Claremont Center for the Mathematical Sciences

An Exact Algorithm for the Unanimous Vote Problem (Feyza Duman Keles, NYU)
2026-05-01 16:15 - 17:15 (America/Los_Angeles) | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: Consider n independent, biased coins, each with a known probability of heads. Presented with an ordering of these coins, flip (i.e., toss) each coin once, in that order, until we have observed both a head and a tail, or flipped all coins. The Unanimous Vote problem asks us to find the ordering that minimizes the expected number of flips. Gkenosis et al. [arXiv:1806.10660] gave a polynomial-time approximation algorithm for this problem, where the approximation ratio is the golden ratio. They left open whether the problem was NP-hard. We answer this question by giving an exact algorithm that runs in time O(n log n). The Unanimous Vote problem is an instance of the more general Stochastic Boolean Function Evaluation problem: it thus becomes one of the only such problems known to be solvable in polynomial time. Our proof uses simple interchange arguments to show that the optimal ordering must be close to the ordering produced by a natural greedy algorithm. Beyond our main result, we compare the optimal ordering with the best adaptive strategy, proving a tight adaptivity gap of 1.2 + o(1) for the Unanimous Vote problem.

From ICON to GenICON: In-Context Operator Learning with Uncertainty Quantification (Siting Liu, UCR)
2026-04-27 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: I will introduce In-Context Operator Networks (ICON), a framework in which a single neural network learns solution operators for differential equations directly from a few prompted input-output examples at inference time, without any weight updates. ICON acts as a few-shot learner across forward and inverse problems for ODEs, PDEs, and mean-field control. I will then present a probabilistic interpretation: under a random differential equation data model, ICON implicitly computes the posterior predictive mean given the context, linking operator learning to Bayesian inference. This motivates GenICON, a generative variant that samples from the posterior predictive for principled uncertainty quantification, yielding a unified Bayesian view of in-context operator learning.

The Secret Life of Turbulent Fluids (Vincent Martinez, Caltech)
2026-04-20 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: Turbulence influences our lives in a multitude of ways, ranging from the mundane (when we stir milk into our coffee) to the spectacular (the formation of galaxies). It is a great achievement of the human intellect that we are able to locate fundamental mechanisms shared by phenomenon with such a dramatic difference in scale and subsequently study them abstractly through mathematics. This talk will present some of the ways for how fluids and turbulent motion can be studied mathematically and introduce a few interesting problems, both theoretical and practical, of ongoing scientific relevance.

Extremal Eigenvalues of Weighted Steklov Problems (Chiu-Yen Kao, CMC)
2026-04-13 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: We study the optimization of Steklov eigenvalues with respect to a boundary density function ρ on a bounded Lipschitz domain. We investigate the minimization and maximization of a Steklov eigenvalue over admissible densities satisfying pointwise bounds and a fixed integral constraint. We establish the existence of optimal solutions and provide structural characterizations: minimizers are bang-bang functions and may have disconnected support, while maximizers are not necessarily bang-bang. On circular domains, the minimization problem admits infinitely many minimizers generated by rotational symmetry, while the maximization problem has infinitely many distinct maximizers that are not symmetry-induced. We also show that an eigenvalue is generally neither convex nor concave with respect to the density function, limiting the use of classical convex optimization tools. To address these challenges, we analyze the objective functional and introduce a Fréchet differentiable surrogate that enables the derivation of optimality conditions. We further design an efficient numerical algorithm, with experiments illustrating the difficulty of recovering optimal densities when they lack smoothness or exhibit oscillations.

The Euler Equations in Function Spaces of Generalized Smoothness (Zachary Radke, OSU)
2026-03-30 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: In this talk, we will describe a well/ill-posedness result for the 2D incompressible Euler equations. We investigate solutions in a setting logarithmically smoother than previously done, in a hope to identify the key dynamics leading to a breakdown of regularity in 2D fluid flow. When order of the logarithmic derivative is sufficiently large one obtains global well posedness, however, below this threshold, one can construct initial data for which the corresponding solution blows up instantaneously in the logarithmic Sobolev norm. In this sense, the result is sharp at this logarithmic scale, but by no means is the story completed by it so we will discuss ways to dive deeper.

A Regularity Criterion for the 3-D Navier-Stokes Equations Based on Finitely Many Observations (Abhishek Balakrishna, USC)
2026-03-23 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: This talk presents a regularity criterion for the three-dimensional Navier–Stokes equations based on finitely many observations of the flow. Motivated by data assimilation, we study a nudging algorithm that incorporates coarse spatial measurements through general interpolation operators.
We show that suitable conditions on the observed data guarantee global regularity of the associated system and yield an exponential tracking property. As a consequence, we obtain a new regularity criterion for weak solutions formulated entirely in terms of finitely many observables, covering modal, volume, and nodal measurements.

An Odd Estimator for Shapley Values (Teal Witter, CMC)
2026-03-09 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: The Shapley value is a ubiquitous framework for attribution in machine learning, encompassing feature importance, data valuation, and causal inference. However, its exact computation is generally intractable, necessitating efficient approximation methods. While the most effective and popular estimators leverage the paired sampling heuristic to reduce estimation error, the theoretical mechanism driving this improvement has remained opaque. In this work, we provide an elegant and fundamental justification for paired sampling: we prove that the Shapley value depends exclusively on the odd component of the set function, and that paired sampling orthogonalizes the regression objective to filter out the irrelevant even component. Leveraging this insight, we propose OddSHAP, a novel consistent estimator that performs polynomial regression solely on the odd subspace. By utilizing the Fourier basis to isolate this subspace and employing a proxy model to identify high-impact interactions, OddSHAP overcomes the combinatorial explosion of higher-order approximations. Through an extensive benchmark evaluation, we find that OddSHAP achieves state-of-the-art estimation accuracy.

Joint work with Fabian Fumagalli, Landon Butler, Justin Singh Kang, and Kannan Ramchandran.

Structure-Preserving Discretizations for Fokker–Planck Equations via the Energy Dissipation Law (Satish Chandran, UCR)
2026-03-02 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: We present a new approach for deriving structure-preserving numerical discretizations of Fokker-Planck equations by establishing a connection between the Fokker-Planck equation and its semi-discrete master equation at the level of the energy-dissipation law. We determine the transition rate in the master equation via the detailed balance condition and the spatial discretization of the continuous energy-dissipation law. This approach ensures that the semi-discrete master equation satisfies the detailed balance condition and converges to the correct equilibrium. In addition to recovering existing transition rates proposed in earlier works, our framework uncovers new transition rates that have not been discussed in the current literature. This work is joint with Dr. Yiwei Wang (UCR).

Explainability and Analysis of Variance (Zijun Gao, USC)
2026-02-16 16:00 - 17:00 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: Existing tools for explaining complex models and systems are associational rather than causal and do not provide mechanistic understanding. We propose a new notion called counterfactual explainability for causal attribution that is motivated by the concept of genetic heritability in twin studies. Counterfactual explainability extends methods for global sensitivity analysis (including the functional analysis of variance and Sobol’s indices), which assumes independent explanatory variables, to dependent explanations by using a directed acyclic graphs to describe their causal relationship. Therefore, this explanability measure directly incorporates causal mechanisms by construction. Under a comonotonicity assumption, we discuss methods for estimating counterfactual explainability and apply them to a real dataset dataset to explain income inequality by gender, race, and educational attainment.
Bio: Zijun Gao is a tenure-track assistant professor in the Department of Data Sciences and Operations at USC Marshall Business School. She received her Ph.D. in Statistics from Stanford University in 2022 supervised by Professor Trevor Hastie. She served as a research associate in the Statistical Lab at the University of Cambridge from 2022 to 2023 hosted by Professor Qingyuan Zhao. Her research focuses on the estimation and inference problems in causal inference with heterogeneity, with side interests in distribution learning, selective inference, and model evaluation. She also works on real-world data motivated topics, with a specific emphasis on the applications in adaptive clinical trial and personalized medication.

A BKM-type criterion for the 3D incompressible Euler equations (Mustafa Aydin, USC)
2026-02-09 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: The three-dimensional incompressible Euler equations describe the motion of an ideal fluid, yet the mechanisms that govern the possible loss of regularity of smooth solutions remain only partially understood. A classical result of Beale, Kato, and Majda shows that if a smooth solution breaks down in finite time, then the time integral of the vorticity’s supremum norm must diverge, providing a sharp conditional criterion for regularity.
In this talk, I will present a new blow-up criterion in the spirit of the Beale–Kato–Majda theorem that emphasizes a different form of control. Instead of requiring bounds on the full vorticity, the criterion involves tangential derivatives of the velocity field, and shows that smooth solutions persist as long as these derivatives remain appropriately bounded in time. The result holds in a variety of settings, including the whole space, periodic domains, and domains with boundaries.

Sampling from the proper colorings of a graph using a number of colors linear in the maximum degree in expected linear time (Mark Huber, CMC)
2026-01-29 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: A proper coloring of a graph is an assignment of colors from ( {1, 2, ldots, k} ) to each node of a graph such that no two nodes connected by an edge receive the same color. Let ( Delta ) denote the maximum degree of the graph. If ( k geq Delta + 1 ) then at least one proper coloring always exists. However, counting the number of proper colorings of an arbitrary graph is a #P-complete problem, even when ( Delta = 3 ). This means finding a polynomial time exact algorithm is unlikely to be found. On the other hand, if a user can sample uniformly at random from the proper colorings of a graph, then it becomes possible to approximately count the number of proper colorings to arbitrary precision in polynomial time. This work presents the first algorithm that has an expected running time that is linear in the size of the graph under the condition that ( k > 3.637 Delta ). Joint work with Kritika Bhandari.

Fractional Brownian Motion: Small Increments and First Exit Time from One-sided Barrier (Qidi Peng, CGU)
2026-01-26 16:00 - 17:00 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: The talk introduces a conjecture on the first exit time of fractional Brownian motion: the upper-tail probability for a fractional Brownian motion to first exit a positive-valued barrier over time T has the exact asymptotic rate T^(H-1), where H is the Hurst parameter of the fractional Brownian motion. The talk tries to understand this conjecture by providing several equivalent statements. We then introduce the best effort made in the current literature towards solving this conjecture.

Structure-Aware Adaptive Nonconvex Optimization for Deep Learning and Scientific Computing (Minxin Zhang, UCLA)
2025-12-01 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: Modern machine learning and scientific computing pose optimization challenges of unprecedented scale and complexity, demanding fundamental advances in both theory and algorithmic design for nonconvex optimization. This talk presents recent advances that address these challenges by exploiting matrix and tensor structures, integrating adaptivity, and leveraging sampling techniques. In the first part, I introduce AdaGO, a new optimizer that combines orthogonalized momentum updates with adaptive learning rates. Building on the recent success of the Muon optimizer in large language model training, AdaGO incorporates an AdaGrad-type stepsize that scales orthogonalized update directions by accumulated past gradient norms. This design preserves the structural advantage of orthogonalized updates while adapting stepsizes to noise and the optimization landscape. We establish optimal convergence rates for smooth nonconvex functions and demonstrate improved performance over Muon and Adam on classification and regression tasks. The second part focuses on zeroth-order global optimization. We develop a theoretical framework for inexact proximal point (IPP) methods for global optimization, establishing convergence guarantees when proximal operators are estimated either deterministically or stochastically. The quadratic regularization in the proximal operator induces a concentrated Gibbs measure landscape that facilitates effective sampling. We propose two sampling-based algorithms: TT-IPP, which constructs a low-rank tensor-train (TT) approximation using a randomized TT-cross algorithm, and MC-IPP, which employs Monte Carlo integration. Both IPP algorithms adaptively balance efficiency and accuracy in proximal operator estimation, achieving strong performance across diverse benchmark functions and applications. Together, these works advance structure-aware adaptive first-order optimization for deep learning and zeroth-order global optimization in scientific computing.

A Signal Separation View of Classification (Ryan O'Dowd, CGU)
2025-11-17 16:15 - 17:15 | Estella 1021 (Emmy Noether Room), Pomona College, Claremont, CA, 91711, United States
Abstract: The problem of classification in machine learning has often been approached in terms of function approximation. In this talk, we propose an alternative approach for classification in arbitrary compact metric spaces which, in theory, yields both the number of classes, and a perfect classification using a minimal number of queried labels. Our approach uses localized trigonometric polynomial kernels initially developed for the point source signal separation problem in signal processing. Rather than point sources, we examine a convex combination of probability distributions representing the various classes from the machine learning classification problem. The localized kernel technique developed for separating point sources is then shown to separate the supports of these distributions. This is done in a hierarchical manner in our MASC algorithm to accommodate touching/overlapping class boundaries. The algorithm works in an active learning paradigm, deciding on points to query for their true class label and extending those labels to nearby points. We illustrate our theory on several simulated and real life data sets, including the Salinas and Indian Pines hyperspectral data sets and a document data set.

To Wait or Not to Wait? A Trade-off Between Population Externality and Signal Quality (Lan-Yi Liu, National Taiwan University)
2025-11-10 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: Transparency is vital for efficiency in social systems, yet individuals with critical information often strategically postpone disclosure, even when required, to benefit themselves.
To study this behavior, we introduce a multi-stage Chinese restaurant game with incomplete information that features system-recommended action rules and varying levels of player foresight. In our model, players initially receive a suggestion to join a queueing group based on their private signal, but can choose to switch groups. Following this, players sequentially select a final resource, balancing the desire to avoid congested externalities with the need to acquire more information.
We prove a closed-form solution for the players’ pure-strategy Nash equilibrium. Our key finding is that players with high-quality signals have no incentive to reveal their information to those with low-quality signals. This suggests that allowing players to strategically determine their decision timing, without further system design, leads to an inefficient equilibrium allocation.
Our results on congested externalities and system suggestions help explain the inherent trade-off between information quality and decision timing in various real-world scenarios, such as the challenges of vaccine distribution during a pandemic, the strategic crowding of factory location selection, and the decision-making faced by political candidates positioning themselves on the spectrum.

Convergence analysis of the Alternating Anderson-Picard method for nonlinear fixed-point problems (Xue Feng, UCLA)
2025-11-03 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: Anderson Acceleration (AA) has been widely used to solve nonlinear fixed-point problems due to its rapid convergence. This talk focuses on a variant of AA in which multiple Picard iterations are performed between each AA step, referred to as the Alternating Anderson-Picard (AAP) method. Despite introducing more `slow’ Picard iterations, this method has been demonstrated to be efficient and even more robust in both linear and nonlinear cases. However, there is a lack of theoretical analysis for AAP in the nonlinear context. In this work, we address this gap by establishing the equivalence between AAP and a multisecant-GMRES method that employs GMRES to solve a multisecant linear system at each iteration. From this perspective, we show that AAP actually “converges” the well-known Newton-GMRES method. These connections also help us understand the convergence behavior of AAP, especially the asymptotic convergence rate.

Estimating Shapley Values for Explainable AI via Richer Model Approximations (Teal Witter, CMC)
2025-10-27 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: Modern machine learning is ultimately a simple process: We iteratively update the weights of machine learning models to minimize a problem-specific loss. When it works well, we deploy the model in human-facing domains like healthcare, finance, or the justice system. But even though we know how models are trained, we don’t understand why they make decisions the decision they do. A particularly compelling approach to explaining AI predictions is the Shapley value, a game-theoretic quantity that measures how each input to the model affects its output. Mathematically, the ith Shapley value is the average change in the ith dimension of a particular function defined on the d-dimensional hypercube. Because the hypercube has 2^d points, exactly computing Shapley values is infeasible. In this talk, we will instead leverage algorithmic insights to develop state-of-the-art approximation methods.

Some New Advances in Similarity-Based Predictive Modeling (Joel A. Dubin, University of Waterloo)
2025-10-20 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: Earlier work has shown that similarity-based predictive models can improve upon predictive performance, as compared to using the entire training data to help build models, particular regarding model discrimination for binary responses. My collaborators and I have some updated results to share, regarding similarity-based modeling for joint consideration of model calibration and discrimination, as well as for dynamic prediction models. Properties of our methods will be investigated in comprehensive simulation studies, and we will demonstrate the methods through separate analyses of a publicly-available intensive care unit (ICU) database.

Modeling drug release for in vitro experiments (Minaya Villasana De Armas, Universidad Simon Bolivar)
2025-10-06 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: It is common to use adjuvants in immunotherapeutic regimens to strengthen the immune response. However, multiple dosages are required making it inconvenient for the patient. Hydrogels have been proposed as a vehicle to administer adjuvant and antigen in a sustained slow release thus reducing the need for re-administration.
In this instance, we use experimental data for stability studies on two different thermosensitive pentablock hydrogels as well as release of three adjuvants suspended in the hydrogels or in PLGA nanoparticles. The goal is to model this release and provide a framework by which these models can describe the various release profiles efficiently.

Bounds and Extremal Examples for the Hot Spots Ratio (Alex Hsu, University of Washington)
2025-09-29 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract: The shape of the fluctuations as heat approaches equilibrium in an insulated body are governed by the first Neumann eigenfunction of the Laplacian. Rauch’s hot spots conjecture states that the extrema of the first nontrivial Neumann Laplacian eigenfunction for a Lipschitz domain lies on the boundary. While this conjecture is false in general, its failure can be measured by the hot spots ratio, defined as the maximum over the entire domain divided by the maximum on the boundary. We determine the supremum of this quantity over all Lipschitz domains in every dimension $d$ and construct a sequence of sets for which the hot spots ratio approach this supremum. As $dto infty$, this maximal ratio converges to $sqrt{e}$, which matches the previously best known upper bounds.

LA City Council Reform: A Statistical Study of Alternatives (Evan Rosenman & Sarah Cannon, Claremont McKenna College)
2025-09-15 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Abstract:
The 2022 Los Angeles City Council scandal intensified public demand for governance reform, leading to the creation of the Los Angeles Charter Reform Commission. The commission is now considering proposals from civic and academic groups. Major recommendations include: eliminating the automatic election of candidates who win a primary majority, expanding the size of the City Council, and adopting alternative electoral systems such as multimember districts and ranked-choice voting.
This project offers a rigorous, data-driven evaluation of these proposals, focusing on their implications for proportionality, racial representation, and electoral responsiveness. We combine methods from Statistics and Computer Science, including Bayesian ethnicity imputation, ecological inference, and advanced graph-sampling algorithms to explore district boundaries. This hybrid approach provides new insights into Los Angeles’s political geography and the challenges of building a fair, representative City Council. By providing empirical evidence on the strengths and weaknesses of various districting systems, our work aims to inform policymaking and advance democratic representation in Los Angeles.

The Shooting Method in the Analysis of Two-Point Boundary-Value Problems (Adolfo J. Rumbos, Pomona College)
2025-09-08 16:15 - 17:15
Abstract:
Two-point boundary-value problems (BVPs) appear frequently in applied mathematics. When looking for solutions of boundary-value problems for some partial differential equations (PDEs) in mathematical physics, two-point BVPs come up as a result of applying the method of separation of variables, for instance. In the case of linear PDEs, the resulting two-point BVPs fall into a class of problems known as Sturm-Liouville eigenvalue problems.
This presentation deals with the use of the shooting method to prove existence of solutions of two-point BVPs. The shooting method is a numerical technique used to estimate solutions of two-point BVPs once a solution is known to exist. In this talk we illustrate how the shooting method can be used to prove existence of eigenvalues of linear Sturm-Liouville problems. We also show how the shooting method can be applied to prove existence and uniqueness of solutions for some nonlinear, two-point BVPs, and existence of eigenvalues for some nonlinear eigenvalue problems.
The presentation describes research conducted with collaborators Vaidehi Srinivasan (Pomona College class of 2027) and Gavin Zhao (Pomona College class of 2029) in the summer of 2025 with the support of the Summer Undergraduate Research Program at Pomona College.

Emer Lopera Arias (HMC)
2024-04-29 16:15 - 17:30 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: The fractional p-Laplacian operator. Motivation for its definition and related boundary value problems
Abstract:
Last decades, nonlocal operators, as the fractional Laplacian, have gained to much attention due to its applications to several physical Phenomena. In this talk we aim to motivate the definition of the fractional laplacian operator through a simple but quite illustrative example related to Long jump random walks. then, we will present a generalization to the fractional p-laplacian operator, p>1, and we will show a new result concerning the existence of solutions for a boundary value problem with this operator.

İrma Hacınlıyan (Istanbul Technical University)
2024-04-22 16:15 - 17:30 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: On Nonlinear Schrödinger Type Equations: Wave Modulation and Mathematical Analysis
Abstract: The nonlinear Schrödinger (NLS) equation describes the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear dispersive media. The NLS equation with soliton solutions is one of the significant and widely pursued research areas on the nonlinear wave motions. Sub-studies on such evolution equations include derivations of those equations; finding exact solutions and developing methods for these; studies of the analytic and geometric structure of the equations as integrable systems; singularities and collapses of solutions; analyzing modulation equations by dynamical systems methods; and related subjects. In this talk, nonlinear modulation of waves propagating in a generalized elastic solid according to different physical scenarios will be discussed. Properties of the solutions of NLS-type equations that are derived using the reductive perturbation method will be examined by mathematical analysis methods.

Ying Wang (University of Oklahoma)
2024-04-08 16:15 - 17:30 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: Numerical studies to the Chaplygin gas system and Pressureless Euler Equations
Speaker: Ying Wang
University of Oklahoma, USA
Abstract
In this talk, we will discuss the numerical solutions to the Riemann problem for Chaplygin gas equation, which is the Euler equations equipped with the state equation p = -1/rho, as well as the 2D zero-pressure Euler Equations. The spatial discretization is performed using WENO reconstruction together with a carefully defined limiter and time integration is achieved using TVD RK. The numerical results confirm high order of accuracy. This is a joint work with Ling Jin.

Chiu-Yen Kao (Claremont McKenna College)
2024-04-01 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: Geometric Optimization Involving Partial Differential Equations and Its Applications
Speaker: Professor Chiu-Yen Kao, Claremont McKenna College, Claremont, CA USA
Abstract: Optimal geometric design for energy functionals, which depends on solutions to partial differential equations, provides a vast number of interesting and challenging mathematical problems. One of the famous problems is that L. Rayleigh conjectured that the disk should minimize the fundamental frequency of vibration (the square root of the first Laplace-Dirichlet eigenvalue) among all shapes of equal area, more than a century ago. In this talk, we will share our recent findings related to geometric optimization for eigenvalue problems. Our approach involves the development of various computational methods to optimize Steklov eigenvalues and Laplace-Beltrami eigenvalues. Additionally, we explore applications of these methods.

Ami Radunskaya (Pomona College)
2024-03-25 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: Can a function tell us how immune cells kill?
Speaker: Prof. Ami Radunskaya (Pomona College, Claremont CA)
Abstract: The immune system is able to fight cancer by mustering and training an army of effector “killer” cells. Mathematical models of tumor-immune interactions must describe the proliferation, recruiting and killing rates of immune cells. Earlier work surprisingly showed that the functions describing the kill rates distinguish between two types of immune cells. The mechanisms behind these differences have been a mystery, however. In an attempt to unravel this mystery, we have created a cell-based fixed-lattice model that simulates immune cell and tumor cell interaction involving tumor recognition and two killing mechanisms. These mechanisms play a big role in the effectiveness of many cancer immunotherapies. Results from model simulations, along with theories developed by ecologists, can help to illuminate which mechanisms are at work in different conditions.

Sigifredo Herron (UNAL)
2024-03-18 16:15 - 17:15 | This event is virtual. Zoom link: https://pomonacollege.zoom.us/my/radzoom, Claremont, CA, 91711, United States

Gabe Chandler (Pomona College)
2024-03-04 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
Title: Graphical Anomaly Detection for High Dimensional and Object Data
Abstract: Anomaly detection is an important task in data analysis, though an agreed upon definition of what constitutes an outlier does not exist. Accordingly, a graphical tool that can highlight interesting observations in a data set that the scientist can then investigate with domain specific knowledge would be of value. The depth quantile function (DQF), a recently introduced feature map that takes data of arbitrary dimension to a function of a single variable while encoding certain geometric information, will provide such a tool. After introducing the DQF, we will discuss adaptations that make it particularly suited to the problem of anomaly detection, particularly the case where the non-anomalous data is living on a lower dimensional manifold in the data space. The DQF is also kernelizable, allowing applications to non-Euclidean data, as will be demonstrated via several examples.

Javier Gonzalez Anaya (Harvey Mudd College)
2024-02-26 16:15 - 17:15 | Emmy Noether Room, Estella 1021, Pomona College,, 610 N. College Ave., Claremont, CA, 91711, United States
This is the continuation of the semester’s joint seminar with the Universidad Nacional de Colombia-Manizales.
Title: Enumerating linearity regions of max-pooling layers in convolutional neural networks
Abstract: Convolutional neural networks (CNN’s) are central tools in the application of machine learning to text, audio and image processing. Their success stems from the ability of these networks to identify key features in complex datasets at a relatively low computational cost. Max-pooling layers (MPL’s) are key components of CNN’s that reduce the number of parameters used by the network while making it more robust to small changes in the input data. From a mathematical point of view, MPLs are piecewise-linear functions, and their number of linearity regions can be interpreted as a measure of complexity of the layer. In this talk I will explain how we can use combinatorial techniques to count these linearity regions, and survey our current results in the area.
