Course: Stochastic Differential Equations
IMPA. August 15 - November 29, 2024. Thursdays and Fridays at 10:30. Room 236.
Exercises: list 1 (due Sept 6), list 2 (due Sept 20), list 3 (due Oct 4), list 4 (due Oct 18), list 5 (due Dec 2)
Monitoria: Fridays at 15:30, room 347. (with Andre Considera, andre.luis@impa.br)
Lecture notes: part 1 (basics), part 2 (Ito integration), part 3 (SDO) (updated 23/Oct)
Contents: Probability spaces, random variables and stochastic processes. Brownian motion. Ito calculus. Martingale. Stochastic differential equations: examples and basic properties. Diffusions. Markov property. Generator. Numerical methods.
Bibliography:
Bernt Oksendal, Stochastic differential equations: an introduction with applications, Springer, 2013.
Evans, Lawrence C. An introduction to stochastic differential equations. American Mathematical Soc., 2012.
Course "Hidden symmetries in fluid dynamics" (with Simon Thalabard, U. Nice)
Wed & Fri, 10/1-2/2 at 15:00-17:00 in room 232.
VIDEOS on IMPA's YouTube Channel:
Lecture 1: Navier-Stokes (NS) and Euler equations. Symmetries. Conservation of energy.
Lecture 2: NS turbulence, Gaussian HIT, Gaussian vs. NS, viscous effects.
Lecture 3: Scaling anomaly, Random-h model, Multifractal theory.
Lecture 4: Solvable intermittent model of turbulence.
Lecture 5: Hidden scaling symmetry in shell models. Anomalous scaling laws.
Lecture 6: Kolmogorov multipliers. Rescaled NS equations.
Lecture 7: Hidden symmetries in NS turbulence.
Lecture 8: Group theory approach to hidden scaling symmetries.
Lecture 9: Group theory approach to quasi-Lagrangian formalism. Perspectives.
Notes: Simon (lecture_1, lecture_2, lecture_3, lecture_6_7) & Alexei (lecture_4_5_6, lecture_8_9)
Análise Complexa (Complex Analysis)
Classes: Monday and Wednesday at 15:30, room 224 (March 13 - June 30)
Monitor: Gabriel Santos Barbosa (gabriel.barbosa@impa.br).
Lists of exercises: List 1, List 2, List 3, List 4, List 5, List 6, List 7: Ch.VII.1: Ex.4,7; Ch.VII.2: Ex.7,9; Ch.VII.3: Ex.4,7.
Introduction to the Theory of Oscillations and Waves
Classes: January-February, Monday/Tuesday/Wednesday, 10:00-12:00. Room 232. Videos of classes on Youtube
Material:
Exercises:
Solutions can be delivered to me or to the monitor at the class.
You can also send them scanned or photographed by e-mail to the monitor (antoine.barlet@impa.br) and me (alexei@impa.br)
Monitor of the course: Antoine Barlet, e-mail: antoine.barlet@impa.br, sala 108 (laboratory of Fluid Dynamics)
Numerical Analysis (March-June 2022):
Wednesday and Friday 13:30-15:00. IMPA, Auditorium 1.
IMPA Summer School course (January-February 2021):
Topics in Mathematics of Turbulence
January-February, Wednesday/Thursday/Friday, 15:00-17:00. IMPA, Auditorium 1.
Linear Algebra and Applications
March-June: Thursday at 10:30, Friday at 9:30.
12/03 (video, notes), 18/03 (video, notes), 19/03 (video, notes), 25/03 (video, notes), 1/04 (video, notes), 08/04 (video, notes), 09/04 (video, notes),
15/04 (video, notes), 16/04 (video, notes), 22/04 (video, notes), 23/04 (video, notes), 29/04 (video, notes), 30/04 (video, notes), 06/05 (video, notes),
07/05 (video, notes), 13/05 (video, notes), 14/05 (video, notes), 20/05 (video, notes), 21/05 (video, notes), 27/05 (video, notes),
28/05 (video1, video2, notes), 4/06 (video, notes), 10/06 (video, notes), 11/06 (video, notes), 17/06 (video, notes), 18/06 (video, notes)
Monitor: Julia Domingues, e-mail: juliahdomingues@hotmail.com
Time for monitoria: please select HERE
Link for online class: it will be sent by email to all registered students, and you can ask me for this link by email.
Introduction to the Theory of Oscillations and Waves
January-February, Wednesday/Thursday/Friday, 10:00-12:00.
Online: feel free to ask me for the link by email.
Álgebra Linear e Aplicações March-June, Tue & Wed 15:30-17:00
Thermal and Statistical Aspects of Fluid Dynamics (together with Simon Thalabard)
January-February, Tue, Wed, Fri 15:00-17:00, room 224 (starting on January 7)
LECTURE NOTES:
Section 1 (by Alexei Mailybaev)
Section 2 (by Simon Thalabard)
August 1: Mini-course on Parametric Resonance
At the São Paulo School of Advanced Sciences on NONLINEAR DYNAMICS
Temporary links: youtube1, youtube2, youtube3, youtube4
March-June: Fluid Dynamics
Tue, Wed 13:30-15:00, room 333
Recommended literature:
ACHESON, D. J. – Elementary Fluid Dynamics, Oxford University Press, 1990.
BATCHELOR, G. – An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
CHORIN, A. J., MARSDEN, J. E. – A Mathematical Introduction to Fluid Mechanics, second edition, Springer-Verlag, 1990.
COURANT, R., FRIEDRICHS, K. O. – Supersonic Flow and Shock Waves, Springer-Verlag, 1976.
LANDAU L.D., LIFSHITZ, E.M. Fluid Mechanics. (Course of T.P., Vol.6).
WHITHAM, G. – Linear and Nonlinear Waves. New York, John Wiley, 1974.
January-February: Introduction to the Theory of Oscillations and Waves
Some rather old notes
Exam questions with literature references
Ordinary Differential Equations
Hydrodynamic Turbulence (reading course)
Spontaneous Stochasticity (mini-course)
Bifurcations in ordinary differential equations
Fluid Dynamics
Introduction to the Theory of Oscillations and Waves
Lectures on Boundaries of Stability Domains (mini-course)
Partial Differential Equations
Measure and Integration
Thermal Flow of Fluids
Stability and Catastrophes in Mechanical Systems
Introduction to the Lattice Boltzmann method for fluid dynamics (reading course)
