**“Notes on the Minkowski Measure, the Minkowski Symmetral, and the Banach-Mazur Distance” **

Xing Huang, Ph.D. student from China

**September 29, 2017**

** 11:20AM- 12:20PM**

** Armitage 123**

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In this talk we derive some basic inequalities connecting the Minkowski measure of symmetry, the Minkowski symmetral and the Banach-Mazur distance. We then explore the geometric contents of these inequalities and shed light on the structure of the quotient of the space of convex bodies modulo affine transformations.

**“The Ins and Outs of the Diffusion Approximation”**

Charles Epstein, Department of Mathematics at UPENN

**October 17, 2017**

** 12:45PM- 1:45PM**

** Science Building Lecture Hall**

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Population Genetics is the study of how the distribution of different types of individuals evolves in a reproducing population. The standard models incorporate effects of random mating, mutation, selection, and migration. In their simplest form these models are discrete Markov chains that model a finite population. These discrete models are very difficult to analyze and compute with, and so methods were developed to replace the discrete models with continuum models allowing for the usage of calculus. We describe both the discrete and continuous models and how they are related, as well as recent work on the analysis and numerics of the continuous models.

**“Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control and more” **

Jesús María Sanz-Serna, Universidad Carlos III de Madrid, Spai

**November 17, 2017**

** 11:20AM- 12:20PM**

** Armitage 123**

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Symplectic Runge-Kutta (RK) schemes were introduced in the 1980’s to integrate numerically Hamiltonian systems of differential equations. As it turns out, unbeknownst to the user, symplectic RK schemes implicitly appear in a number of applications, including automatic differentiation, optimal control, etc. The talk explains in very general terms the links between symplectic integration and those application areas.

**“Stochastic Simulation of Multiscale Reaction-Diffusion Models via First Exit Times” **

Lina Meinecke, Department of Mathematics at UC Irvine

**December 15, 2017**

** 11:20AM- 12:20PM**

**Armitage 123**

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Mathematical models are important tools in systems biology, since the regulatory networks in biological cells are too complicated to understand by biological experiments alone. However, analytical solutions can be derived only for the simplest models and hence numerical simulations are often necessary.

This talk focuses on the mesoscopic simulation level, which captures both, space dependent behavior by diffusion and the inherent stochasticity of cellular systems. Space is partitioned into compartments by a mesh and the number of molecules of each species in each compartment gives the state of the system. We first examine how to compute the jump coefficients for a discrete stochastic jump process on unstructured meshes from a first exit time approach guaranteeing the correct speed of diffusion. Furthermore, we analyze different methods leading to non-negative coefficients by backward analysis and derive a new method, minimizing both the error in the diffusion coefficient and in the particle distribution.

The second part investigates macromolecular crowding effects. A high percentage of the cytosol and membranes of cells are occupied by molecules. This impedes the diffusive motion and also affects the reaction rates. Most algorithms for cell simulations are either derived for a dilute medium or become computationally very expensive when applied to a crowded environment. Therefore, we develop a multiscale approach, which takes the microscopic positions of the molecules into account, while still allowing for efficient stochastic simulations on the mesoscopic level. Finally, we compare on- and off-lattice models on the microscopic level when applied to a crowded environment.