Project Description:<\/b> How do deterministic physical systems give rise to probabilities? In classical probability theory, one can show that some differential equations governing the dynamical evolution of physical systems lead to \u201cuniversal behavior\u201d in the limit of large times. More specifically, the system almost always approaches the same final distribution for certain physical variables, independent of the distribution of initial conditions one began with. This is called the classical method of arbitrary functions\u2014where the arbitrary functions refer to the possible initial conditions, which turn out to be irrelevant to the long time behavior. Such results are thought to explain why physical systems display the probabilities they do, e.g., why a tossed coin lands heads with probability 1\/2 or a roulette wheel has equal probability of landing on each number. In this project, we aim to extend the method of arbitrary functions to systems described by quantum physics, where both the dynamical equations and conceptual framework for probability have novel features. We will explore the behavior for concrete physical models in the limit of large times and the classical limit of small values of Planck\u2019s constant. <\/p>\n
\nProject Level:<\/b> Advanced: students who have taken multiple upper-level mathematics courses
\nAdditional Course Requirements:<\/b> Some background in linear algebra, analysis, and abstract algebra will be helpful for this project. No background in physics is required.
\nProgramming Requirements:<\/b> <\/p>\n
\n Ballistic aggregation <\/h2>\n
Faculty Mentor:<\/b> Krzysztof Burdzy <\/h4>\n
Project Description:<\/b> Ballistic aggregation is a random accumulation model. Typical clusters of discs have three or four arms. The project will focus on four-armed clusters because they seem to be very stable. The goal of the project is to collect statistics from simulations that could support the claim of stability. The project will involve programing (possibly using AI), statistics, and elementary probability. <\/p>\n
\nProject Level:<\/b> Intermediate: students who have taken Math 300
\nAdditional Course Requirements:<\/b> Undergraduate probability and statistics classes would be helpful but are not necessary
\nProgramming Requirements:<\/b> basic programing in any language; AI programing is OK <\/p>\n
\n Billiards, Computation, and Music <\/h2>\n
Faculty Mentor:<\/b> Jayadev Athreya <\/h4>\n
Project Description:<\/b> We will explore the language of billiards in regular polygons: a point mass, moving at unit speed, with no friction, and bouncing off walls with angle of incidence = angle of reflection. The associated language is given by labeling all the sides, and considering the set of all possible codings of trajectories. We will explore this with coding and music, by assigning musical notes to the sides, and seeing what musical phrases we can get. <\/p>\n
\nProject Level:<\/b> Advanced: students who have taken multiple upper-level mathematics courses
\nAdditional Course Requirements:<\/b> algebra
\nProgramming Requirements:<\/b> python <\/p>\n
\n Calculus Visuals – Interactive modeling with Desmos, 3D prints, and other tools <\/h2>\n
Faculty Mentor:<\/b> Andy Loveless <\/h4>\n
Project Description:<\/b> Each student would complete 3\u20134 short \u201cvisual vignettes\u201d (or 1\u20132 larger ones) over the course of the term. The workflow would be…
\nThey choose a topic (from my very long list). Topics will range from 100\u2013200 level UW courses (not just math) to nature, motion, or physics.
\n– Create a 1\u20132 page write-up with context and references
\n– Link to an interactive Desmos (or similar) model via QR code
\n– Possibly include a 3D print if appropriate
\n– Summarize discoveries and further questions
\nWe would display results on a “gallery” page on my website.
\nSet out copies of reports and prints during the end of term poster session. <\/p>\n
\nProject Level:<\/b> Open: students who have taken Math 126
\nAdditional Course Requirements:<\/b>
\nProgramming Requirements:<\/b> Some desmos or geogebra experience would be nice <\/p>\n
\n Number Theory and Noise <\/h2>\n
Faculty Mentor:<\/b> Matthew Conroy <\/h4>\n
Project Description:<\/b> Students will learn some elementary number theory (and other mathematics) via the creation of audio representations of integer sequences. The project page is here: https:\/\/sites.math.washington.edu\/~conroy\/WXML\/integerSequenceNoise\/project.htm Students should be open to listening carefully to noisy sounds and doing a little programming (zero experience required or expected!). By listening to the sounds and asking questions about phenomena heard therein, we will investigate number theoretic topics (and topics related to hearing, digital audio, fourier analysis, etc.) Students will contribute to the mathematical community by adding to a library of sounds publicly available to other researchers. <\/p>\n
\nProject Level:<\/b> Open: students who have taken Math 126
\nAdditional Course Requirements:<\/b>
\nProgramming Requirements:<\/b> None, but some python or PARI\/GP programming will happen. Students will be given much assistance and guidance in this. No previous knowledge necessary. <\/p>\n
\n The Mathematics of Medical Imaging <\/h2>\n
Faculty Mentor:<\/b> Larry Pierce <\/h4>\n
Project Description:<\/b> Students will use matlab or python to model photon transport from a radioactive object and virtual detectors to detect those photons. Finally, students will solve the inverse problem of creating an image from the photons detected. <\/p>\n
\nProject Level:<\/b> Open: students who have taken Math 126
\nAdditional Course Requirements:<\/b> There are close ties to Math 208, but not necessary to have taken it
\nProgramming Requirements:<\/b> python or matlab ; preferably all students would program in the same language <\/p>\n
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Quantum Probability via Arbitrary Functions Faculty Mentor: Dr. Benjamin Feintzeig Project Description: How do deterministic physical systems give rise to probabilities? In classical probability theory, one can show that some differential equations governing the dynamical evolution of physical systems lead to \u201cuniversal behavior\u201d in the limit of large times. More specifically, the system almost always…<\/p>\n