Math Colloquium - Fall 2023

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September 8

Victor Matveev, NJIT

Accuracy of Deterministic vs. Stochastic Simulation of Neurotransmitter and Hormone Release

Most cell processes exhibit high variability due to the fundamental stochasticity of all biochemical reactions. Quantifying the impact of stochastic effects is necessary for a deeper understanding of physiological processes, and helps in the choice of an efficient approach for the computational modeling of a particular process. This is especially true in the case of synaptic neurotransmitter release and endocrine hormone release, which are triggered by the calcium-sensitive transmembrane proteins that fuse the secretory vesicle membrane with the cell membrane upon calcium ion binding. It is known that stochastic calcium channel gating is one of the primary sources of vesicle fusion variability, and is computationally inexpensive to implement when simulating this mechanism. However, a more fundamental reason for the variability of secretory vesicle fusion is that a relatively small number of calcium ions enter the synaptic (or endocrine) terminal through a calcium channel during a brief cell stimulation. This entails large fluctuations in local calcium ion concentration caused by the calcium diffusion and its binding to calcium buffers and vesicle release sensors. This understanding leads to the widely held view that solving deterministic reaction-diffusion equations for continuous concentration fields may not provide sufficient accuracy when modeling calcium-dependent cell processes.

However, several comparative studies show a surprising close agreement between deterministic and stochastic simulations of calcium-dependent biochemical pathways, if the calcium channel gating is not calcium-dependent. This is a surprising result, deserving careful investigation. In this talk I will present further analysis and comparison of stochastic vs. deterministic modeling of biochemical processes regulating vesicle fusion, showing that the discrepancy between the two approaches can be surprisingly small even when as few as 40-50 ions enter the cell per single channel-vesicle complex. The reason is that the discrepancy between deterministic and stochastic approaches is determined by the size of the correlation between the local calcium concentration and the state of the vesicle release sensor, rather than the fluctuation amplitude per se. Although diffusion and buffering increase fluctuation amplitude, they effectively average out correlations between reactant fluctuations. Therefore, the mass-action reaction-diffusion equations for calcium concentration coupled to mean-field description of vesicle release sensors provide an accurate estimate of the probability distribution of vesicle release latency. These results are general and apply to the modeling of any biochemical pathway that involves at most second-order reactions between molecules of different species.

September 15

Chun Liu, IIT Chicago
Hosted by: Yuan-Nan Young

Dynamic Boundary Conditions and Motion of Grain Boundaries

I will present the dynamic boundary conditions in the general energetic variational approaches. The focus is on the coupling between the bulk effects with the active boundary conditions.In particular, we will study applications in the evolution of grain boundary networks, in particular, the dragof trip junctions. This is a joint work with Yekaterina Epshteyn (University of Utah) and Masashi Mizuno (Nihon University).

September 22

Matthieu Labousse, ESPCI Paris
Hosted by: Anand Oza

Soft Violation of Bell's Inequality

Walking drops on Faraday waves are one of the rare examples of non-quantum wave-particle duality. A series of striking experiments with one walking drop has led to behaviors that were thought to be peculiar to the quantum scale. I will present a recent numerical and experimental investigation involving the coupling of two walking drops.To our great surprise, we found that the statistical behavior of this system shares some non-expected features of collective emission of photons in quantum optics, including superradiance and violation of Bell's inequality. This result is very intriguing as the quantum counterpart is the signature of non-separable states which in our case,is the result of a collective wave self-organization.

September 29

Shidong Jiang, Flatiron Institute

A Dual-space Multilevel Kernel-splitting Framework for Discreteand Continuous Convolution

We introduce a new class of multilevel, adaptive, dual-space methods for computing fast convolutional transforms. These methods can be applied to a broad class of kernels, from the Green's functions for classical partial differential equations (PDEs) to power functions and radial basis functions such as those used in statistics and machine learning. The DMK (dual-space multilevel kernel-splitting) framework uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied.The main novelty of DMK is that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables, but without requiring the FFT for its asymptotic performance. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per gridpoint, even in a fully adaptive context. For continuous source distributions, the evaluation of local interactions is further accelerated by approximating the kernel at the finest level as a sum of Gaussians with a highly localized remainder.The Gaussian convolutions are calculated using tensor product transforms, and the remainder term is calculated using asymptotic methods. We illustrate the performance of DMK for both continuous and discrete sources with extensive numerical examples in two and three dimensions.This is joint work with Leslie Greengard. 

October 6

Arnold Mathijssen, University of Pennsylvania
Hosted by: Enkeleida Lushi

Title: TBA

Abstract: TBA

October 13

Nick Trefethen, Harvard University
Hosted by: Linda Cummings

Polynomials and Rational Functions

Much of my past twenty years has been spent working with polynomials(Chebfun, Approximation Theory and Approximation Practice) and then rational functions (AAA and AAA-LS approximation, lightning Laplace solver).  This talk will start with a broad discussion of the role of polynomials and rational functions in computational mathematics. Polynomials are everywhere, rational functions not so much.Then we will turn to some of the new developments that suggest rational functions may be more important in the future.  The talk will include numerical demos and also a new theorem quantifying the power of rational functions for solving Laplace problems in the plane.

October 20

TBA

October 27

Vu Thai Luan, Mississippi State University
Hosted by: David Shirokoff

Title: TBA

Abstract: TBA

November 3

Anna Balazs, University of Pittsburgh
Hosted by: Lou Kondic

Title: TBA

Abstract: TBA

November 10

TBA

November 17

Niall Mangan, Northwestern University
Hosted by: Amitabha Bose

Title: TBA

Abstract: TBA

November 24

No Colloquium - Thanksgiving Break

December 1

Yoichiro Mori, University of Pennsylvania
Hosted by: Enkeleida Lushi

Title: TBA

Abstract: TBA

December 8

TBA

Updated: September 25, 2023