**Speaker - Institution
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**Speaker - Institution
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**Alina ChertockNorth** - Carolina State University (Host: **Alexander Kurganov**)

**Abstract:
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Many system of hyperbolic conservation and balance laws contain uncertainties in model parameters, initial or boundary data due to modeling or measurement errors. Quantifying these uncertainties is important for many applications since it helps to conduct sensitivity analysis and to provide guidance for improving the models. Among the most popular numerical methods for uncertainty quantification are stochastic spectral methods. Such methods decompose random quantities on suitable approximation bases. Their attractive feature is that they provide a complete probabilistic description of the uncertain solution.

A classical choice for the stochastic basis is the set of generalized Polynomial Chaos (gPC) spanned by random polynomials, continuous in the stochastic domain and truncated to some degree. It is well-known, however, that when applied to general nonlinear (non-symmetric) hyperbolic systems, such approximations result in systems for the gPC coefficients, which are not necessarily globally hyperbolic since their Jacobian matrices may contain complex eigenvalues. In this talk, I will present a splitting strategy that allows one to overcome this difficulty and demonstrate the performance of the proposed approach on a number of numerical examples including systems of shallow water and compressible Euler equations.

**Location: Gibson Hall 414
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**Time: 3:30**

**Speaker - Institution
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**Speaker - Institution
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**Time:**

**Slawomir Kwasik - Tulane University
**

**Abstract:
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**Location: Gibson Hall 310
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**Time: 12:30**

**Speaker - Institution
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**Time:**

**Holly Swisher** - Oregon State University (Host: **Victor Moll**)

**Abstract:
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Nearly 100 years after his untimely death, Ramanujan's legacy is still intriguing mathematicians today. One of the last obsessions of Ramanujan were what he called mock theta functions. In this talk, we will begin by discussing Ramanujan's work on integer partitions and how they connect to objects called modular forms and mock theta functions. Then we will continue by exploring some recent work in this area, including the construction of a table of mock theta functions with some interesting properties, including what is called quantum modularity. Part of this work is joint with Sharon Garthwaite, Amanda Folsom, Soon-Yi Kang, and Stephanie Treneer. The rest is joint with Brian Diaz and Erin Ellefsen from their undergraduate REU project this summer.

**Location: Gibson Hall 414
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**Time: 3:30**

**Speaker - Institution
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**Speaker - Institution
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**Location:
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**Time:**

**Fang Sun - Tulane University
**

**Abstract: N/A
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**Location: Gibson Hall 310
**

**Time: 12:30**

**Selvi Beyarslan - Tulane University
**

**Abstract:
**

**Location: Stanley Thomas 316
**

**Time: 4:15**

PROBABILITY and STATISTICS

**Melanie Jensen - Tulane University
**

**Abstract:
**

In recent years, particle tracking experiments have provided new insights into interactions between our immune system and foreign bodies like viruses and bacteria. Sam Lai (UNC-Chapel Hill, Pharmacy) and collaborators have demonstrated that certain types of antibodies have the capacity to essentially immobilize Herpes virus in mucus and it is believed that a similar effect will be true for HIV. Because antibodies are too small to be directly observed in these experiments, the physical mechanism underlying this effect remains unclear. Using particle tracking data for Herpes virions, we construct a multi-scale model for virion movement and virion-antibody-mucin reaction kinetics to investigate the impact of varying antibody concentrations on virion movement.

First, we develop a classification system for path data to distinguish among diffusive, subdiffusive, and stationary motions. We use a continuous-time Markov model to describe virion-antibody-mucin kinetics and introduce a multi-scale approximation to compute important properties of the system that help us predict what fraction of a virion population should be immobilized at a given time, and how long a virion's immobilization should last. To specify our model with the data, we use identifiability analysis to set mathematically optimal and biological feasible parameter values. Finally, we compare theoretical immobilization times with observed immobilization times to determine whether prominent qualitative features of the data are predicted by our linear stochastic model.

**Location: Gibson Hall 310
**

**Time: 3:00**

**Nantel Bergeron** - York University (Host: **Mahir Can**)

**Abstract:
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Given a family of combinatorial objects we often have operations that allow us to combine them to create larger objects and/or ways to decompose them into smaller members of the family. In the best situation we have in fact an algebraic structure, i.e. a graded Hopf algebra. I will give example of such structure using graphs, trees, set partitions, etc.

The antipode is a map from the Hopf algebra into itself that is defined recursively, with a lot of cancelation and is difficult to compute in general. I will motivate why we should care about the antipode and why we should aim to find a cancelation free formula.

An important example is the combinatorial Hopf algebra of graphs. In this case, a cancelation free formula of for its antipode is given by, Humpert and Martin. We will see that such formula gives a structural understanding of certain evaluations of the combinatorial invariants for graphs. In particular we recover very nicely a classical theorem of Stanley for the evaluation of the chromatic polynomial at -1.

I will discuss some generalization of this example.

**Location: Gibson Hall 414
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**Time: 3:30**

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**Shilpa Khatri - University of California-Merced, Applied Mathematics
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**Abstract:
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**Location: Stanley Thomas 316
**

**Time: 3:00**

**Alexej Gossmann - Tulane University
**

**Abstract:
**

**Location: Stanley Thomas 316
**

**Time: 4:15**

**Lukasz Sikora - Tulane University
**

**Abstract:
**

An ongoing effort in the study of microparticle movement in biofluids is the proper characterization of subdiffusive processes i.e. processes whose mean-squared displacement scales as a sublinear power law. In order to describe phenomena that lead to subdiffusive behavior, a few models have been developed: fractional Brownian motion, the generalized Langevin equation, and random walks with dependent increments. We will present perhaps a simpler model that leads to subdiffusion and is designed to characterize systems where a regularly diffusive particle intermittently becomes trapped for long periods of time.

To start with, we introduce the rigorous model of switching diffusion. We present a stochastic differential equation perspective and, using heavy tail analysis methods, we will present a proof that switching diffusion with heavy-tailed immobilization times is asymptotically subdiffusive.

**Location: Gibson Hall 310
**

**Time: 3:00**

**Zhenheng Li - University of South Carolina, Aiken
**

**Abstract:
**

We show in this talk how to construct a monoid from a highest weight representation of a Kac-Moody group over the complex numbers. The unit group of the monoid is the image of the Kac-Moody group under the representation, multiplied by the nonzero complex numbers. We then show that this monoid has similar properties to those of a J-irreducible reductive linear algebraic monoid. More specifically, the monoid is unit regular and has a Bruhat decomposition, and the idempotent lattice of the generalized Renner monoid of the Bruhat decomposition is isomorphic to the face lattice of the convex hull of the Weyl group orbit of the highest weight.

**Location: Gibson Hall 414
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**Time: 2:00**

**Speaker - Institution
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**Abstract:
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**STUDENT COLLOQUIUM**

**Lukasz Sikora - Tulane University
**

**Abstract:
**

An ongoing effort in the study of microparticle movement in biofluids is the proper characterization of subdiffusive processes i.e. processes whose mean-squared displacement scales as a sublinear power law. In order to describe phenomena that lead to subdiffusive behavior, a few models have been developed: fractional Brownian motion, the generalized Langevin equation, and random walks with dependent increments. We will present perhaps a simpler model that leads to subdiffusion and is designed to characterize systems where a regularly diffusive particle intermittently becomes trapped for long periods of time.

To start with, we introduce the rigorous model of switching diffusion. We present a stochastic differential equation perspective and, using heavy tail analysis methods, we will present a proof that switching diffusion with heavy-tailed immobilization times is asymptotically subdiffusive.

**Location: Stanley Thomas 316
**

**Time: 4:15**

**John Fricks - Arizona State University
**

**Abstract:
**

In living cells, Brownian forces play a dominant role in the movement of small and not so small particles, such as vesicles, organelles, etc. However, proteins and other macromolecules bind to one another, altering the underlying Brownian dynamics. In this talk, classical approaches in the biophysical literature to time series observations which switch between bound and unbound states will be presented, and an alternative approach using stochastic expectation-maximization algorithm (EM) combined with particle filters will be proposed along with extensions for non-quadratic potentials when the particle is bound

**Location: Gibson 310
**

**Time: 3:00**

GEOMETRY & TOPOLOGY

**Margaret Symington - Mercer University
**

**Abstract:
**

Blowing up and down are fundamental surgeries in symplectic geome- try. In dimension four, equivariant blow-ups of symplectic four-manifolds equipped with a T^{2}-action or an S^{1}-action are well understood. It is less clear how to perform these operations while preserving the structure of a manifold with an S^{1} R-action. In this talk I will describe a class of com- pletely integrable Hamiltonian systems with two degrees of freedom that are \nice" enough for study via surgeries and then explain the topology, sym-plectic geometry and the Hamiltonian aspects of two types of surgeries in this setting: nodal trades and blowing up or down.

**Location: Tilton Hall 305
**

**Time: 12:30**

**Prof Gerardo Chowell** - GEORGIA STATE UNIVERSITY (Host: **James Hyman**)

**Abstract:
**

There is a long tradition of using mathematical models to generate insights into the transmission dynamics of infectious diseases and assess the potential impact of different intervention strategies. The increasing use of mathematical models for epidemic forecasting has highlighted the importance of designing reliable models that capture the baseline transmission characteristics of specific pathogens and social contexts. More refined models are needed however, in particular to account for variation in the early growth dynamics of real epidemics and to gain a better understanding of the mechanisms at play. Here, we review recent progress on modeling and characterizing early epidemic growth patterns from infectious disease outbreak data, and survey the types of mathematical formulations that are most useful for capturing a diversity of early epidemic growth profiles, ranging from sub-exponential to exponential growth dynamics. Specifically, we review mathematical models that incorporate spatial details or realistic population mixing structures, including meta-population models, individual-based network models, and simple SIR-type models that incorporate the effects of reactive behavior changes or inhomogeneous mixing. In this process, we also analyze simulation data stemming from detailed large-scale agent-based models previously designed and calibrated to study how realistic social networks and disease transmission characteristics shape early epidemic growth patterns, general transmission dynamics, and control of international disease emergencies such as the 2009 A/H1N1 influenza pandemic and the 2014-2015 Ebola epidemic in West Africa.

**Location: Gibson Hall 414
**

**Time: 3:30**

**Xiaoming Zheng** - CENTRAL MICHIGAN UNIVERSITY (**Host KUN ZHAO**)

**Abstract:
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This talk presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by the projection of mesh nodes onto the interface and the insertion right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. This is a joint work with John Lowengrub of University of California at Irvine.

**Location: Gibson Hall 325
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**Time: 3:30**

**Speaker - Institution
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**Cooper Boniece - Tulane University
**

**Abstract:
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Or integrate 2/3 times? Or, more generally, can we find

a family of operators that `interpolate' the integral and derivative operators?

And what sort of questions can we answer with such tools?

Fractional Brownian motion (fBm) is a generalization of Brownian motion that allows for correlated increments. In general, fBm lacks a key property in stochastic integration theory -- it is not a semimartingale -- and so much of the machinery from classical theory is unavailable

when considering integration questions related to fBm. In this talk, we examine some of the historical developments of fractional calculus and explore its connections with fBm and related integration theory.

**Location: Stanley Thomas 316
**

**Time: 4:15**

**Veronica Ciocanel - Brown University (Host Scott McKinley)
**

**Abstract:
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Messenger RNA (mRNA) localization is essential during the early development of many organisms, including during development of frog egg cells into embryos. This accumulation of RNA at the cell periphery is not well understood, but is thought to depend on diffusion, bidirectional movement and anchoring mechanisms. Our goal is to test these proposed mechanisms using dynamical systems and stochastic models and analysis, informed by parameter estimation. These methods allow us to extract asymptotic quantities such as effective velocity and diffusion, and to conclude that the PDEs considered have approximate traveling wave solutions. We confirm the hypothesis of bidirectional transport, and use the parameter estimates in numerical studies of localization.

**Location: Gibson Hall 310
**

**Time: 3:00**

**Arindam Banerjee - Purdue University
**

**Abstract:
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Stillman's question asks whether one can find an upper bound for protective dimension of homogeneous ideals depending only on the degree sequence of its generating set, independent of the number of variables in the polynomial ring. it is believed that this question has a positive answer and many researchers have proved it for different special cases in recent years. In this joint work with Giulio Caviglia, we study a variant of this question; i.e, we ask if one conjectures a bound for a a fixed degree sequence, whether that can be algorithmically verified. We answer this in affirmative and in the process we also prove some other related and interesting results.

**Location: Gibson Hall 414
**

**Time: 2:00**

**Prof. Sorin Mitran** - UNIVERSITY OF NORTH CAROLINA AT CHAPEL HILL (Host: **LISA FAUCI**)

**Abstract:
**

Microscopic organisms exhibit various modes of propulsion: ciliary or flagellar beating, lamellipodium protrusion followed by attachment/detachment to a substrate, taking over control of actin production in a host cell. High-throughput computational simulation can provide detailed description of specific motility aspects, but their cost and complexity is an impediment to furthering biological understanding of organism propulsion. This talk presents work on the transformation of detailed computational simulation into tractable reduced-order models. Two specific cases are considered: (1) ciliary propulsion to illustrate model reduction of a deterministic system, and (2) propulsion of Listeria monocytogenes to illustrate aspects of stochastic model reduction. In ciliary propulsion, molecular dynamics level computation is used to furnish a detailed description of mechanical behavior of the microtubule constituents of a cilium. The data is used to construct a finite element model that is markedly different from the Euler-Bernoulli beam models typically used in cilia studies. L. monocytogenes moves by taking over the production of actin within a host cell. Stochastic modeling of the growth of the host cytoskeleton catalyzed by L. monocytogenes is used to construct a statistical model of the flight/forage behavior that can be used to infer infection virulence. Model reduction in this case involves consideration of the differential geometry of probability distributions, a field of study known as information geometry. The model reduction procedures are presented at a conceptual level, avoiding technical details, concentrating on the goal of arriving at correct models of direct utility to biology.

**Location: Gibson Hall 414
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**Time: 3:30**

**Speaker - Institution
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**Yuanzhen Cheng - Tulane University
**

**Abstract:
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Shallow water models are widely used to describe and study fluid dynamics phenomena where the horizontal length scale is much greater than the vertical length scale, for example, in the atmosphere and oceans. Since analytical solutions of the shallow water models are typically out of reach, development of accurate and efficient numerical methods is crucial to understand many mechanisms of atmospheric and oceanic phenomena. In this dissertation, we are interested in developing simple, accurate, efficient and robust numerical methods for two shallow water models --- the Saint-Venant system of shallow water equations and the two-mode shallow water equations.

We first construct a new second-order moving-water equilibria preserving central-upwind scheme for the Saint-Venant system of shallow water equations. Special reconstruction procedure and source term discretization are the key components that guarantee the resulting scheme is capable of exactly preserving smooth moving-water steady-state solutions and a draining time-step technique ensures positivity of the water depth. Several numerical experiments are performed to verify the well-balanced and positivity preserving properties as well as the ability of the proposed scheme to accurately capture small perturbations of moving-water steady states. We also demonstrate the advantage and importance of utilizing the new method over its still-water equilibria preserving counterpart.

We then develop and study numerical methods for the two-mode shallow water equations in a systematic way. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches---two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme---and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method for this system.

**Location: Gibson Hall 325
**

**Time: 3:00**

**Laura Matusevich - Texas A&M
**

**Abstract:
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**Location: Gibson Hall 414
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**Time: 2:00**

**Michael Anshelevich** - TEXAS A&M (Host: **MAHIR CAN**)

**Abstract:
**

Limit theorems for sums of independent random variables (or, equivalently, for convolutions of measures) are a cornerstone of classical probability theory. Distributions arising as limits in these theorems are called infinitely divisible.

We will discuss limit theorems for repeated composition of functions on the upper half-plane. Note that unlike addition or convolution, composition is a non-commutative operation. What are the limit theorems? Which functions arise as limits? We will see both parallels and differences from the usual setting. This is joint work with John D. Williams.

**Location: Gibson Hall 414
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**Time: 3:30**

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PROBABILITY & STATISTICS

**Sean Lawley - University of Utah
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**Abstract:
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**Location: Gibson 310
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**Time: 3:00**

**Jonathan Montaño - University of Kansas
**

**Abstract:**

**Location: Gibson Hall 414
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**Time: 2:00**

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**Geordie Richards - UTAH STATE UNIVERSITY (Host: NATHAN GLATT-HOLTZ)
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**Abstract:
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We will survey some recent results on the construction and proof of invariance for certain canonical measures, such as the Gibbs measure, under the flow of dispersive Hamiltonian PDEs. Proving the invariance of these measures is often nontrivial due to the low regularity of functions belonging to their support. Focus will be placed on the generalized Korteweg-de Vries (gKdV) equations; Bourgain proved invariance of the Gibbs measure for KdV and mKdV, which have quadratic and cubic nonlinearities, respectively. Previously, we proved invariance of the Gibbs measure for the quartic gKdV by exploiting a nonlinear smoothing induced by initial data randomization. More recently, in joint work with Tadahiro Oh (Edinburgh) and Laurent Thomann (Nantes), we have established this invariance for gKdV with any odd power (defocusing) nonlinearity using a probabilistic construction of solutions.

**Location: Gibson Hall 414
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**Time: 3:30**

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**Prof. Leo Rebholz - Clemson University (Host Kun Zhao)
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**Abstract:
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**Location: Gibson Hall 414
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**Time: 3:30**

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**Gregory Lyng - UNIVERSITY OF WYOMING (HOST: VINCENT MARTINEZ)
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**Abstract:
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In this talk we give an overview of a body of results pertaining to the stability of detonation waves. These are particular, dramatic solutions to systems modeling mixtures of reacting gases. They are known to have delicate stability properties. On the mathematical side, the centerpiece of the program is the Evans function. This is a spectral determinant whose zeros agree in location and multiplicity with the eigenvalues of the linearized operator about the wave; it enters the analysis at both the nonlinear and linear/spectral levels. We discuss both theoretical aspects of the Evans function and also issues related to its practical computation. On the physical side, much of the novelty of this body of work stems from the inclusion of oft-neglected diffusive effects (e.g., viscosity, heat conductivity, species diffusion) in the analysis. Indeed, this modeling choice sometimes leads to surprising results.

**Location: Gibson Hall 414
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**Time: 3:30**

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