MCAIM hosts a seminar in which graduate students present their work. Talks are aligned with MCAIM’s mission: to initiate and foster research interactions between mathematics, the sciences, and engineering.

**2022-202**3

**April 7, 2023****Speaker:** Varun Goyal, University of Michigan**Title:** Nonlinear Model for the Mechanics and Dynamics of Cochlear Outer Hair Cell Stereocilia**Abstract: **Sound waves vibrating the eardrum excite the ossicles in the middle ear ultimately driving waves in the cochlea. Cochlear vibrations are processed by inner hair cells and outer hair cells (OHCs). Our focus is on the OHCs that nonlinearly amplify the sound converting a time-varying motion of its apically adorned hair bundle (HB) to an alternating current. The OHC HB consists of roughly three rows of stereocilia arranged according to their heights. Understanding how the bundle stiffness, sensitivity, and transduction current depend on the physiology and anatomy of the stereocilia is crucial and an open question. Therefore, we are developing a three-row model of an isolated HB to quantify each row’s contribution to the passive and active mechanics of the HB. The derived equations of motion include the nonlinear kinematics, viscoelastic HB mechanics, and the nonlinear response of the mechano-electric transducer channels coupled to an adaptation mechanism. We also linearize the model to conduct stability analysis and determine the dependence of the responses on the rate constants. Our preliminary results show a higher current influx through the middle row than the shortest row and closely predicts the biophysical parameters like the sensitivity and stiffness of the bundle.

**March 31, 2023****Speaker:** Hongyi Zhou, University of Michigan**Title:** Quantifying the Dissipation Enhancement of Cellular Flows**Abstract: **We study the dissipation enhancement by cellular flows. Previous work by Iyer, Xu, and Zlatoš produces a family of cellular flows that can enhance dissipation by an arbitrarily large amount. We improve this result by providing quantitative bounds on the dissipation enhancement in terms of the flow amplitude, cell size and diffusivity. Explicitly we show that the mixing time is bounded by C(ε^2/κ + |ln δ|^2/(ε^2A)). Here κ is the diffusivity, ε is the cell size, A/ε is the flow amplitude, and δ = (κ/A)^(1/2) is the thickness of the boundary layer. The above agrees with the optimal heuristics. We also prove a general result relating the dissipation time of incompressible flows to the mixing time. The main idea behind the proof is to study the dynamics probabilistically and construct a successful coupling.

**March 17, 2023****Speaker:** Shirlyn Wang, University of Michigan**Title:** Modeling CTL-Mediated Tumor Cell Death Mechanisms and the Activity of Immune Checkpoints in Immunotherapy**Abstract: **Immunotherapy has dramatically transformed the cancer treatment landscape. Of the variety of types of immunotherapies available, immune checkpoint inhibitors (ICIs) have gained the spotlight. Although ICIs have shown promising results for some patients, the low response rates in many cancers highlight the challenges of using immune checkpoint blockade as an effective treatment. Cytotoxic T lymphocytes (CTLs) execute their cell-killing function via two distinct mechanisms. The first process is fast-acting and perforin/granzyme-mediated, and the second is a slower, Fas ligand (FasL)-driven killing mechanism. There is also evidence suggesting that the preferred killing mechanism by CTLs depends on the antigenicity of tumor cells. To determine the key factors affecting responses to checkpoint blockade therapy, we constructed an ordinary differential equation model describing in vivo tumor-immune dynamics in the presence of active or blocked PD-1/PD-L1 immune checkpoint. Specifically, we analyzed which aspects of the tumor-immune landscape affect the response to ICIs with endpoints of tumor size and composition in the short and long term. By generating a virtual cohort with heterogeneous tumor and immune attributes, we also simulated the therapeutic outcomes of immune checkpoint blockade in a largely diverse population. In this way, we identified key tumor and immune characteristics that are associated with tumor elimination, dormancy and escape. Our analysis sheds light on which fraction of a population potentially responds well to ICIs and ways to enhance therapeutic outcomes with combination

therapy.

**February 10, 2023****Speaker:** Jiajia Guo, University of Michigan**Title:** Anisotropic Active Brownian Particle with a Fluctuating Propulsion Force**Abstract: **The active Brownian particle (ABP) model describes a swimmer, synthetic or living, whose direction of swimming is a Brownian motion. The swimming is due to a propulsion force, and the fluctuations are typically thermal in origin. We present a 2D model where the fluctuations arise from nonthermal noise in a propelling force acting at a single point, such as that due to a flagellum. We take the overdamped limit and find several modifications to the traditional ABP model. Since the fluctuating force causes a fluctuating torque, the diffusion tensor describing the process has a coupling between translational and rotational degrees of freedom. An anisotropic particle also exhibits a noise-induced drift. We show that these effects have measurable consequences for the long-time diffusivity of active particles, in particular adding a contribution that is independent of where the force acts. This is joint work with Prof. Jean-Luc Thiffeault.

**December 2, 2022****Speaker:** Chris Stith, University of Michigan**Title:** Trapped Surface Formation in General Relativity**Abstract: **Trapped surfaces are a central topic of study in mathematical general relativity. Penrose’s incompleteness theorem (1965) tells us that the presence of these surfaces in a suitable class of spacetimes implies that the spacetime is geodesically incomplete, thus tying the concept of trapped surfaces to the study of incompleteness and singularity formation in GR. In this talk, we will introduce the concept of trapped surfaces and discuss Christodoulou’s breakthrough result from 2009 showing that closed trapped surfaces can form in vacuum via the focusing of incoming gravitational radiation.

**November 18, 2022****Speaker: **Reebhu Bhattacharyya, University of Michigan**Title:** Complex-Valued Hamiltonians in Classical and Quantum Mechanics**Abstract: **The Hamiltonian, which is for many systems the energy function, plays an important role in both classical and quantum evolution of systems. Generally, it is a positive real-valued function in classical mechanics corresponding to a positive self-adjoint operator in quantum mechanics. But what happens if we allow it to be complex-valued (i.e., non-Hermitian in the quantum sense)? In this talk, we will explore some interesting geometric aspects of this question. In particular, we will give a quick overview of the Hamiltonian formulation of classical mechanics and how to get a quantum description using a quantization scheme. Then, we will describe the (classical) time evolution of a system under a complex Hamiltonian and finally, we will make a few remarks about what this tells us about quantum evolution.

**November 4, 2022****Speaker: **April Nellis, University of Michigan**Title:** A Neural Network Approach to High-Dimensional Optimal Switching Problems with Jumps**Abstract: **We develop a backward-in-time machine learning algorithm that uses a sequence of neural networks to solve optimal switching problems in energy production, where electricity and fossil fuel prices are subject to stochastic jumps. We then apply this algorithm to a variety of energy scheduling problems, including novel high-dimensional energy production problems. Our experimental results demonstrate that the algorithm performs with accuracy and experiences linear to sub-linear slowdowns as dimension increases, demonstrating the value of the algorithm for solving high-dimensional switching problems.

**October 21, 2022****Speaker: **Leonardo Heveling, Radboud University**Title:** Causality and Time in General Relativity**Abstract: **Understanding the causal relationships between spacetime events is an important aspect of General Relativity. In this talk, I will introduce some basic concepts in causality theory, such as the causal and chronological relations, time functions, and causality

conditions. Towards the end, I will focus on the most important causality condition, namely global hyperbolicity, and its different

characterizations. Time permitting, I will include a recent characterization obtained by Burtscher and myself.

#### 2021-2022

**April 15, 2022****Speaker: **Neophytos Charalambides, University of Michigan**Title:** Approximate Matrix Multiplication and Laplacian Sparsifiers**Abstract:** A ubiquitous operation in numerical analysis and scientific computing is matrix multiplication. However, it presents a major computational bottleneck when the matrix dimension is high, as can occur for large data size or feature dimension. A common approach in approximating the product, is to subsample row vectors from the two matrices, and sum the rank-1 outer products of the sampled pairs. We propose a sampling distribution based on the leverage scores of the two matrices. We give a characterization of our approximation in terms of the Euclidean norm, analogous to that of a L_2-subspace embedding. We then show connections between our algorithm; CR-multiplication, with Laplacian spectral sparsifiers, which also have numerous applications in data science, and how approximate matrix multiplication can be used to devise sparsifiers.

**April 1, 2022****Speaker: **Kashvi Srivastava, University of Michigan**Title:** Deterministic and Stochastic Modeling of Dynamical Systems in Chemical Kinetics**Abstract:** Several significant chemical reactions include molecules called enzymes which act as catalysts for product formation. Mathematical models have been developed to study such enzyme kinetics to understand the behavior of chemical species involved and the conditions which influence them. Singular perturbation methods can be applied to further simplify the governing differential equations of the reactions. In this talk, we will discuss one such significant system in chemical kinetics called the Michaelis-Menten reaction. We will introduce Fenichel theory and apply it to the system to obtain approximations. We will also talk about the validity of these approximations in stochastic environments. The talk will include brief introductions to the theory of perturbation problems, dynamic bifurcations and stochastic simulation algorithms.

**March 11, 2022****Speaker: **Alexis Flores-Betancourt**Title:** Fast Finite Elements for Atomic Force Microscopy**Abstract:** Atomic force microscopy is a type of scanning probe microscopy with resolution on the order of fractions of a nanometer and is widely used to collect data on various mechanical, functional, and electrical properties. As a powerful imaging and measurement technique, its use in nanoscale research and industrial R&D is critical. The relationship between an AFM cantilever and its application mode is crucial in determining the success of an AFM measurement. Although a very versatile and reliable instrument, the forces acting at the tip of the cantilever are still not quantifiable. Determining such forces will not only make the AFM a more quantitative instrument but will also allow the distinguishment of true signals from spurious ones. With the use of finite element analysis, we are creating a model that can quickly and efficiently simulate the dynamic behavior of an AFM cantilever under realistic conditions – giving us a tool to fit real data to extract more directly the forces acting at the tip and along the cantilever. This work will allow people to use AFM to directly determine mechanical properties of materials at the nanoscale as well as electromechanical properties and be used as a tool to train neural nets and similar tools.

**February 4, 2022****Speaker: **Michael Wadas, University of Michigan**Title:** Formation and Scaling of Vortex Rings Generated from Shock-Accelerated Interfaces**Abstract:** Vortex rings are known to emerge in a variety of flows relevant to astrophysics, high energy density physics, and inertial confinement fusion, where they can significantly affect the flow through the transport of vorticity. We systematically study the generation and scaling of such rings utilizing a numerical platform involving a shock passing through an interface separating two dissimilar fluids along which there is a hole filled with the heavier fluid. As the shock passes through the interface, it deposits baroclinic vorticity that induces a complex phase inversion process ultimately resulting in the ejection of a ring from the hole. By modulating the aspect ratio of the hole, the amount of vorticity in the flow available to the ring is controlled. Based on the aspect ratio of the ring, we find that two distinct flow regimes emerge. For small aspect ratios, a single ring is generated that contains the majority of the vorticity deposited by the shock. Beyond a critical hole aspect ratio, however, the circulation of the ejected ring saturates, and the additional vorticity in the flow accumulates in a jet that trails the leading ring. This behavior suggests the existence of a fundamental formation number governing the scaling of rings generated from shock-accelerated interfaces, including those in Richtmyer-Meshkov flows. This work is funded by the U.S. Department of Energy (DOE) NNSA Center of Excellence under cooperative agreement number DE-NA0003869 and by the U.S. DOE NNSA Stewardship Science Graduate Fellowship under grant DE-NA0003960.

**November 12, 2021****Speaker: **Guanhua Sun, University of Michigan**Title:** Why Do We Sleep: Three Synaptic Hypothesis**Abstract:** Why do we sleep? What happens in our brain during sleep? How does synaptic activity affect our sleep and vice-versa? Scientists have been keen to answer these questions. In this talk, I will survey through three synaptic hypotheses of sleep: 1. Synaptic Homeostasis Hypothesis(SHY) 2. Network Homeostasis Hypothesis 3. Phosphorylation Hypothesis. And I will present current computational works of a calcium plasticity model that can potentially test one or more of those three hypotheses.

**October 29, 2021Speaker: **Preetham Mohan, University of Michigan

**Title**

*:*A Brief History of Quantum Learning Theory

**Abstract:**With the success of machine learning algorithms and the early promise of quantum computing, it is natural to ask whether efficient quantum learning is possible. Are there learning contexts where we observe a quan¬tum (exponential) speed-up? Are there instances where it is not possible for quantum computers to (significantly) outperform classical ones? In this talk, we begin a discussion on these questions by reviewing the results for several learning problems. In particular, we begin with quantum exact, PAC, and agnos¬tic learning of classical Boolean functions. Thereafter, we will consider statisti¬cal and adversarial learning of quantum states. We end with a brief overview of broader quantum machine learning results, including the quest for a quantum neural network.

**October 8, 2021Speaker: **Som Phene, University of Michigan

**Title:**Black Hole Information and Mirror Symmetry

**Abstract:**We explore how an understanding of Black Hole Information through the perspective of Reflection Symmetry prescription gives rise to a natural interpretation of quantum extremal surface. A comparison of quantum information-theoretic tools in the context of quantum entanglement entropy, with the classical setting (Shannon entropy), will be drawn in order to understand Ryu-Takayanagi prescription and it applications to Holography.

**2020-2021 **

**April 5, 2021****Speaker: **Boyang Wu, University of Michigan**Title: **On the Wave Turbulence Theory: Deriving the Kinetic Equation and Time Scale for the Six-Wave Interactions in the Alpha-Fermi-Pasta-Ulam System**Abstract: **Wave Turbulence is a non-equilibrium statistical system of randomly interacting waves. Kinetic equations of Wave Turbulence describe the evolution of wave energy in Fourier space. In this talk, we begin by introducing the basic idea of Wave Turbulence formalism and making a summary of the work done for the alpha-Fermi-Pasta-Ulam system, and figure out that there is no rigorous proof for the kinetic equation and the resulting timescale. We then derive the kinetic equation for the six-wave Interactions in the alpha-Fermi-Pasta-Ulam system with N = 16, 32, and 64 masses, which has a slightly different model whose four-wave resonances cannot be removed by any nondivergent canonical transformations compared to the common six-wave Hamiltonian system, following the formal steps.

**March 29, 2021****Speaker:** Rishi Sonthalia, University of Michigan**Title: **How to Train Linear Stacked Denoising Autoencoders**Abstract: **Learning a good representation is a crucial step in many machine learning pipelines. One common technique to do so is to use Stacked Denoising Autoencoders (SDAEs). In this talk I will go over the basics of Stacked Denoising Autoencoders. We will that are superior to many forms of pre-training. I will then present how Random Matrix Theory can be used to understand their superior performance.

**March 8, 2021****Speaker:** Caleb Meyer, University of Michigan**Title:** Modeling Circadian Phase in Humans**Abstract: **Circadian rhythms are endogenous oscillations of approximately 24 hours, which are largely entrained by the light/dark cycle. Understanding the rhythmic processes that determine circadian phase has many practical applications (related to jet lag, shift work, chronotherapy, and overall rest/alertness, for example). In this talk, we will detail the primary light-based ODE models that have been used to predict circadian phase in humans. We present some of our work in adapting these models to differing datasets and populations, including shift workers and cancer patients. Finally, we discuss alternate techniques to extract parameters of circadian rhythm from physiological data collected via wearable devices.

**February 15, 2021****Speaker: **Chris Stith, University of Michigan**Title: **The Cauchy Problem for the Einstein Equations**Abstract: ** The Einstein equations are a system of PDE governing the curvature of a pseudo-Riemannian manifold (spacetime). It is natural to ask if these equations have a formulation as an initial value problem, and if they can be solved in this context. The full answer to this question arose more than 30 years after Einstein first wrote down the equations; in this talk we will discuss Choquet-Bruhat’s breakthrough work on the subject from the mid-20th century. Time permitting, some related topics in the rich history of this problem may be discussed.

**January 25, 2021****Speaker:** Saibal De, University of Michigan**Title:** Quantum Computing for Continuous Optimization Problems**Abstract:** Optimization is one of the most common learning tasks in many scientific and industrial applications. However, in many cases, these problems are so large and complex that it takes days, if not weeks, to obtain an answer even with the fastest supercomputers. Quantum computing has recently attracted a lot of attention based on its potential for accelerating specific computational tasks well beyond classical means. For instance, last year Google AI announced their achievement of quantum supremacy; they performed a computation that would take tens of thousands of years on even the fastest classical supercomputer. In this talk, I will introduce some of the basic concepts of quantum computing and focus on how we can potentially use it to solve optimization problems.

**December 7, 2020****Speaker:** Alana Huszar, University of Michigan**Title:** Non-commutative Rank and the Null Cone**Abstract: **How would we figure out what the rank of a matrix is, when we’re considering its rank over a free skew field? This non-commutative version of the “Edmonds’ problem”, tackled by many computer scientists, has rich connections to invariant theory. In this example based talk, we’ll go through different formulations of non-commutative rank, and tools we can use from invariant theory, like the “null cone”, to determine this rank. No previous familiarity (besides linear algebra) is required.

**November 30, 2020 ****Speaker: **Kevin Whitley, University of Michigan**Title: **Time-Domain Signatures of Supermassive Black Hole Binaries**Abstract: **Modern observational techniques and technologies are insufficient for identifying close-separation supermassive black hole (SMBH) binaries, despite their paramount importance to the evolution of SMBHs and the galaxies which host them. Current methods focus on resolving spectral signatures such as distinct emission line regions or multiple HI absorption lines in jets, but these are still unable to probe the extremely close separations relevant to gravitational wave explorations. In this work, we focus on the potential for identification of sub-parsec SMBH binaries via periodic electromagnetic emission. We use a 2D hydrodynamics simulation of a close-separation (100 AU), low mass-ratio (m/M = 0.01) SMBH binary to explore the characteristic frequencies with which optical, UV, and X-ray emissions will vary in these sources. We find strong variability across the electromagnetic spectrum with periods corresponding to the binary orbital time. There is also a significant enhancement of the continuum X-ray emission over a single SMBH model due to shocks excited along the accretion streams of the secondary’s accretion disk and where they strike the primary’s accretion disk and the circumbinary disk. These features can enable the identification of SMBH binary candidates through both single-epoch observations and long-term time monitoring.

**November 9, 2020 ****Speaker:** Luis Álvarez, City University of New York**Title:** Symmetries Uncover the Minimal Regulatory Network for Logical Computations in Bacteria**Abstract:** The use of symmetries in physics to reduce a complex system to its underlying components and interactions is a widely known useful tool. Recently we have shown 1 that relevant symmetries of biological networks allow for a systematic reduction of the networks that preserves information flow. Symmetry fibrations, which consists of grouping nodes that share an isomorphic input tree into equivalence relations called fibers, allows for the collapse of the network while preserving information flow. Further reducing the network by the k-core decomposition of the collapsed network gives the minimal network driving the dynamics of the entire network. In gene transcriptional regulatory networks (TRN), where fibers consist of genes that are synchronized, this gives the minimal transcriptional regulatory network. This minimal structure is understood as a combination of genetic circuits which perform core logical computations from outside inputs and the current state. Hence, symmetry principles unveil the minimal TRN that corresponds to the core computational machinery.

**October 26, 2020****Speaker:** Jorge Arce Garro, University of Michigan**Title:** Control Theory Methods for Qubits in Open Quantum Systems**Abstract:** Control of closed (Hamiltonian) quantum systems has been widely studied in the last few decades from both theoretical and interdisciplinary points of view. Recently, there has been a growing interest in control of open (dissipative, non-Hamiltonian) quantum systems because of their applications to physics, chemistry and quantum computing. In this talk, we will give an introduction to the control of open quantum systems, specializing to the case of the qubit: a 2-level quantum system widely studied in quantum computing and other applications. In particular, we will focus on mathematical tools used to describe and manipulate decoherence, which we will identify as the effect on a quantum state due to interactions with the environment.

**October 5, 2020Speaker: **Yutong Wang, PhD Candidate | Department of Electrical Engineering and Computer Science

**Title:**Multi-Class Support Vector Machines and Ordered Partitions

**Abstract:**Classification is a central problem in supervised learning, where the goal is to learn a decision function that accurately assigns labels to instances. The support vector machine (SVM) is a learning algorithm that is popular in practice and also has strong theoretical properties. However, most of the theory developed is for the binary classification setting, where there are only two possible labels to choose from. Our work is concerned with the multiclass setting where there are three or more possible labels for the decision function to choose from. Multiclass SVMs have been formulated in a variety of ways. A recent empirical study by Dogan et al. compared nine such formulations and recommended the variant proposed by Weston and Watkins (WW). Despite the superior empirical performance of the WW multiclass SVM, its theoretical properties remain poorly understood. Towards bridging this gap, we establish a connection between the hinge loss used in the WW multiclass SVM with ordered partitions. We use this connection to justify the recent empirical findings.