Colloquium

How do we turn million voxel brain images and tens of thousands of genes into a handful of features that still capture what matters? This talk begins with an application-first tour of two threads in my recent work.

I'll start neuroimaging with a shift robust tensor model that learns small spatial misalignments while it fits, producing crisp, interpretable coefficient maps that highlight clinically meaningful regions. I'll also show a Bayesian SKPD extension for mixed type outcomes that brings principled uncertainty to both predictions and effect maps. As a brief detour, I'll touch on a likelihood based reconstruction method for positronium lifetime imaging—a PET technique that extracts microstructure sensitive contrast from timing data (here "lifetime" refers to particle physics, not patient survival).

We'll then pivot to high /ultrahigh dimensional cancer genomics with Fréchet distance covariance sufficient dimension reduction (Fréchet dCov SDR). This model free approach distills thousands of predictors into a few low dimensional indices that preserve complex, possibly nonlinear dependence with outcomes—including distribution valued responses such as survival profiles. I'll show how it plugs into standard classifiers and regressors, scales to large feature sets, and leads to visual summaries that make downstream decisions easier.

Active Learning techniques (IBL, POGIL, …) rely on teamwork to promote student learning. Far too often, assessments don’t follow suit and instead ask students to perform computational tasks without assessing whether or not students can solve problems, apply their knowledge to new settings, or think creatively. This talk provides a detailed examination of how the University of Minnesota has revised its precalculus sequence to incorporate communication skills as a significant component of the grading scheme.

A Formal Bayesian Approach to Enhance fMRI k-Space Measurements Improves Reconstructed Image Quality
John Bodenschatz

Hypothesis Testing for High Dimensional Problems—Application on Age and Cancer Related Circular RNAs
Edward Liu

Hybrid of Functional and Vector PCA
Soroush Mahmoudiandehkordi

Data Assimilation Methods for High Dimensional and Non-Linear Systems
Sylvester Mensah

Bayesian Spatiotemporal Modeling of Syphilis in Wisconsin
Navid Mohseni

Fast Optimization of Implicit Neural Representations for 3D CT Reconstruction
Mahrokh Najaf

Regularized Multivariate Two-way Functional Principal Component Analysis
Mobina Pourmoshir

VPPE: Application of Scaled Vecchia Approximations to Parallel Partial Emulation
Josh Seidman

Spatially Varying Coefficient Regression with Spike-and-Slab Group Lasso
Qishi Zhan

In the 1960s, mathematician Ján Černý suggested that the length of the shortest reset word for a synchronizing automaton with n states is no more than (n-1)². This idea, now known as the Černý Conjecture, has been an interesting open problem in automata theory for over six decades. In this talk, we will become acquainted with synchronizing automata, explore theoretical and computational efforts to validate or invalidate the Černý Conjecture, and survey applications of synchronizing automata. You may even see a magic trick!

A sequence of operators \(T_n : H(\mathbb{C}) \to H(\mathbb{C})\) on the space \(H(\mathbb{C})\) of entire functions is said to be universal if there exists a vector \(f \in H(\mathbb{C})\) such that \(\{T_n f : n \ge 0\}\) is dense in \(H(\mathbb{C})\). Chan, Hofstad, and Walmsley showed that every non-scalar linear operator commuting with differentiation admits a hypercyclic vector \(f(z)\) of the form of an infinite product of linear factors: \[f(z) = \prod_{j=1}^{\infty} \left( 1 - \frac{z}{d_j} \right) \quad \text{where} \quad d_j \in \mathbb{C} \setminus \{0\}.\] However, it remains open whether there exists a universal vector of the above form whenever the sequence of automorphic composition operators \(C_{\sigma_n} : H(\mathbb{C}) \to H(\mathbb{C})\) is universal, where the automorphisms \(\sigma_n(z) = a_n z + b_n\) with \(a_n, b_n \in \mathbb{C}\) and \(a_n \neq 0\). In this talk, we provide two sufficient conditions on the sequences \(a_n\) and \(b_n\) that ensure the existence of a universal vector of the above form for the composition operators \(C_{\sigma_n} : H(\mathbb{C}) \to H(\mathbb{C})\).

We will consider Bayesian Decision Theoretic methodologies for multiple hypothesis testing when there is skewness in the alternative hypotheses. An application of this for gene expression data will be considered to detect targeted genes regulated by microRNAs.

In this talk, we explore the Electrical Impedance Tomography (EIT) inverse problem. EIT is an emerging non-invasive medical imaging modality that is relatively low-cost and portable with many promising applications, including breast cancer detection. In EIT, known harmless currents are injected through electrodes placed on the skin and the resulting voltages are measured at the electrodes and used to recover the internal conductivity. The inverse problem is severely ill-posed and reconstructed images are generally low resolution, in part due to the diffusive nature of the electrical current. In this talk, we focus on tackling the problem from three directions. First, we seek to determine whether machine learning tools can be used for breast tissue classification from measurement data alone, thus bypassing the image reconstruction task. Next, we tackle image reconstruction. The image reconstruction problem is generally optimized for either speed or accuracy, and increasing the speed generally requires making linearizing assumptions. In the second avenue, we seek to harness the speed of a certain class of reconstruction algorithms involving Complex Geometrical Optics (CGO)-based methods and determine whether further increasing the non-linearity of existing methods results in higher resolution reconstructions on simulated and experimental data. Finally, in the third approach, we investigate embedding task-specific a priori information into the CGO-based reconstruction algorithms to improve spatial resolution.

With the support of our Graduate Student TAs, I present a broad variety of mathematical puzzles, some with logical content. Their solutions require precise and correct reasoning, in line with what is commonly known as pure mathematics.

UK law demands that we are Net Zero by 2050 and that part of the delivery to that requires 500,000 hectares of new woodland to be planted, an area covering 2.5% of the UK. But where can we plant them, what species should we plant and when do we put them in the ground? Moreover, how can policy be used to incentivize private landowners, those that own more than 70% of the nation's land, to forego existing land use for tree planting? In this talk, I present the work of the ADD-TREES, a project funded under the AI for Net Zero program that seeks to offer interactive decision support to landowners and policymakers. I will advocate for an ethical approach to decision support that avoids simply treating net zero as another multi-objective optimization problem and rather seeks to engage users with net zero compatible decision spaces. I will show some of our key innovations, including field-level climate downscaling, linked deep Gaussian process modelling for networks, efficient uniform sampling for tiny subspaces and clustering of planting strategies with preference weighted stochastic neighborhood embeddings.

This talk won't answer the question in the title because of two reasons: first, because
the question is very vague, and second, because the question is very hard. Instead, I will
take you on a 100-year journey of exploration from the perspective of computability theory.

This point of view begins with the development of a mathematical definition of algorithm, which provides a clear framework: simple means computable, and complicated means incomputable. The story immediately deepens with the question, how incomputable? — leading us, after many twists and turns, to one of the central open problems in computability theory: Martin's conjecture. Along the way, we’ll see how philosophical considerations and technical definitions interact to deepen our understanding.

We developed a Bayesian multimodal model to detect biomarkers (or neuromarkers) using resting-state functional and structural data while comparing a late-life depression group with a healthy control group. Biomarker detection helps determine a target for treatment intervention to get the optimal therapeutic benefit for treatment-resistant patients. The borrowing strength of the structural connectivity has been quantified for functional activity while detecting the biomarkers. In the biomarker searching process, thousands of hypotheses are generated and tested simultaneously using our novel method to control the false discovery rate for small samples. Several existing statistical approaches, frequently used in analyzing neuroimaging data have been investigated and compared via simulation with the proposed approach to show its excellent performance. Results are illustrated with a live data set generated in a late-life depression study. The role of detected biomarkers in terms of cognitive function has been explored.

Large-scale composite null hypothesis testing is a commonly utilized technique in modern genetics research. Loosely speaking, a genetic composite null hypothesis occurs when attempting to determine whether all null hypotheses in a set of individual tests should simultaneously be rejected. For example, in a lung cancer replication study of two independent cohorts, finding a replicated effect means that a mutation is associated with lung cancer in both - not just at least one - of the two datasets. Other common composite null study designs include mediation analysis and pleiotropy analysis. This talk will first give a brief overview of challenges in the aforementioned settings and show how genetics topics can be mapped to statistical language. We then introduce the conditionally symmetric multidimensional Gaussian mixture model (csmGmm), an empirical Bayes approach to composite null testing with key interpretability advantages. For example, the csmGmm provably harmonizes frequentist and Bayesian significance rankings in composite null settings. Such a feature is important in the p-value dominated genetics literature. We demonstrate application of the csmGmm on a collection of translational lung cancer genetic association studies that motivated this work.

The importance of timely decisions in smart grid has led to the integration of intelligent networks, which require advanced machine learning, and optimization tools to learn, infer and control their operation. Due to stochastic nature of load and renewables, the renewable generation output or the load power may not be accurately acquired. In this talk, I will discuss modeling generation, load uncertainty using chance constraints, and propose efficient and distributed algorithms for energy storage sizing.