Week of January 25 - January 21

**Mengyang Gu - Johns Hopkins (Host: Gustavo Didier)**

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**Timothy Daley - Stanford (HOST: Michelle Lacey)
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In modern applications of high-throughput sequencing technologies researchers may be interested in quantifying the molecular diversity of a sample (e.g. T-Cell repertoire, transcriptional diversity, or microbial species diversity). In these sampling-based technologies there is an important detail that is often overlooked in the analysis of the data and the design of the experiments, specifically that the sampled observations often do not give a fully representative picture of the underlying population. This has long been a recognized problem in statistical ecology and in the broader statistics literature, and is commonly known as the missing species problem.

In classical settings, the size of the sample is usually small. New technologies such as high-throughput sequencing have allowed for the sampling of extremely large and heterogeneous populations at scales not previously attainable or even considered. New algorithms are required that take advantage of the scale of the data to account for heterogeneity, but are also sufficiently fast and scale well with the size of the data. I will discuss a moment-based approach for estimating the missing species based on an extension of Chao's moment-based lower bound (Chao, 1984). We apply results from the classical moment problem to show that solutions can be obtained efficiently, allowing for estimators that are simultaneously conservative and use more information. By connecting the rich theory of the classical moment problem to the missing species problem we can also clear up issues in the identifiability of the missing species.

**Location:** Stanley Thomas

**Time: ** 3:30

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We consider a setup where particles are released into a domain and diffuse freely. Part of the boundary is absorbing, where the particles can escape the domain, another part is reflecting. The rest of boundary consists of capture regions that switch between being reflecting and absorbing. After capturing a particle, the capture region becomes reflecting for an exponentially distributed amount of time. This non-zero recharge time correlates the particles' paths, complicating the mathematical analysis of this system. We are interested in the distribution of the number of particles that are captured before they escape.

Our results are derived from considering our system in several ways: as a full spatial diffusion process with recharging traps on the boundary; as a continuous-time Markov process approximating the original system; and lastly as a system of ODEs in a mean-field approximation. Considering the full spatial diffusion process, we prove that the total expected number of the captured particles has an upper-bound of the order of log n. We then apply our approximations to investigate time courses for the expected number and higher ordered statistics of captured particles. We find that the amount of variation observed in the total number of captured particles varies non-monotonically with the mean recharge time. Lastly, we combine these results together to predict stochastic properties of intracellular signals resulting from receptor activation.

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Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu