Week of March 29 - March 25

**Jose Castillo** - Computational Science Research Center - San Diego State University

**Abstract: TBA**

**Location:** Gibson Hall 126

**Time: ** 3:30

**Jennifer Wilson - University of Michigan Host (Mentor Stafa)**

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This talk will illustrate some patterns in the homology of the configuration space F_k(M), the space of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena -- relationships between unstable homology classes in different degrees -- established in joint work with Jeremy Miller. This project was inspired by work-in-progress of Galatius--Kupers--Randal-Williams.

**Location:** Dinwiddie 102

**Time: ** 3:30

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**Ilya Timofeyev - University of Houston**

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Radical shifts in the genetic composition of large cell populations are rare events with quite low probabilities, which direct numerical simulations generally fail to evaluate accurately. We develop an applicable large deviations framework for a class of Markov chains used to model genetic evolution of bacterial populations. We illustrate this framework by computing the most likely evolutionary paths describing emergence of genotypes with lower fitness in realistic parameter settings.

**Location:** Dinwiddie 102

**Time: ** 3:00

**Karl Hofmann** - TU Darmstadt

**Abstract:** Gibson Hall 310

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**Time:** 3:00

**Nathan Bedell - Tulane University
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In this talk, I will introduce the notion of an "Olog" (short for ontology log), as popularized by David Spivak in his book "Category Theory for the Sciences". An Olog is a natural format for knowledge representation given (usually) by a free category generated by a directed graph, quotiented by some domain-specific relationships expressed as equalities between morphisms in the category. A functor from this category to some other category is interpreted as an implementation, or model of the Olog. I then discuss some of the latest findings in my theory of graded categories, and discuss how this framework might prove useful in extending Spivak's Ologs to a "dynamic" theory of ontologies, what I am tentatively calling a theory of "meta-ontologies", with an eye towards applications in both metamathematics and artificial intelligence, as well as potentially in fields such as biology, psychology, and the social sciences, where a study of such "dynamic ontologies" might yield interesting results which would not otherwise be obvious.

**Location:** Stanley Thomas 316

**Time: ** 4:30

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**Robert Walker - University of Michigan
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This is joint work with Irena Swanson found on arXiv:1806.03545. Given a polynomial ring C over a field and proper ideals I and J whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of I+J into a collection of primes described in terms of the associated primes of select powers of I and of J. We discuss applications to constructing primary decompositions for powers of I+J, and to attacking the persistence problem for associated primes of powers of an ideal.

**Location:** Gibson Hall 310

**Time: ** 3:00

**Jinsu Kim - UC Irvine**

**Abstract:
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A reaction network is a graphical configuration that describes an interaction between species (molecules). If the abundances of the network system is small, then the randomness inherent in the molecular interactions is important to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain. One of challenging issues facing researchers who study biological systems is the often extraordinarily complicated structure of their interaction networks. Thus, how to characterize network structures that induce characteristic behaviors of the system dynamics is one of the major open questions in this literature. In this talk, I will provide recent issues and discoveries on stochastically modeled reaction networks. Specifically, we will focus on existence of stationary distributions and the convergence rate for the process to its stationary distribution.

**Location: ** Dinwiddie 102

**Time: ** 3:00

**Vaishavi Sharma - Tulane University
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In this talk, I will start from scratch and introduce the p-adic numbers and discuss the field Qp. I'll go over some examples and applications and then finally talk a little about the problems we're working on.

**Location:** Stanley Thomas 316

**Time:** 4:30

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Week of March 15 - March 11

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**Karl Hofmann - Tulane/TU Darmstadt**

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One of the drawings of mine, produced for a book on topological semigroups and exhibited in our Common Room, gave rise to the "Tulane Mathematics Coffee Mug"—which in turn suggested that I give a talk meandering through my encounters with the arts when they become applied in the professional life of mathematicians. So I hope to present a picture book of examples created when I was called upon to provide illustrations to journals and books–a few more at any rate than are exhibited in the Mathematics Common Room and the Mathematics Library.

I studied studio arts in the graphic domain at the University of Tübingen (Germany) on the side of my academic training in Mathematics prior to my joining the Tulane Mathematics Department in 1960.

**Location:** Tilton Hall 305

**Time: ** 1:00pm

**Marius Vladoiu - Purdue University and the University of Bucharest**

**Abstract:
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Naturally, most famous classes of toric ideals come equipped with a rich algebraic and homological structure, but they also have a common combinatorial feature, namely, equality of various special combinatorial subsets. In particular, one such class is represented by strongly robust toric ideals, for which the Graver basis is a minimal generating set. In this talk we aim to discuss a few open questions related to strongly robust toric ideals, arising from combinatorial commutative algebra, algebraic geometry, and a surprising connection to combinatorics. The talk is based on joint works with Sonja Petrovic and Apostolos Thoma, and on an ongoing project with Apostolos Thoma.

**Location:** Gibson Hall 310

**Time:** 3:00

**Ray Bai - Institution**

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We introduce the partial-sum limit theorems for a class of stationary sequences exhibiting both heavy tails and long-range dependence. The stationary sequences are constructed using heavy-tailed multiple stochastic integrals and conservative null dynamical systems. The limits are represented by multiple stables integrals, where the integrands involve the local times of the intersections of stationary stable regenerative sets. This is a joint work with Takashi Owada and Yizao Wang.

**Location:** Dinwiddie 102

**Time:** 3:00

**Olivia du Roure - PMMH, ESPCI Paris, France**

**Abstract:
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The dynamics of an individual flexible objects in viscous flows is key to deciphering the rheological behavior of many complex fluids and soft materials. It also underlies a wealth of biophysical processes from flagellar propulsion to intracellular streaming. The interaction between an elongated object and a given flow depends strongly on the properties of the object - flexibility, aspect ratio and dimensions - and the flow geometry. This interaction is governed by the elasto-viscous number that compares elastic and viscous forces. In this seminar I will discuss dome of the configurations of this problem we have studied ,during the last years, by combining microfluidics, microfabrication and microscopy.

**Location: ** Stanley Thomas 316

**Time: ** 3:00

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Week of March 8 - March 3

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Week of March 1 - February 25

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**Cynthia Anhalt -** University of Arizona

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This presentation will engage participants in considering ways to develop pedagogical strategies and ways of thinking for actively and purposefully engaging students in mathematics classrooms. We will examine research-based recommendations recently highlighted in the mathematical Association of America (MAA) publication, Instructional Practices Guide (2017). These strategies call for inclusion of multiple ways to express and represent mathematical ideas in order to develop connections for learning. Diversifying ways to represent the mathematical ideas during instruction translates into access to mathematics for more students.

**Location:** Tilton Hall 305

**Time:** 1:00

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**Tewodros Amdeberhan - Tulane University**

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WZ stands for Wilf-Zeilberger. We will explain the ideas behind this meta-mathematics and explore further implications, including some of our own work.

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**Diego Villamizar - Tulane University
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We will discuss the combinatorial definition and some properties of Feynman diagrams based on the book "A combinatorial perspective of Quantum field theory" of Karen Yeats. Also, we will show some connections with Stirling numbers.

**Location:** Stanley Thomas 316

**Time:** 4:30

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Week of February 22 - February 18

**Hayriye Gulbudak - University of Louisiana at Lafayette**

**Abstract: **

A current challenge for disease modeling and public health is understanding pathogen dynamics across scales since their ecology and evolution ultimately operate on several coupled scales. This is particularly true for vector-borne diseases, where within-vector, within-host, and between vector-host populations all play crucial roles in diversity and distribution of the pathogen. Despite recent modeling efforts to determine the effect of within-host virus-immune response dynamics on between-host transmission, the role of within-vector viral dynamics on disease spread is overlooked. Here we formulate an age-since-infection structured model where epidemic model parameters are governed by ODE systems describing within-host immune-pathogen dynamics and within-vector viral kinetics. We define the basic reproduction number, R0 and study the threshold dynamics of the system. Numerical results suggest that within-vector-viral kinetics may play a substantial role in epidemics. Finally, we address how the model can be utilized to better predict the success of control strategies such as vaccination, or Wolbachia-based biocontrol.

**Location:** Gibson Hall 126

**Time:** 3:30

**Gerardo Chowell - Georgia State (Host: James Hyman and Zhuolin Qu)**

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**Location:** Dinwiddie 102

**Time:** 3:30

**Ricardo Cortez - Tulane University**

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**Location: Gibson Hall 400D
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**Time: 11:00
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**Aram Bingham - Tulane University**

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**Location: ** Gibson Hall 310

**Time:** 3:00

**Sankhaneel Bisui - Tulane University
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**Abstract: **

I am going to present our joint work with Dr. Tai Huy Ha, Dr. A.V. Jayanthan and Abu C. Thomas. Our interest is to investigate the resurgence number of fiber product of projective schemes. In this talk, we will also see how resurgence number corresponding to the ideal of the fiber product of the schemes depends on that of the original schemes. While considering the asymptotic resurgence the resurgence number of the fiber product follows a nice relation with the resurgence of the original schemes. We will also see the relationship and we will also see how there is a possibility of the resurgence number becoming arbitrarily large.

**Location:** Stanley Thomas 316

**Time:** 4:30

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Week of February 15 - February 11

**Max Yaremchuk - Navy Research Laboratory**

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Improving the quality of global ocean weather forecasts is a primary task of oceanographic research. The problem is currently treated as a statistically and dynamically consistent synthesis of the data streams arriving from satellites and autonomous observational platforms. Due to the immense size of the ocean state vector (10^8-10^9), operational algorithms combine variational optimization techniques with limited-size (10^2-10^3) ensembles simulating statistical properties of the error fields. A brief overview of current situation in operational state estimation/forcasting is presented with a special focus on selected problems requiring applied math research. These include efficient linearization and transposition of the operators describing evolution of the ocean state, sparse approximation of the inverse correlation matrices and consistent treatment of the state vector components with non-gaussian error statistics.

**Location: ** Gibson Hall

**Time:** 10:00

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**Chris Miles - NYU**

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Cells send and receive signals in the form of diffusing particles that search for target sites called receptors. This mechanism has received great theoretical interest for over 40 years, but a more recent fact that receptors themselves diffuse along a surface has largely been neglected. This raises the natural question: does the target diffusing help or hurt the ability of a diffusing particle to locate it? We'll modify a classical PDE model into a PDE with stochastic boundary conditions, which we'll study using matched asymptotic analysis and Monte Carlo simulations.

**Location:** Dinwiddie 102

**Time: 3:00
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Week of February 9 - January 28

Zach Bradshaw - University of Arkansas

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Local energy solutions to the Navier-Stokes equations, that is, weak solutions which are uniformly locally square integrable, but not necessarily globally square integrable, have proven a useful class for studying regularity and uniqueness. In this talk we survey several recent results concerning the existence and properties of local energy solutions, including applications to self-similar solutions.

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**Carola Wenk - Tulane University**

**Abstract: **

We introduce new distance measures for comparing embedded graphs based on the Fréchet distance and the weak Fréchet distance. These distances take the combinatorial structure as well as the geometric embeddings of the graphs into account. Our distance measures are motivated by the comparison of road networks, for instance, to evaluate the quality of a map construction algorithm, a task for which suitable distance measures are currently missing. We present a general algorithmic approach for computing these distances, which yields polynomial time algorithms if the graphs are trees and, if the weak Fréchet distance is used, if the graphs are plane. For general embedded graphs, however, deciding the distances is NP-complete. This work has been conducted in collaboration with Hugo A. Akitaya, Maike Buchin, Bernhard Kilgus, and Stef Sijben. The preprint is available at https://arxiv.org/abs/1812.09095 .

**Location: Hebert Hall
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**Abu Thomas - Tulane University
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**Location: Gibson Hall 310
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**Time: 3:00
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We prove a central limit theorem for the components of the eigenvectors corresponding to the d largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and furthermore the mean and the covariance matrix of each row are functions of the associated vertex's block membership. Together with prior results for the eigenvectors of the adjacency matrix, we then compare, via the Chernoff information between multivariate normal distributions, how the choice of embedding method impacts subsequent inference. We demonstrate that neither embedding method dominates with respect to the inference task of recovering the latent block assignments.

**Location:** Gibson Hall 126A

**Time:** 3:30

**Villamarin Gomez, Sergio Nicolas - Tulane University
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The Eulerian polynomials were introduced by Euler when studying the sum of alternating consecutive numbers with a fix exponent. Afterwards he introduced its coefficients as the Eulerian numbers and gave many interesting combinatorial identities, some of which relate to the Riemann Zeta function. In our talk I’ll introduce them and present some of its properties and also some generalizations along with generalizations of the Stirling numbers of the second kind.

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**Time:** 4:30

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Week of February 1 - January 28

**Andrei Tarfulea - University of Chicago**

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Kinetic equations model gas and particle dynamics, specifically focusing on the interactions between the micro-, meso-, and macroscopic scales. Mathematically, they demonstrate a rich variety of nonlinear phenomena, such as hypoellipticity through velocity-averaging and Landau damping. The question of well-posedness remains an active area of research.

In this talk, we look at the Landau equation, a mathematical model for plasma physics arising from the Boltzmann equation as so-called grazing collisions dominate. Previous results are in the perturbative regime, or in the homogeneous setting, or rely on strong a priori control of the solution (the most crucial assumption being a lower bound on the density, as this prevents the elliptic terms from becoming degenerate).

We prove that the Landau equation has local-in-time solutions with no additional a priori assumptions; the initial data is even allowed to contain regions of vacuum. We then prove a "mass spreading" result via a probabilistic approach. This is the first proof that a density lower bound is generated dynamically from collisions. From the lower bound, it follows that the local solution is smooth, and we establish the mildest (to date) continuation criteria for the solution to exist for all time.

**Location: ** Greenbaum House – GR – MacLaren Room

**Time:** 3:30

**Abstract:
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The (semiparametric) Gaussian copula model consists of distributions that have dependence structure described by Gaussian copulas but that have arbitrary marginals. A Gaussian copula is in turn determined by an Euclidean parameter $R$ called the copula correlation matrix. In this talk we study the normal scores (rank correlation coefficient) estimator, also known as the van der Waerden coefficient, of $R$ in high dimensions. It is well known that in fixed dimensions, the normal scores estimator is the optimal estimator of $R$, i.e., it has the smallest asymptotic covariance. Curiously though, in high dimensions, nowadays the preferred estimators of $R$ are usually based on Kendall's tau or Spearman's rho. We show that the normal scores estimator in fact remains the optimal estimator of $R$ in high dimensions. More specifically, we show that the approximate linearity of the normal scores estimator in the efficient influence function, which in fixed dimensions implies the optimality of the normal scores estimator, holds in high dimensions as well.

**Location:** Dinwiddie 102

**Time:** 3:30

**Sankhaneel Bisui - Tulane University**

**Abstract:
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The symbolic powers of a homogeneous ideal are a well-studied object. In the study of symbolic power it is natural to ask when these symbolic powers contain ordinary powers and vice versa. We can easily check that for any, n I^n \subset I^{(n)}. It is a subtle and generally open problem to determine for which positive integers m,r we have $I^{(m)} \subset I^r $. It is now known that for $m > Nr $ that containment holds. Now, what can be said about the bounds for a specific ideal? This question leads to the definition of resurgence number and the asymptotic versions of this number. In this talk, I will introduce the resurgence number and the asymptotic versions of the number.

Our interest is to investigate the resurgence number of fiber product of projective schemes. We will also see how resurgence number corresponding to the ideal of the fiber product of the schemes depends on that of the original schemes. While considering the asymptotic resurgence the resurgence number of the fiber product follows a nice relation with the resurgence of the original schemes. I am going to present the relation and we will also see how there is a possibility of the resurgence number becoming arbitrarily large.

**Location:** Gibson Hall 310

**Time:** 3:00

**Sushovan Majhi - Tulane University**

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We eat pizza and talk about Topological Data Analysis (TDA). TDA is an emerging subfield of applied mathematics. Various topological concepts such as homology, persistent homology, discrete Morse theory are becoming widely useful in analyzing and visualizing data. Applications include reconstruction of shapes, 3D printing, feature detection, medical imaging etc. We briefly touch upon Morse theory and its discrete analog. Also, we discuss some of its major applications in TDA.

**Location: ** Stanley Thomas 316

**Time:** 4:30

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

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Model calibration or data inversion involves using experimental or field data to estimate the unknown parameters of a mathematical model. This task is complicated by the discrepancy between the model and reality, and by possible bias in field data. The model discrepancy is often modeled by a Gaussian stochastic process (GaSP), but it was observed in many studies that the calibrated mathematical model can be far from the reality. Here we show that modeling the discrepancy function via a GaSP often leads to an inconsistent estimation of the calibration parameters even if one has an infinite number of repeated experiments and an infinite number of observations in each experiment. In this work, we develop the scaled Gaussian stochastic process (S-GaSP), a new stochastic process to model the discrepancy function in calibration. We establish the explicit connection between the GaSP and S-GaSP through the orthogonal series representation. We show the predictive mean estimator in the S-GaSP calibration model converges to the reality at the same rate as the one by the GaSP model, and the calibrated mathematical model in the S-GaSP calibration converges to the one that minimizes the L2 loss between the reality and mathematical model, whereas the GaSP calibration model does not have this property. The scientific goal of this work is to use multiple interferometric synthetic-aperture radar (InSAR) interferograms to calibrate a geophysical model for Kilauea Volcano, Hawaii. Analysis of both simulated and real data confirms that our approach is better than other approaches in prediction and calibration. Both the GaSP and S-GaSP calibration models are implemented in the "RobustCalibration" R Package on CRAN.

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**Victor Bankston - Institution**

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That which is physical decays, but that which is mathematical persists. Since computation bridges the corporeal and the ideal, schemes of "error correcting codes" are needed. Beyond solving this necessary engineering problem, error correcting codes serve as examples of extremal combinatorial objects. We will discuss the Hamming Code, linear codes, convolutional codes and theoretical bounds.

**Location:** Stanley Thomas, 316

**Time:** 4:30

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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

**Abstract: **

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.

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