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Disease progression can be tracked via a cascade of changes in biomarkers and clinical measurements over the disease time course. For example, in progressive neurodegenerative diseases (ND), such as Alzheimer's Disease, changes in biomarkers (neuroanatomical images, cerebrospinal fluid) may precede clinical measurements (cognitive batteries) by months or years. Viewing repeated measurements of biomarkers and clinical measurements as a multivariate time series composed of continuous and discrete values, successful modeling of disease progression balances capturing stereotypic patterns in disease progression across subjects with subject-level variability in timing, acceleration, and shape of disease progression trajectories. A Bayesian model combining parametric and constrained semi-parametric disease progression models is proposed for curve alignment and covariate-dependent smoothing of exponential family outcomes across the disease time course, allowing for the characterization of typical disease progression as well as heterogeneity in the timing, speed, ordering, and shape of disease progression at the population-level and at the subject-level via random effects structure that partitions phase and amplitude variance. The framework will accommodate continuous and count outcomes, allowing for the incorporation of measurements ranging from neuroimaging features to sensitive sub-scales of cognitive batteries. A working example is provided from patients experiencing progressive ND.

TBA

Abstract: There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimension, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton.

This is joint work with S. Cao. 

In this talk, I will present a Julia package, HypersurfaceRegions.jl, for computing all connected components in the complement of an arrangement of real algebraic hypersurfaces in R^n. The package is based on a modified implementation of the algorithm from the paper "Smooth Connectivity in Real Algebraic Varieties" by Cummings et al. I will outline the theory behind the algorithm and our implementation. I will demonstrate the use of the package through various examples.

Modern machine learning has shown remarkable promise in multiple applications. However, brute force use of neural networks, even when they have huge numbers of trainable parameters, can fail to provide highly accurate predictions for problems in the physical sciences. We present a collection of ideas about how enforcing physics, exploiting multifidelity knowledge and the kernel representation of neural networks can lead to significant increase in efficiency and/or accuracy. Various examples are used to illustrate the ideas.

Dissertation Defense: Bartu Bingol

Ph.D. Candidate, Mathematics

Dissertation Title: Obstructed deformation problems and (mod I) congruences between elliptic curves

Advisor: Tom Weston

TBD

TBD

Algebraic matroids and secant varieties.