# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### A classification of affine smooth spherical varieties

F. Knop has translated Delzant's conjecture (1990) on compact multiplicity free spaces (also known as noncommutative completely integrable systems) into a conjecture on affine smooth spherical varieties. If $G$ is a reductive algebraic group (over $C$) and $X$ is an affine $G$-variety (over $C$), then $X$ is called spherical if its coordinate ring $C\left[X\right]$ is multiplicity free as a $G$-module. Knop's conjecture states that if moreover $X$ is smooth, the module structure of $C\left[X\right]$ determines the $G$-variety $X$ up to isomorphism. As a first step towards proving the conjecture, we have classified these varieties, "up to pesky tori and connected components".

### Ergodic theory and the distribution of ${n}^{2}x$ modulo one

I will discuss some recent results on the distribution of certain unipotent orbits on hyperbolic surfaces and their implication on the randomness of the fractional parts of the sequence ${n}^{2}x$.

### Floquet solutions for linear ODEs with quasiperiodic coefficients

We investigate the local reducibility of linear analytic multifrequency cocycles on SL(2,R) using renormalisation. In this way we show that some linear ODEs with coefficients depending quasiperiodically on time can be conjugated to others with constant coefficients. Much in the same way as Floquet theory for periodic coefficients.

### A lower bound to the spectral threshold in curved tubes

Motivated by the theory of quantum waveguides, we consider the Laplacian in curved tubes of arbitrary cross-section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube. This is a joint work with Pavel Exner and Pedro Freitas.

### The $N$-Membranes Problem for Nonlinear Degenerate Systems

We consider the elliptic variational inequality associated with quasilinear nonlinear systems, including those of $p$-Laplacian type, and with the ordering constraint of the $N$-membrane problem. We extend the Lewy-Stampacchia inequalities for the solution, obtaining new regularity results for the derivatives of the solution, including in the case of linear operators ( $p=2$) integrability of second derivatives for all $q>1$. Considering the $N$-membrane problem as coupled $\left(N-1\right)$-obstacle problem we obtain also the corresponding conditions for the stability of the coincidence sets for the variation of external forces.

### Asymptotic properties of the ground-state solutions of singularly perturbed elliptic Hamiltonian systems

We study the shape of the positive solutions of an elliptic system of the form
 $-{\epsilon }^{2}\Delta u+u=g\left(v\right),-{\epsilon }^{2}\Delta v+v=f\left(u\right)$

as the parameter $\epsilon$ goes to zero, namely the appearance of "spike-layer patterns" for the solutions of the system having minimal energy. Here the nonlinear terms are assumed to have superlinear and subcritical growth at infinity and we consider both cases of Neumann and Dirichlet boundary conditions over a given bounded open set $\Omega \subset {R}^{n}$. The proofs combine standard estimates in elliptic equations with new ideas in the calculus of variations (variational methods and Morse theory). This is joint work with J. Yang (to appear in Trans. Amer. Math. Soc.) and with A. Pistoia (to appear in J. Differential Equations).

### Noncommutative topology and quantales

The classical Gelfand representation theorem tells us that any unital (complex) $C$*-algebra is, up to isomorphism, the algebra of continuous complex valued functions on a compact Hausdorff space. The noncommutative analogue of this result is the Gelfand-Naimark theorem, which shows how any $C$*-algebra can be concretely realized as an algebra of bounded operators on some Hilbert space. However, there is a sense in which this noncommutative "analogue" fails to provide a characterization of those operators on the Hilbert space that actually lie in the given $C$*-algebra, and in 1971 Giles and Kummer (and also Akemann, in a related but independent way) introduced a notion of noncommutative topology in terms of which, essentially, every $C$*-algebra becomes an algebra of "continuous functions".

But lacking in their approach is a self-contained (i.e., independent of $C$*-algebras) characterization of what should be meant by such a noncommutative topology, and it was partly in an attempt to answer this question that quantales were proposed by Mulvey in 1983 as possible candidates for such topologies. In this talk I shall focus on the connections between quantales and $C$*-algebras, in particular addressing as an example, if time allows, Connes' noncommutative space of Penrose tilings.

### Topological entropy and homological growth on graphs

We establish a precise relation between the topological entropy and entropies arising from the homological growth and the exponential growth rate of the number of periodic points for a piecewise monotone graph map showing that the first one is the maximum of the latter two. This nontrivially extends a result of Milnor and Thurston on piecewise monotone interval maps. For this purpose we generalize the concept of Milnor-Thurston zeta function involving Lefschetz zeta function.

### Iterates of nonpolynomial maps

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