Mathematics & Statistics Department

The Euler characteristic satisfies an inclusion-exclusion principle χ(X∪Y)=χ(X)+χ(Y)-χ(X∩Y), which lets one regard it as a measure – a peculiar one where a point has measure 1 while a circle has measure 0. One can use it to integrate simple functions (in the sense of measure theory), and the resulting integral calculus has deep roots in algebraic geometry and has recently found surprising applications to data analysis. However, since it is only finitely – not countably – additive, it does not fit into the framework of Lebesgue integration, and there are problems integrating even the most elementary non-simple functions. I will describe a new approach to this calculus, based on differential geometry, which makes it possible to integrate a large class of non-simple functions, and which hints at new ways to apply differential geometry to data analysis.

The Curry-Howard correspondence formalizes an analogy between computer programs and mathematical proofs. This talk will introduce alternative foundations for mathematics animated by this analogy. The basic object is called a type, which can be simultaneously interpreted as something like a set or as something like a mathematical proposition. Homotopy type theory refers to the recent discovery that a type can also be interpreted as something like a topological space. We will discuss the implications of this homotopy theoretic interpretation for the so-called univalent foundations of mathematics.

Government agencies across the world are estimating important population quantities using data collected under a complex sample design.  With the advent of big data and advancements in technology, a wealth of additional data sources, such as remotely sensed data and administrative data, are available.  In this talk, I present a class of survey estimators that seek to combine existing survey data with these new sources of data, often using machine learning methods to do so.  I then examine the utility of these modern techniques in estimating official statistics.

Increased attention is being paid to the promotion of healthy habits via mobile applications, which foster online fitness communities and offer users the ability to track their physical fitness. At the intersection of social media and activity tracking, these platforms represent affordable, scalable technologies for capturing information about both physical activity and peer-to-peer interactions. Importantly, the large-scale behavioral trace data archived by these platforms offers researchers a novel and powerful opportunity to examine health-related behaviors. In this study, we explore the characteristics and dynamics of social exercise (i.e. fitness activities with at least one peer physically co-present), using data collected from an online fitness community popular with cyclists and runners. Our analysis of the personal networks of the online fitness users uncovers a strong gender-based homophily between users, as well as distinct generative processes for network formation across genders. We further examine how performance and social feedback vary by the number/gender of people participating in an activity; our results indicate that when peers are physically co-present for fitness activities (i.e. when users participate in group workouts) (i) exercise tends to be of higher performance (e.g., longer distance, higher heart rate), and (ii) users receive more online feedback from other users. These effects of co-present activity partners are true for both men and women, though we do find that women are proportionately more likely to engage in group activities. Altogether, these results improve our understanding of how technological solutions such as mobile apps and online communities may be utilized in developing affordable and large-scale health intervention strategies.

Topology, and particularly algebraic topology, seeks to develop computable invariants that describe the overall shape of abstract spaces. This talk will be about how such invariants (e.g., the Euler Characteristic and the Homology of a space) can be used to analyze scientific data sets. I will use several examples to illustrate how these ideas have real applications.

The Boltzmann equation models the motion of a rarefied gas in which particles interact through binary collisions. Instead of tracking each particle separately, the Boltzmann equation describes the evolution of the particle density function. The focus of this talk will be the tail behaviour of such density functions (how they behave for large velocities). We will show that they have exponential decay. An interesting aspect of our result is in the use of Mittag-Leffler functions, which are a generalization of exponential functions. This is based on joint works with Alonso, Gamba and Pavlovic.