Upcoming MAMS Seminar Series
Spring 2025
4/22/2025, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Yitong Zheng (Case Western Reserve University)
Title:Magnitude and l_1-intrinsic volumes
Abstract: Magnitude is a numerical isometric invariant of metric spaces. We will discuss how to define the magnitude of finite metric spaces and compact positive definite spaces. Then I will introduce the l_1-intrinsic volumes which is analogous to the classical intrinsic volumes of a convex body; for instance, there is a Hadwiger-type theorem for l_1-intrinsic volumes. Finally, we will show that in l_1^n, magnitude can be used to recover l_1-intrinsic volumes of convex bodies.
4/15/2025, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Richard Huber’s (DTU, Denmark)
Title: The L2-Optimal Discretization of Tomographic Projection Operators
Abstract: Tomographic inverse problems are a cornerstone of medical investigations, allowing the visualization of patients’ interior features. While the infinite-dimensional operators modeling the measurement process (e.g., the Radon transform) are well understood, in practice one can only observe finitely many measurements and employ finitely many computations in reconstruction. Thus, proper discretization of these operators is crucial; principle criteria are the approximation quality (discretization error) and the computational complexity. For iterative reconstruction approaches, the computation of the forward operator and the adjoint (called the backprojection) is often a major driver of the methods’ computational complexity. Different discretization approaches show distinct strengths regarding the approximation quality of the forward- or backward projections, respectively, which commonly leads to the use of the ray-driven forward and the pixel-driven backprojection (creating a non-adjoint pair of operators). Using such unmatched projection pairs in iterative methods can be problematic, as theoretical convergence guarantees of many iterative methods are based on matched operators. We present a novel theoretical framework for designing an L2-optimal Galerkin discretization of the forward projection. Curiously, this optimized scheme is also the optimal discretization for the backprojection. In particular, this approach grants a matched discretization framework for which both the forward and backward discretization (being the optimal choices) converge in the strong operator topology, thus eliminating the need for unmatched operator pairs. In the parallel beam case, this optimal discretization is the well-known strip model for discretization, while in the fanbeam case, a novel weighted strip model is optimal.
4/8/2025, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Isaac Hubbard (Case Western Reserve University)
Title:M-Ellipsoids
Abstract: We prove the existence of M-ellipsoids, originally due to V. Milman: for any symmetric convex body K, there is an ellipsoid whose volume is comparable to that of K (and likewise for their respective polar duals) with a number of other useful properties, allowing it to act as a stand-in for K in proving certain volume inequalities. To illustrate, we prove reverse versions of the Brunn-Minkowski and Santaló inequalities.
4/1/2025, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Brandon Oliva (Case Western Reserve University)
Title: Quantitative Metric Density and the Hausdorff Metric
Abstract: Last semester, I discussed the Lebesgue Density Theorem, proved that there isn’t a Quantitative version of Lebesgue Density with a counterexample, and proved a “Quantitative Metric Density Theorem” on doubling metric measure spaces. For this talk, I will be providing an application of this theorem using the Hausdorff distance to show that if one moves a closed ball, there won’t be many large gaps that the balls has to skip over.
The references I will be using are my Masters’ thesis, and “Notes on Pointed Gromov-Hausdorff Convergence” by Jansen.
3/25/2025, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Diliya Yalikun (Case Western Reserve University)
Title: Ball and spindle convexity with respect to a convex body
Abstract: Let C be a convex body, we introduce two notions of convexity relative to C. We will look at how some fundamental properties of classical convex sets can be applied to the C-ball convex sets and C-spindle convex sets.
3/18/2025, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Effrosyni Chasioti (Case Western Reserve University)
Title: Dvoretszky’s Theorem
Abstract: We will state and present the proof of Dvoretszky’s Theorem. This theorem is one of the influential and fundamental results in convex geometry and the local theory of Banach spaces. It has been proved first by Dvoretszky in the 1961 and later was proven by V. Milman in 1971 using the concentration of measure phenomenon and connected this theorem with the local theory of Banach spaces in a more direct and revolutionary way.
This theorem basically says that every sufficiently high dimensional normed vector space has low dimensional subspaces that are approximately Euclidean, or in the language of convex geometry , that every high dimensional compact symmetric convex subset of $\mathbb{R}^N$ has a low dimensional sections that resemble ellipsoids.
3/4/2025, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Mengchun Cai (Case Western Reserve University)
Title: Convergence Rate for some Classical Ensembles
Abstract: In this talk, I will introduce some classical ensembles in the random matrix theory and discuss their convergence rates to their limiting point processes with respect to the W_1 distance.
2/25/2025, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Fabio Isaza (Case Western Reserve University)
Title: Approximating Continuous Functions with Neural Networks
Abstract: We will discuss the formal set up of neural networks as layered compositions of fixed non-linear and varying affine operators, the classical results by Hornik, Stinchcombe, and White; and more recent results by Boris Hanin and Mark Sellke connecting the NN functional approximations to approximation by free-node splines and more. We will also explore the theoretical underpinnings of Rectified Linear Unit (ReLU) NNs as universal approximators of continuous functions and discuss practical considerations in their implementation for function approximation tasks.
2/11/2025, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Michael Roysdon (Case Western Reserve University)
Title: Comparison problems for Radon transforms
Abstract: In 1956 Herbert Busemann and Clinton M. Petty created a list of 10 problems in Convex Geometry, among which only the first had been solved fully in 1999. A key feature of the solution to the first problem is that it (and in fact, the majority of the Busemann-Petty problems) can be reformulated into the language of Harmonic Analysis. Inspired by the Busemann-Petty problems and their connection to Harmonic Analysis, we consider the following natural question for various Radon transforms: Let p>1. Given a pair of nonnegative, even and continuous functions f,g such that the Radon transform of f is pointwise smaller than the Radon transform of g, is it necessarily true that the L^p-norm of f is smaller than the L^p-norm of g? As it turns out, this simple question has a deep connection to the Busemann-Petty problem and the slicing problem of Bourgain. As a consequence of our investigation, we show that this implies reverse Oberlin-Stein type estimates for the spherical Radon transform when p >1; this is complementary to a recent work of Johnathan Bennett and Terence Tao in which similar reverse estimates were proven in the case 0 < p <= 1. We will discuss a similar problem for the dual Radon transform.
2/4/2025, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Matthew Wascher (Case Western Reserve University)
Title: The effects of individual-level behavioral responses on SIS epidemic persistence
Abstract: The contact process (SIS epidemic model) has long been studied as a model for the spread of infectious disease through a population. One important question concerns the long-term behavior of the epidemic–does it result in a large outbreak, or does the infection die out quickly? There is a large literature on the effects of the underlying population structure on this long-term behavior. However, the role of individual-level behavioral responses on the epidemic is less studied. In this talk, I will introduce the contact process, some key ideas used to analyze it, and a few notable results. I will then discuss my recent work on modified versions of the contact process that include individual-level behavioral responses to the epidemic. I will present some results on how individual-level behavioral responses can influence the long-term behavior of an epidemic and discuss why analyzing these models is mathematically challenging.
Spring 2025
4/23/2025, Wed. 4-5 pm in Sears 439
Speaker: Professor David Gurarie (CWRU)
Title: Immune driven evolution in host-parasite systems and agent-based modeling of malaria
Abstract: Parasite evolution in response to challenges can result in outbreaks and pandemics (Influenza, Covid, Measles), drug resistance (failure of treatment), persistent spread in endemic areas. Several factors can affect parasite evolution and selection in such systems, but foremost are host immunity and transmission environment. The talk will outline some basic questions and concepts of evolutionary biology in host-parasite systems, and modeling approaches. We specifically highlight the case of malaria, and propose an agent-based model (ABM) which takes detailed account of in-host biology, genetically structured parasite makeup, and mosquito vector. Our ABM allows us to simulate and explore a wide range of questions related to parasite spread in host communities, its long-term evolution, and the effect of chemotherapy. Further development will leverage ABM with malaria therapy data and machine learning, to design an effective computational framework for malaria simulation on different scales. We shall demonstrate some preliminary work along these lines.
3/5/2025, Wed. 4-5 pm in Sears 439
Speaker: Yiming Ying (University of Sydney)
Title: Optimal Rates for Generalization of Gradient Descent Methods with ReLU Neural Networks
Abstract: The neural tangent kernel (NTK) perspective establishes a fundamental connection between neural networks (NNs) trained by gradient descent (GD) and kernel methods. A key question arising from this insight is whether GD-based training of NNs can achieve optimal statistical rates comparable to kernel methods.
In this talk, I will present our recent work addressing this question, offering new insights into the theoretical guarantees of gradient-based deep learning models. For binary classification, we show that gradient descent (GD) applied to training two-layer ReLU neural networks with the logistic loss can achieve the optimal margin bound, provided the data is NTK-separable. Our analysis highlights the intricate interplay between optimization and generalization, leveraging a reference model and a refined estimation of Rademacher complexity. For least-squares regression, we demonstrate that both GD and SGD with two-layer ReLU NNs can attain optimal minimax rates under polynomial overparameterization. Finally, I will discuss how these results extend to deep ReLU networks in the regression setting.
2/26/2025, Wed. 4-5 pm in Sears 439
Speaker: Anuj Abhishek (Case Western Reserve University)
Title: A probabilistic interpretation of Landweber iteration for a linear ill-posed inverse problem
Abstract: The contents of this talk are still at a very early proof-of-concepts stage. We will first describe the so-called Gaussian-beam Radon transform that is used to model the observed data in many optical tomography applications, such as optical coherence tomography (OCT), which use optical-beams instead of X-rays. X-rays can be modeled as traveling along straight lines, and as such, the model for the observed data in CT imaging is the Radon transform. However, monochromatic laser beams do not have a uniform shape intensity like X-rays. Since a Gaussian beam describes the characteristics of an electromagnetic beam, such as a laser: in many laser optics applications, the laser beam is assumed to be a Gaussian beam with a Gaussian intensity profile. It has been recently shown that in OCT, when a reconstruction algorithm does not account for the Gaussian beam (and instead the data is modeled by the usual Radon transform), the reconstruction quality greatly suffers. In this project, we model the observed data in OCT by a Gaussian-beam Radon transform and propose to use a Landweber type iterative method for reconstruction. In the second part of the talk, we present a probabilistic perspective on this iterative method. In this analysis, we will consider both the input (starting guess) as well as the output of the iterative method to be probability distributions. Thus each iteration of the method will be thought of as ‘pushforwarding’ a probability distribution. At the conclusion of the iterative method, we will use the mean of the output distribution as our estimate of the solution and use the output distribution to quantify the uncertainty in the estimate. We will also give some preliminary justification of the validity of the proposed approach. This is joint work with Souvik Roy (UT Arlington) and Anwesa Dey (University of Utah).
2/5/2025, Wed. 4-5 pm in Sears 439
Speaker: Professor Sakshi Arya (CWRU)
Title: Semi-Parametric Batched Global Multi-Armed Bandits with Covariates
Abstract: The multi-armed bandits (MAB) framework is a widely used approach for sequential decision-making, where a decision-maker selects an arm in each round with the goal of maximizing long-term rewards. Moreover, in many practical applications, such as personalized medicine and recommendation systems, feedback is delayed and provided in batches, contextual information is available at the time of decision-making, and rewards from different arms are related rather than independent.
We propose a novel semi-parametric framework for batched bandits with covariates and a shared parameter across arms, leveraging the single-index regression (SIR) model to capture relationships between arm rewards while balancing interpretability and flexibility. Our algorithm, Batched single-Index Dynamic binning and Successive arm elimination (BIDS), employs a batched successive arm elimination strategy with a dynamic binning mechanism guided by the single-index direction. We consider two settings: one where a pilot direction is available and another where the direction is estimated from data, deriving theoretical regret bounds for both cases. When a pilot direction is available with sufficient accuracy, our approach achieves minimax-optimal rates (with $d = 1$) for nonparametric batched bandits, circumventing the curse of dimensionality. Extensive experiments on simulated and real-world datasets demonstrate the effectiveness of our algorithm compared to the nonparametric batched bandit method.