Tue, January 29, 2019
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9:00am [9:30am] Prof. Eduard Feireisl, Czech Academy of Sciences
Name of the instructor: Prof. Eduard Feireisl. Affiliation: Czech Academy of Sciences. Mode of instruction: via videoconference. Title of the mini-course: Mathematical Aspects of Euler Equations. Venue: A1A2 hall, CDEEP, IIT Bombay. We consider the phenomenon of oscillations in the solution families to partial differential equations. To begin, we briefly discuss the mechanisms preventing oscillations/concentrations and make a short excursion in the theory of compensated compactness. Pursuing the philosophy "everything what is not forbidden is allowed" we show that certain problems in fluid dynamics admit oscillatory solutions. This fact gives rise to two rather unexpected and in a way contradictory results: (i) many problems describing inviscid fluid motion in several space dimensions admit global-in-time (weak solution); (ii) the solutions are not determined uniquely by their initial data. We examine the basic analytical tool behind these rather ground breaking results - the method of convex integration applied to problems in fluid mechanics and, in particular, to the Euler system.

3:00pm [3:30pm] Radhika Gupta (Technion, Israel)
Geometry Seminar Speaker: Radhika Gupta (Technion, Israel). Title: `Cannon-Thurston maps for CAT(0) groups with isolated flats'. Time: 15:30 - 16:30, Tuesday, January 29, 2019. Venue: Ramanujan Hall. Abstract: Consider a hyperbolic 3-manifold, called a mapping torus, that fibers over a circle with fiber a closed orientable surface of genus at least 2. Cannon and Thurston showed that the inclusion map from the surface into the 3-manifold extends to a continuous, surjective map between the visual boundaries of the respective universal covers. This gives a surjective map from a circle to a 2-sphere. Mj showed that a Cannon-Thurston map also exists for a hyperbolic group and its normal hyperbolic subgroups. In this talk, we will explore what happens when we consider the mapping torus of a surface with boundary, which is not hyperbolic but CAT(0) with isolated flats under some conditions.