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DTSTART;TZID=America/Los_Angeles:20260914T161500
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UID:10002252-1789402500-1789406100@colleges.claremont.edu
SUMMARY:Adaptive Experimental Design using Shrinkage Estimators (Evan Rosenman\, Claremont McKenna College)
DESCRIPTION:In multi-armed trials\, adaptive designs are a popular way to increase estimation efficiency or identify optimal treatments. Many recent papers have proposed adaptive methods for minimizing the error of unbiased estimators that consider each treatment arm in isolation. This approach may be inefficient\, because it fails to borrow shared information across the treatment arms.\nWe consider adaptivity in a sequential trial with K active treatments and a control\, and suggest the use of Stein-like shrinkage estimators to obtain the final causal estimates. These estimators share information\, yielding reductions in expected estimation error\, relative to estimating each causal effect in isolation. Moreover\, for a group of candidate estimators\, the expected error can be expressed as the expectation of ratios of Gaussian quadratic forms\, and can thus be computed efficiently. Hence\, we suggest a simple algorithm for sequential adaptivity: assign each new arrival to the arm that will minimize the estimated shrinker loss.\nThrough simulations\, we demonstrate that this approach reduces estimation error\, especially in the low signal-to-noise regime. We also characterize how our adaptive algorithm assigns treatments differently than would a classical Neyman allocation and suggest a method for constructing shorter confidence intervals at the conclusion of the trial.
URL:https://colleges.claremont.edu/ccms/event/adaptive-experimental-design-using-shrinkage-estimators-evan-rosenman-claremont-mckenna-college/
LOCATION:Emmy Noether Room\, Estella 1021\, Pomona College\,\, 610 N. College Ave.\, Claremont\, CA\, 91711\, United States
CATEGORIES:Applied Math Seminar
ORGANIZER;CN="Ami Radunskaya":MAILTO:aradunskaya@pomona.edu
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DTSTART;TZID=America/Los_Angeles:20260921T161500
DTEND;TZID=America/Los_Angeles:20260921T171500
DTSTAMP:20260913T231319Z
CREATED:20260913T231319Z
LAST-MODIFIED:20260913T231319Z
UID:10002329-1790007300-1790010900@colleges.claremont.edu
SUMMARY:Number Theory meets Differential Equations: Using Homotopy Continuation to Compute Shabat Polynomials
DESCRIPTION:A Belyĭ pair \(\bigl( S\,\\, z(w) \bigr)\) is a compact\, connected Riemann surface \(S\) together with a rational function \(z \colon S \to \mathbb{P}^1(\mathbb{C})\) which has at most three critical values \(\bigl\{ 0\,\\, 1\,\\, \infty \bigr\}\). The preimage \(z^{-1}\bigl( [0\,1] \bigr) \subseteq S \hookrightarrow \mathbb{R}^3\) is a bipartite graph called a Dessin d’Enfant. When \(S = \mathbb{P}^1(\mathbb{C}) \simeq S^2(\mathbb{R})\) is the Riemann sphere\, a tree—namely\, a planar graph with one face—can be realized as the Dessin d’Enfant of a polynomial. Named in honor of Georgii Borisovich Shabat\, a Belyĭ map \(z \colon S \to \mathbb{P}^1(\mathbb{C})\) which is also a polynomial is called a Shabat polynomial. \nIn this project\, we seek to compute more examples of Shabat polynomials. We introduce a trick which reduces the computation of such polynomials to finding solutions to a system of polynomial equations. We explore computing these solutions via two methods: (1) Newton’s method\, to find approximate values\, and (2) the homotopy continuation method\, to deform known solutions into desired yet unknown ones. Simply put\, in this talk we give an introduction to using differential equations to answer questions in number theory. \nThis project was completed through the Research Experience for Undergraduate Faculty (REUF) at AIM/ICERM in Summers 2025 and 2026 (DMS‑2015462).
URL:https://colleges.claremont.edu/ccms/event/number-theory-meets-differential-equations-using-homotopy-continuation-to-compute-shabat-polynomials/
LOCATION:Emmy Noether Room\, Estella 1021\, Pomona College\,\, 610 N. College Ave.\, Claremont\, CA\, 91711\, United States
CATEGORIES:Applied Math Seminar
ORGANIZER;CN="Ami Radunskaya":MAILTO:aradunskaya@pomona.edu
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DTSTART;TZID=America/Los_Angeles:20260928T161500
DTEND;TZID=America/Los_Angeles:20260928T171500
DTSTAMP:20260913T232258Z
CREATED:20260913T232258Z
LAST-MODIFIED:20260913T232258Z
UID:10002330-1790612100-1790615700@colleges.claremont.edu
SUMMARY:Brain Imaging and Mathematical Modeling of Alzheimer's Disease (Peter Hinow\, U. Wisconsin\, Milwaukee)
DESCRIPTION:A wide variety of medical imaging methods exist for research and diagnostic procedures on the Central Nervous System (CNS)\, among them Magnetic Resonance Imaging (MRI) and Optical Imaging. Mathematical modeling and simulation contribute significantly to an improved understanding of image data and the intricate transport\, diffusion\, and metabolic processes in the brain. This is of particular importance in the study of pathologies such as Alzheimer’s Disease\, which\, 125 years after its description\, remains poorly understood and for which only moderate success has been made in its treatment. In the first work to be presented\, we investigate tracer diffusion and transport in the glymphatic system by combining optical imaging in rats with a partial differential equation of reaction-advection-diffusion type. In a second work\, we collect observations on the accumulation of amyloid-β\, changes in neuronal density\, and a decline in cognitive performance in TgF344-AD and wild-type rats. We develop a compartmental ordinary differential equation model and determine its parameters by fitting the output to the experimental observations. In the long run\, our mathematical modeling effort is intended to bridge AD research in rodent models and the human condition of AD. This is joint work with Dr. Muna Aryal (North Carolina Agricultural and Technical State University) and Micah Hesketh\, MS (graduate student\, UWM).
URL:https://colleges.claremont.edu/ccms/event/brain-imaging-and-mathematical-modeling-of-alzheimers-disease-peter-hinow-u-wisconsin-milwaukee-2/
LOCATION:Emmy Noether Room\, Estella 1021\, Pomona College\,\, 610 N. College Ave.\, Claremont\, CA\, 91711\, United States
CATEGORIES:Applied Math Seminar
ORGANIZER;CN="Ami Radunskaya":MAILTO:aradunskaya@pomona.edu
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