# Math Colloquium - Fall 2023

Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall I, unless noted otherwise.

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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

**Collective Functionalities Emerging in Microbial Active Matter**

Microbes often cooperate to withstand predators and compete even with multicellular organisms. Together, they can achieve functionalities that alone they cannot. However, this puzzle of how biological self-organization emerges from the collective dynamics of individual constituents remains unsolved. In this talk, I will discuss some of these collective functionalities, including communication, navigation, and cooperative nutrient transport. First, we focus on ultra-fast communication through “hydrodynamic trigger waves”, signals between cells that propagate hundreds of times faster than their swimming speed [1]. Second, we will explore how bacteria can reorient against flows and contaminate reservoirs upstream [2]. Third, we consider how bacteria generate their own flows to transport nutrients [3], and how “active carpets” like biofilms can lead to enhanced non-equilibrium diffusion [4]. Together, these ideas help us understand emergent self-organization in biological systems and the design space of active materials.

[1] Mathijssen et al. “Collective intercellular communication through ultra-fast hydrodynamic trigger waves,” Nature 571, 560-564 (2019)

[2] Mathijssen et al. “Oscillatory surface rheotaxis of swimming E. coli bacteria,” Nat. Commun. 10, 3434 (2019)

[3] Jin et al. “Collective entrainment and confinement amplifies transport by schooling micro-swimmers”, Phys. Rev. Lett. 127: 088006 (2021)

[4] Guzman-Lastra et al. “Active carpets drive non-equilibrium diffusion and enhanced molecular fluxes,” Nat. Commun. 12: 1906 (2021)

Arnold Mathijssen was named ‘30 under 30’ by Scientific American and was awarded the Sir Sam Edwards PhD Thesis Prize for his work in group of Julia Yeomans FRS at the University Oxford (2016). Supported by an HFSP cross-disciplinary fellowship, he moved to the lab of Manu Prakash at Stanford University, where the American Physical Society presented him the Charles Kittel Award (2019). He is now Assistant Professor of Physics & Astronomy at UPenn, director of the working group for Environmental and Biological Fluid Dynamics, and co-chair of the 2024 CUWiP Conference for Undergraduate Women in Physics.

#### 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

**Pavel Lushnikov**, University of New Mexico

Hosted by: Mike Siegel

**Stokes Waves, Riemann Surface Sheets and Wavebreaking of Surface Dynamics**

A fully nonlinear surface dynamics of the time dependent potential flow of ideal incompressible fluid with a free surface is considered in two dimensional geometry. Arbitrary large surface waves can be efficiently characterized through a time-dependent conformal mapping of a fluid domain into the lower complex half-plane. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian system for the pair of new conformal variables. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets including Stokes wave which is the fully nonlinear wave propagating with constant velocity. The analytical structure of Stokes wave is analyzed. For non-limiting Stokes wave the only singularity in the physical sheet of Riemann surface is the square-root branch point. The corresponding branch cut defines the second sheet of the Riemann surface if one crosses the branch cut. The infinite number of pairs of square root singularities is found corresponding to infinite number of non-physical sheets of Riemann surface. Each pair belongs to its own non-physical sheet of Riemann surface. Increase of the steepness of the Stokes wave means that all these singularities simultaneously approach the real line from different sheets of Riemann surface and merge together forming 2/3 power law singularity of the limiting Stokes wave. It is shown that non-limiting Stokes wave at the leading order consists of the infinite product of nested square root singularities which form the infinite number of sheets of Riemann surface. Stokes waves of small amplitude have well-known Benjamin-Fair instability. We find that that for steeper Stokes waves, an instability caused by disturbances localized at the wave crest vastly surpasses the growth rate of the modulational instability. The nonlinear evolution of the instability leads to the formation of plunging breakers. If we consider initial conditions with short branch cuts then fluid dynamics is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including the pairs of moving square root branch points resulting in wavebreaking.

#### October 27

**Vu Thai Luan**, Mississippi State University

Hosted by: David Shirokoff

**Advanced Time Integrators for Multiphysics Problems and Applications**

In many applications in science and engineering, there is a high demand for fast and accurate computational methods that can simulate complex multi-physical processes and their interactions taking place at a wide range of temporal scales. A notable example of such an application is numerical weather prediction (NWP) and climate modeling, which rely on computational solution of primitive equations used to predict the behavior of the atmosphere, oceans, land surface, etc. Numerical solutions of such multiphysics problems remain a challenging task due to the presence of multiple time scales in the system where different processes take different amounts of time to complete. For instance, acoustic and gravity waves travel much faster than advection/convection processes, thereby preventing the use of longer time steps which poses difficulties for real-time simulation of weather and ocean conditions. With the significantly increased computational power in recent years, the challenge will increase with modern simulations which include more physical processes (e.g., adding smaller scales) in an attempt to capture higher-order modeling effects that were omitted from earlier models. As such, developing fast and accurate time integration methods is crucial for a wide range of applications that rely on large-scale simulations of complex systems. This talk aims to introduce innovative exponential and multirate time integrators recently developed and to showcase their performance across several applications in NWP, computer graphics, and computational biology.

#### November 3

**Lou Kondic**, NJIT

**From Materials Science to Computational Topology: Interaction Networks in Particulate Matter**

We will discuss interaction networks that spontaneously form in particulate-based systems. These networks, most commonly known as `force chains' in granular systems, are dynamic structures that are by now known to be of fundamental importance for revealing the underlying physical causes of a number of physical phenomena involved in statics and dynamics of particulate-based systems. The presentation will focus on applications of algebraic topology, and in particular of persistent homology (PH) to analysis of such networks found in both simulations and experiments. PH allows for a simplified representation of complex interaction fields in both two and three spatial dimensions in terms of persistent diagrams (PDs)that are essentially point clouds. These point clouds could be compared meaningfully, meaning that they allow for the analysis of the underlying systems' static and dynamic properties. It is important to point out that such representation is robust concerning small perturbations, which is crucial in applying the method to the analysis of experimental data. In the second part of the talk, we will focus on interaction networks in both simulated and experimental systems experiencing stick-slip, intermittent-type of dynamics, with a focus on exploring the predictability potential of the considered topological measures.

#### November 10

**Indranil SenGupta**, Florida International University

Hosted by: Sunil Dhar

**Risk Management, Data Science-based Improvements, and Financial Applications**

In this presentation, at first we present some results for stochastic modeling in the financial markets. There are some well-known drawbacks for the existing stochastic models in the literature. An improvement of the stochastic model through various machine and deep learning algorithms will be proposed. This refinement of the stochastic model based on data science-based approaches will lead to the extraction of a deterministic parameter from the financial data set. The analysis is implemented to the commodity market and some promising new results will be obtained. The results will also be implemented in the portfolio optimization and hedging techniques corresponding to the risk management of commodity markets. We also connect the relevance of this research to the environmental finance.

#### November 17

**Niall Mangan**, Northwestern University

Hosted by: Casey Diekman

**Identifying Models From Data**

Building models for biological, chemical, and physical systems has traditionally relied on domain specific intuition about which interaction and features most strongly influence a system. Statistical methods based in information criteria provide a framework to balance likelihood and model complexity. Recently developed for and applied to dynamical systems, sparse optimization strategies can select a subset of terms from a library that best describe data, automatically interfering model structure. I will discuss my group's application and development of data driven methods for model selection to 1) find simple statistical models to use wastewater surveillance to track the COVID pandemic and 2) recover chaotic systems models from data with hidden variables. I'll briefly discuss current preliminary work and roadblocks in developing new methods for model selection of biological metabolic and regulatory networks.

#### November 24

**No Colloquium - Thanksgiving Break**

#### December 1

**Ian Tobasco**, Rutgers

Hosted by: Travis Askham

**Towards Homogenization of Mechanism-based Mechanical Metamaterials **

Mechanical metamaterials are many-body elastic systems that deform in unusual ways, due to the interactions of essentially rigid building blocks. Examples include origami patterns with many folds, or kirigami patterns made by cutting material from a thin elastic sheet. In either case, the local deformations of the pattern involve internal degrees of freedom which must be matched with the usual global Euclidean invariances --- e.g., groups of four origami panels move by coordinated rotations and translations, but it is still possible to bend the whole origami pattern into an overall curved shape. This talk will introduce the homogenization problem for kirigami and origami metamaterials to a broad audience and describe our recent results. Our goal is to explain the link between the design of the individual cuts/folds and the bulk deformations and geometries they can produce. This is joint work with Paul Plucinsky (U. Southern California, Aerospace and Mechanical Engineering) and Paolo Celli (Stony Brook U., Civil Engineering).

#### December 8

**Paul Hand**, Northeastern University

Hosted by: Enkeleida Lushi

**Signal Recovery with Generative Priors**

Recovering images from very few measurements is an important task in imaging problems. Doing so requires assuming a model of what makes some images natural. Such a model is called an image prior. Classical priors such as sparsity have led to the speedup of Magnetic Resonance Imaging in certain cases. With the recent developments in machine learning, neural networks have been shown to provide efficient and effective priors for inverse problems arising in imaging. In this talk, we will discuss the use of neural network generative models for inverse problems in imaging. We will present a rigorous recovery guarantee at optimal sample complexity for compressed sensing and other inverse problems under a suitable random model. We will see that generative modelsenable an efficient algorithms for phase retrieval and spiked matrix recovery from generic measurements with optimal sample complexity. In contrast, no efficient algorithm is known for this problem in the case of sparsity priors. We will discuss strengths, weaknesses, and future opportunities of neural networks and generative models as image priors. This talk is intended to be pedagogical in nature and will involve technical mathematical results using nonasymptotic random matrix theory.

*Updated: November 28, 2023*