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`2021-05-14 15:00:00``2021-05-14 16:30:00``Maseeh Mathematics + Statistics Colloquium``Friday, May 14th, 3:00PM Register to Join Zoom Meeting https://pdx.zoom.us/meeting/register/tZcld-mhpzMtGN3MqS7-UE9Lns2BoLnB3hVy Ellen Eischen, PhD Department of Mathematics University of Oregon Title: An Introduction to the Bernoulli Numbers, from Pythagoras to Present Abstract: Consider these basic questions: What can we say about finite sums of powers of consecutive whole numbers? What can we say about whole number solutions to polynomial equations? What about factorizations into primes? What about values of the Riemann zeta function? In interesting families of examples — elementary and sophisticated, ancient and modern — "Bernoulli numbers" unify these seemingly unrelated questions. After an introduction to the Bernoulli numbers, we will explore related developments for these intertwined problems, which lead to central challenges in number theory and beyond. This talk will be appropriate for undergraduates, graduate students, and faculty in all areas of math.``Register to Join Zoom Meeting https://pdx.zoom.us/meeting/register/tZcld-mhpzMtGN3MqS7-UE9Lns2BoLnB3``Kathie Leck, Fariborz Maseeh Department of Mathematics + Statistics``Kathie Leck, Fariborz Maseeh Department of Mathematics + Statistics``America/Los_Angeles``public`Register to Join Zoom Meeting https://pdx.zoom.us/meeting/register/tZcld-mhpzMtGN3MqS7-UE9Lns2BoLnB3

Kathie Leck, Fariborz Maseeh Department of Mathematics + Statistics

**Friday, May 14th, 3:00PM**

Register to Join Zoom Meeting https://pdx.zoom.us/meeting/register/tZcld-mhpzMtGN3MqS7-UE9Lns2BoLnB3hVy

**Ellen Eischen**, PhD

Department of Mathematics

University of Oregon

**Title**: An Introduction to the Bernoulli Numbers, from Pythagoras to Present

**Abstract**: Consider these basic questions: What can we say about finite sums of powers of consecutive whole numbers? What can we say about whole number solutions to polynomial equations? What about factorizations into primes? What about values of the Riemann zeta function? In interesting families of examples — elementary and sophisticated, ancient and modern — "Bernoulli numbers" unify these seemingly unrelated questions. After an introduction to the Bernoulli numbers, we will explore related developments for these intertwined problems, which lead to central challenges in number theory and beyond. This talk will be appropriate for undergraduates, graduate students, and faculty in all areas of math.

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### Tatha Goswami Ph.D. Mathematical Sciences Dissertation Defense

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`2021-05-14 09:00:00``2021-05-14 11:15:00``Tatha Goswami Ph.D. Mathematical Sciences Dissertation Defense``Ph.D. in Mathematical Sciences Dissertation Defense Tatha Goswami Numerical techniques and simulations for studying various high power optical fiber amplifiers MS Mathematics, Indian Institute of Technology, Bombay, 2014 BS Mathematics, St. Xavier's College Kolkata, 2012 Defense date: 5/14/2021 Time: 9:00 - 11:15 a.m.Zoom Meeting: https://pdx.zoom.us/j/86303013790 Committee chair: Jay Gopalakrishnan, PhD Committee members: Jeffrey Ovall, PhD Panayot Vassilevski, PhD Jacob Grosek, PhD Erik Sanchez, PhD Abstract In this dissertation, we present a simplified scalar numerical model for the fiber laser amplifier simulations using Coupled mode theory. A new scale model referred to as an equivalent short fiber, which captures some of the essential characteristics of a longer fiber is introduced. Computations using models of two commercially available fibers – one doped with ytterbium, and the other with thulium show the practical utility of the concept. Fiber quantum defect heating is included in the model. We solve the heat equation coupled with our CMT equation to get the solution. Transverse mode instability is observed in both ytterbium and thulium-doped fiber. Various power thresholds are presented for TMI. Also, to find the root cause of TMI and investigate how to mitigate this chaotic process, we experimented with different refractive index gratings. Finally, this dissertation uses numerical simulations of a thulium-doped optical fiber amplifier to predict various performance characteristics such as peak temperatures, expected output powers, and efficiencies, presence of amplified spontaneous emission (ASE), etc. using Single- and two-tone configurations.``Zoom Meeting: https://pdx.zoom.us/j/86303013790``Kathie Leck, Fariborz Maseeh Department of Mathematics + Statistics, mathstatadmin@pdx.edu``Kathie Leck, Fariborz Maseeh Department of Mathematics + Statistics, mathstatadmin@pdx.edu``America/Los_Angeles``public`