. As a new example, we examine the theories $BP^{((C_{2^n}))}\langle m,m\rangle$. These spectra serve as natural equivariant generalizations of connective integral Morava $K$-theories. We provide a complete computation of the $a_{\sigma}$-localized slice spectral sequence of $i^*_{C_{2^{n-1}}}BP^{((C_{2^n}))}\langle m,m\rangle$, where $\sigma$ is the real sign representation of $C_{2^{n-1}}$. To achieve this computation, we establish a correspondence between this localized slice spectral sequence and the $H\mathbb{F}_2$-based Adams spectral sequence in the category of $H\mathbb{F}_2 \wedge H\mathbb{F}_2$-modules. Furthermore, we provide a full computation of the $a_{\lambda}$-localized slice spectral sequence of the height-4 theory $BP^{((C_{4}))}\langle 2,2\rangle$. The $C_4$-slice spectral sequence can be entirely recovered from this computation., Comment: Improved exposition. 56 pages, 15 figures

We study modules over the commutative ring spectrum $H\mathbb F_2\wedge H\mathbb F_2$, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator $\xi_k$ in the category of associative algebras freely kills the higher generators $\xi_{k+n}$. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative $H\mathbb F_2\wedge H\mathbb F_2$-algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum.

Effectively measuring, understanding, and improving mobile app performance is of paramount importance for mobile app developers. Across the mobile Internet landscape, companies run online controlled experiments (A/B tests) with thousands of performance metrics in order to understand how app performance causally impacts user retention and to guard against service or app regressions that degrade user experiences. To capture certain characteristics particular to performance metrics, such as enormous observation volume and high skewness in distribution, an industry-standard practice is to construct a performance metric as a quantile over all performance events in control or treatment buckets in A/B tests. In our experience with thousands of A/B tests provided by Snap, we have discovered some pitfalls in this industry-standard way of calculating performance metrics that can lead to unexplained movements in performance metrics and unexpected misalignment with user engagement metrics. In this paper, we discuss two major pitfalls in this industry-standard practice of measuring performance for mobile apps. One arises from strong heterogeneity in both mobile devices and user engagement, and the other arises from self-selection bias caused by post-treatment user engagement changes. To remedy these two pitfalls, we introduce several scalable methods including user-level performance metric calculation and imputation and matching for missing metric values. We have extensively evaluated these methods on both simulation data and real A/B tests, and have deployed them into Snap's in-house experimentation platform., Comment: WSDM '21: Proceedings of the 14th International Conference on Web Search and Data Mining

In this paper, we introduce the localized slice spectral sequence, a variant of the equivariant slice spectral sequence that computes geometric fixed points equipped with residue group actions. We prove convergence and recovery theorems for the localized slice spectral sequence and use it to analyze the norms of the Real bordism spectrum. As a consequence, we relate the Real bordism spectrum and its norms to a form of the $C_2$-Segal conjecture. We compute the localized slice spectral sequence of the $C_4$-norm of $BP_\mathbb{R}$ in a range and show that the Hill--Hopkins--Ravenel slice differentials is in one-to-one correspondence with a family of Tate differentials for $N_1^2 H{\mathbb{F}}_2$., Comment: Accepted version

With the rapid growth and prevalence of social network applications (Apps) in recent years, understanding user engagement has become increasingly important, to provide useful insights for future App design and development. While several promising neural modeling approaches were recently pioneered for accurate user engagement prediction, their black-box designs are unfortunately limited in model explainability. In this paper, we study a novel problem of explainable user engagement prediction for social network Apps. First, we propose a flexible definition of user engagement for various business scenarios, based on future metric expectations. Next, we design an end-to-end neural framework, FATE, which incorporates three key factors that we identify to influence user engagement, namely friendships, user actions, and temporal dynamics to achieve explainable engagement predictions. FATE is based on a tensor-based graph neural network (GNN), LSTM and a mixture attention mechanism, which allows for (a) predictive explanations based on learned weights across different feature categories, (b) reduced network complexity, and (c) improved performance in both prediction accuracy and training/inference time. We conduct extensive experiments on two large-scale datasets from Snapchat, where FATE outperforms state-of-the-art approaches by ${\approx}10\%$ error and ${\approx}20\%$ runtime reduction. We also evaluate explanations from FATE, showing strong quantitative and qualitative performance., Comment: Accepted to KDD 2020 Applied Data Science Track

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum $MGL$ and of the Real bordism spectrum $MU_{\mathbb R}$. These brackets are "red-shifting" in the sense that while the terms in the bracket will be of some chromatic height $n$, the bracket itself will be of chromatic height $(n+1)$. Using these, we deduce a family of exotic multiplications in the $\pi_{(\ast,\ast)}MGL$-module structure of the motivic Morava $K$-theories, including non-trivial multiplications by $2$. These in turn imply the analogous family of exotic multiplications in the $\pi_{\star}MU_\mathbb R$-module structure on the Real Morava $K$-theories.

We provide a necessary and sufficient condition for a type D Temperley-Lieb algebra ${\rm TLD}_n(\delta)$ being semi-simple by studying branching rule for cell modules. As a byproduct, our result is used to study the so-called forked Temperley-Lieb algebra, which is a quotient algebra of ${\rm TLD}_n(\delta)$., Comment: 26 pages

We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{\mathbb{R}}$, and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of $MU_{\mathbb{R}}$., Comment: Minor changes. Accepted version

As online platforms are striving to get more users, a critical challenge is user churn, which is especially concerning for new users. In this paper, by taking the anonymous large-scale real-world data from Snapchat as an example, we develop \textit{ClusChurn}, a systematic two-step framework for interpretable new user clustering and churn prediction, based on the intuition that proper user clustering can help understand and predict user churn. Therefore, \textit{ClusChurn} firstly groups new users into interpretable typical clusters, based on their activities on the platform and ego-network structures. Then we design a novel deep learning pipeline based on LSTM and attention to accurately predict user churn with very limited initial behavior data, by leveraging the correlations among users' multi-dimensional activities and the underlying user types. \textit{ClusChurn} is also able to predict user types, which enables rapid reactions to different types of user churn. Extensive data analysis and experiments show that \textit{ClusChurn} provides valuable insight into user behaviors, and achieves state-of-the-art churn prediction performance. The whole framework is deployed as a data analysis pipeline, delivering real-time data analysis and prediction results to multiple relevant teams for business intelligence uses. It is also general enough to be readily adopted by any online systems with user behavior data., Comment: Published at KDD 2018

While mobile social apps have become increasingly important in people's daily life, we have limited understanding on what motivates users to engage with these apps. In this paper, we answer the question whether users' in-app activity patterns help inform their future app engagement (e.g., active days in a future time window)? Previous studies on predicting user app engagement mainly focus on various macroscopic features (e.g., time-series of activity frequency), while ignoring fine-grained inter-dependencies between different in-app actions at the microscopic level. Here we propose to formalize individual user's in-app action transition patterns as a temporally evolving action graph, and analyze its characteristics in terms of informing future user engagement. Our analysis suggested that action graphs are able to characterize user behavior patterns and inform future engagement. We derive a number of high-order graph features to capture in-app usage patterns and construct interpretable models for predicting trends of engagement changes and active rates. To further enhance predictive power, we design an end-to-end, multi-channel neural model to encode temporal action graphs, activity sequences, and other macroscopic features. Experiments on predicting user engagement for 150k Snapchat new users over a 28-day period demonstrate the effectiveness of the proposed models. The prediction framework is deployed at Snapchat to deliver real world business insights. Our proposed framework is also general and can be applied to other social app platforms., Comment: 9 pages. Accepted to KDD 2019 Research Track

In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a "10/8+4"-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the $j$-based Atiyah-Hirzebruch spectral sequence., Comment: 116 pages, 15 figures. Minor revisions. Comments welcome!

We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{((C_4))}\langle 2 \rangle$. After periodization and $K(4)$-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory $E_4$ with $C_4$-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ is 384-periodic., Comment: 111 pages, 45 figures. To appear in Memoirs of the American Mathematical Society

We show, at the prime 2, that the Picard group of invertible modules over $E_n^{hC_2}$ is cyclic. Here, $E_n$ is the height $n$ Lubin--Tate spectrum and its $C_2$-action is induced from the formal inverse of its associated formal group law. We further show that $E_n^{hC_2}$ is Gross--Hopkins self-dual and determine the exact shift. Our results generalize the well-known results when $n = 1$., Comment: v3: Version to appear Algebraic and Geometric Topology

Online controlled experiments (a.k.a. A/B testing) have been used as the mantra for data-driven decision making on feature changing and product shipping in many Internet companies. However, it is still a great challenge to systematically measure how every code or feature change impacts millions of users with great heterogeneity (e.g. countries, ages, devices). The most commonly used A/B testing framework in many companies is based on Average Treatment Effect (ATE), which cannot detect the heterogeneity of treatment effect on users with different characteristics. In this paper, we propose statistical methods that can systematically and accurately identify Heterogeneous Treatment Effect (HTE) of any user cohort of interest (e.g. mobile device type, country), and determine which factors (e.g. age, gender) of users contribute to the heterogeneity of the treatment effect in an A/B test. By applying these methods on both simulation data and real-world experimentation data, we show how they work robustly with controlled low False Discover Rate (FDR), and at the same time, provides us with useful insights about the heterogeneity of identified user groups. We have deployed a toolkit based on these methods, and have used it to measure the Heterogeneous Treatment Effect of many A/B tests at Snap.