Definition of homotopy groups, homotopy theory of CW complexes, Huriewich theorem and Whitehead's theorem, Eilenberg-Maclane spaces, fibration and cofibration sequences, Postnikov towers, and obstruction theory. Prerequisite: MATH 7800.

Seminars

Lenny Ng (Duke) - Studying topology through symplectic geometry

Studying topology through symplectic geometry

Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe a way to use this approach to construct a rather powerful invariant of knots called "knot contact homology", which has unexpected relations to string theory and mirror symmetry.

Lenny NgStudying topology through symplectic geometry

Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe a way to use this approach to construct a rather powerful invariant of knots called "knot contact homology", which has unexpected relations to string theory and mirror symmetry.

" class="addtocalendar" target="_new">Add to Google CalendarDavid Saltman (Center for Communications Research, Princeton) - Genus one curves from division algebras of degree 3 (time 1-2pm)

If D/F is a division algebra of degree 3, then the Severi-Brauer variety of D, call it X, is a form of the projective plane. The line bundle O(3) is defined on X, which says it makes sense to talk about cubic curves on X. Since X has no rational points, these are genus one curves and not elliptic curves. However, they are principle homogeneous spaces over their Jacobians E, which are elliptic curves. Which j invariants occur? One attacks this via the action of PGL_3 on cubic forms over the algebraic closure. However, this PGL_3 action has a generic nontrivial stabilizer, which we show is a feature and not a bug.

Add to Google CalendarMimi Dai (UIC) - Regularity for the 3D Navier-Stokes equations and related problems

Regularity for the 3D Navier-Stokes equations and related problems

As one of the most significant problems in the study of partial differential equations arising in fluid dynamics, Leray's conjecture in 1930's regarding the appearance of singularities for the 3-dimensional (3D) Navier-Stokes equations (NSE) has been neither proved nor disproved. The problems of blow-up have been extensively studied for decades using different techniques. By using a method of wavenumber splitting which originated from Kolmogorov's theory of turbulence, we obtained a new regularity criterion for the 3D NSE. The new criterion improves the classical Prodi-Serrin, Beale-Kato-Majda criteria and their extensions. Related problems, such as the well/ill-posedness, will be discussed as well.

Mimi DaiRegularity for the 3D Navier-Stokes equations and related problems

As one of the most significant problems in the study of partial differential equations arising in fluid dynamics, Leray's conjecture in 1930's regarding the appearance of singularities for the 3-dimensional (3D) Navier-Stokes equations (NSE) has been neither proved nor disproved. The problems of blow-up have been extensively studied for decades using different techniques. By using a method of wavenumber splitting which originated from Kolmogorov's theory of turbulence, we obtained a new regularity criterion for the 3D NSE. The new criterion improves the classical Prodi-Serrin, Beale-Kato-Majda criteria and their extensions. Related problems, such as the well/ill-posedness, will be discussed as well.

" class="addtocalendar" target="_new">Add to Google CalendarAlexey Cheskidov (UIC) - Long time behavior of the Navier-Stokes equations and turbulence

Long time behavior of the Navier-Stokes equations and turbulence

Turbulence is often referred to as the last unsolved problem in classical physics, even though it is widely believed that turbulent flows are governed by the Navier-Stokes equations. Even when the velocity field in a turbulent flow is chaotic, experimental and numerical evidence shows that the averaged velocity still displays some regular structure. In this talk we will discuss the long time behavior of solutions to the Navier-Stokes equations and review some recent advances in the mathematical description of turbulence.

Alexey CheskidovLong time behavior of the Navier-Stokes equations and turbulence

Turbulence is often referred to as the last unsolved problem in classical physics, even though it is widely believed that turbulent flows are governed by the Navier-Stokes equations. Even when the velocity field in a turbulent flow is chaotic, experimental and numerical evidence shows that the averaged velocity still displays some regular structure. In this talk we will discuss the long time behavior of solutions to the Navier-Stokes equations and review some recent advances in the mathematical description of turbulence.

" class="addtocalendar" target="_new">Add to Google CalendarUzi Vishne (Bar Ilan) - TBA (time 1-2pm)

http://u.cs.biu.ac.il/~vishne/

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MATH 7840

Definition of homotopy groups, homotopy theory of CW complexes, Huriewich theorem and Whitehead's theorem, Eilenberg-Maclane spaces, fibration and cofibration sequences, Postnikov towers, and obstruction theory. Prerequisite: MATH 7800.