Tuesday, November 7 |
07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
09:00 - 09:30 |
Gerard Misiolek: On continuity properties of solution maps of the SQG family ↓ The SQG family are active scalar equations that interpolate between the 2D Euler and the standard surface quasi-geostrophic equations. Not surprisingly, they share a lot of analytic (and geometric) properties with the latter though to exhibit them is technically somewhat more complicated. I will describe how the corresponding solution maps of the SQG family are at best continuous. This is joint work with Truong Vu. (Online) |
09:45 - 10:15 |
Stephen Preston: Liouville comparison theory for blowup of Euler-Arnold equations ↓ We describe a new method for proving blowup of certain Euler-Arnold equations, partial differential equations which represent geodesics on groups of diffeomorphisms under right-invariant metrics. It is based on using momentum conservation to treat the equation as a first-order ODE on a Banach space, then using proving that solutions breakdown based on a comparison theorem using the known exact solution f of the classical Liouville equation. Applications are given for the right-invariant H2 Sobolev metric on the group of diffeomorphisms of Rn, where we show that solutions can break down in finite time if n≥3. (TCPL 201) |
10:30 - 11:00 |
Coffee Break (TCPL Foyer) |
11:00 - 11:30 |
Theodore Drivas: Irreversible features of the 2D Euler equations ↓ We will discuss aspects of the long term dynamics of 2d perfect fluids. As an application of a certain stability of twisting for general hamiltonian flows, we will show generic loss of smoothness near stable steady states, the existence of many wandering neighborhoods, aging of the Lagrangian flow, along with other examples of complex behavior such as indefinite perimeter growth for special vortex patches. (TCPL 201) |
11:45 - 13:15 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
13:15 - 13:45 |
Peter Michor: Regularity and Completeness of half Lie groups ↓ Half Lie groups exist only in infinite dimensions: They are smooth manifolds and topological groups such that right translations are smooth. Main examples are Sobolev Hr-diffeomorphism groups of compact manifolds, or Ck-diffeomorphism groups, or semidirect products of a Lie group with kernel an infinite dimensional representation space (investigated by Marquis and Neeb). Here, we investigate mainly Banach half Lie groups, the groups of their Ck-elements, extensions, and right invariant strong Riemannian metrics on them: Here surprisingly the full Hopf Rinov theorem holds which is wrong in general even for Hilbert manifolds. (TCPL 201) |
14:00 - 14:30 |
Alexander Shnirelman: Geometric structures on the group of volume preserving diffeomorphisms ↓ We consider the group D=SDiff(M) of volume preserving diffeomorphisms of a bounded domain M⊂\Rn. Every map f∈D can be tautologically regarded as a map f:M→\Rn; if we consider these maps in the Sobolev Hs metric, then we can regard D as a subset of X=Hs(M,\Rn). The geometric properties of the set D⊂\Rn depend a great deal on the Sobolev exponent s. We consider two cases. (1) If s>n/2+2 then D is a smooth submanifold of X. Moreover, D is a "Quasiruled manifold", i.e. it can be uniformly approximated by the "Ruled manifolds", which are the unions of a finite-dimebsional family of affine planes in X. Such manifolds form a category QL(X); the morphisms in this category are "Fredholm Quasiruled maps" (FQR-maps). The main result here is the following Theorem: Let ft∈D be the family of flow maps defined by the Lagrange-Euler equations of theideal incompressible fluid. Then for any t, ft:D→D is an FQR-map wherever it is defined; for n=2 the maps ft are defined all over D for any t. This theorem justifies the use of topological methods in the study of the fluid kinematics. (2) If s=0 (i.e. X=L2(M,\Rn)), then D⊂X is not a smooth manifold; it is quite a singular set in X. If n=2, the intrinsic metric of D is unrelated to the metric induced from X (in particular, D is not metrically connected); for n>2 D is metrically connected but the relation between the intrinsic and the induced metrics looks like for the the Carnot-Caratheodori spaces. The set D does not have a true tangent space TfD at a point f∈D, but we are able to define its substitute Yf; using this structure, we can define a class of dissipative weak solutuions for the Euler equations for n>2 which may have a physical sense. (TCPL 201) |
14:45 - 15:15 |
Coffee Break (TCPL Foyer) |
15:15 - 15:45 |
Anton Izosimov: Geometry of generalized fluid flows ↓ Arnold showed that solutions to the hydrodynamical Euler equation can be interpreted as geodesics on the group of volume-preserving diffeomorphisms. Motivated by insolvability of the two-point problem for the Euler equation (i.e. non-existence of minimizing geodesics between certain pairs of fluid configurations), Brenier and Shnirelman considered its relaxed version, leading to so-called generalized flows. In the talk I will describe the geometry behind such flows. This is based on joint work with Boris Khesin. (TCPL 201) |
16:00 - 16:15 |
Patrick Heslin: Geometry of the generalized SQG equations ↓ The generalized SQG family of equations interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation; a well known model for the three dimensional Euler equations. These equations can be realised as geodesic equations on groups of diffeomorphisms. I will present some results pertaining to Fredholmness of the corresponding exponential maps, as well as the distribution of conjugate points. This is joint work with Martin Bauer, Gerard Misiołek and Stephen Preston. (TCPL 201) |
16:15 - 16:30 |
Levin Maier: On Mañé's critical value for the Hunter-Saxton system ↓ We will study magnetic deformations of the Hunter-Saxton system, in the
sense of magnetic geodesic flows. We represent this system as a
Hamiltonian flow on an infinite dimensional Lie group and use this to
study blow ups and construct weak solutions of this system of partial
differential equations. Furthermore we will use the global weak magnetic
flow on the infinite dimensional Lie group to prove that any two points
in there can be connected by a magnetic geodesic as long as the strength
of the magnetic field is less then Mañé's critical value. (TCPL 201) |
16:30 - 16:45 |
Luke Volk: Simple Unbalanced Optimal Transport ↓ Joint work with B. Khesin and K. Modin. A simple model capturing the key features of unbalanced optimal transport will be explored with an emphasis on a Riemannian submersion of the diffeomorphism group to the space of volume forms of arbitrary mass, as well as a finite-dimensional analogue of this submersion. (TCPL 201) |
16:45 - 17:00 |
Archishman Saha: Symmetry and Reduction for Stochastic Differential Equations ↓ SDEs on manifolds are formulated using Schwartz operators in the context of second order differential geometry. Conversely, most results in second order differential geometry have a probabilistic interpretation. Within this framework, Stratonovich differential equations provide a counterpart to first order differential geometry and ODEs, and their reduction and reconstruction have been studied by Lazaro-Camí and Ortega and others. We aim to generalise these ideas to Schwartz operators. (TCPL 201) |
17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |