Partial Differential Equations - Summer SchoolGulbenkian Foundation, LISBON — JULY 4 to 8, 2016
sketch for the panel Começar by José de Almada Negreiros - image reproduced with kind permission of the Gulbenkian Foundation ProgramThis school comprises three lecture courses (in English) aimed at 1st- or 2nd-year undergraduate students of mathematics, complemented by problem sessions.CoursesMathematical modeling in hydrodynamicsby André Nachbin (IMPA)Abstract: These lectures will use Fluid Dynamics to introduce mathematical modeling through partial differential equations (PDEs). In other words from a physical model (namely some physical hypothesis) we deduce our mathematical model of interest, namely a set of PDEs. Further mathematical modeling can be done to simplify the problem and attempt to solve it analytically. We will see an example where this simplification naturally leads to using Complex Variables. As time permits we will explore mathematical models related to Waves in Fluids and their underlying PDEs. Prerequisites: multivariable calculus (Gauss, Green's and Stokes Theorem), linear algebra and some complex variables (analytic function/complex derivatives, Cauchy-Riemann equations). No Fluid Dynamics is required. Systems of conservation lawsby Esteban Tabak (NYU)Abstract: This course will study systems of conservation laws in one and more dimensions. Models used to illustrate the discussion include traffic flow, flood waves, flows in rivers and lakes and gas dynamics. Mathematics topics include characteristics, simple waves (including shock waves and rarefactions), weak solutions, the Riemann problem, entropy conditions, and some basic numerics (Godunov's method). Time-allowing, there will be some additional discussion of characteristics of general first-order equations and dispersive waves and/or other topics such as non-convex kinematic laws and non-standard jump conditions. No pre-requisites are needed for this class beyond multivariable calculus and linear algebra. What do heating your living room, financial investments, and image processing have in common?by Xavier Cabré (ICREA and UPC Barcelona)Abstract: In these lectures we will see, in a very elementary way, that there is a striking resemblance on the modeling of heat distributions and that of option prices in Finance. In both cases the basic object is the same and it is called "the Laplacian" (or Laplace operator). We will see that it is is also a central object in image processing. We will study the basic properties of both the continuous and the discrete Laplacian and their solutions. Useful background: basic Calculus and Linear Algebra Recommended bibliography:
Schedule
Friday, July 8h at 17h30, after the problem session, there will be a meeting in room PA1 (same room as the problem session) where the participants will be able to talk to the lecturers about academic life in a relaxed setting. On Friday we will go out to dinner at 8pm at Restaurante d'Bacalhau at Parque das Nações. Google provides directions to the restaurant (subway red line to "Parque das Nações", walk to the river, and then north along the river). Here is a handout with directions. Mini-conferenceOn Saturday, July 9th, there will be a student mini-conference in the same venue as the school.All school participants are invited to attend this mini-conference. Here is the schedule of the mini-conference. AssistantSimão Correia ()VenueThe school takes place at the main (Alameda) campus of Instituto Superior Técnico, in central Lisbon, with lectures and problem sessions in lecture hall PA2 (floor -1 of the Mathematics building). |
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