Mathematics Problem Archive

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

For a Birkhoff billiard inside a closed smooth strictly convex hypersurface $S\subset\mathbb{R}^d$, let $T$ act on the space $\mathbb{A}$ of oriented ...

Let $\gamma$ be a smooth convex plane billiard table symmetric about an axis $l$, and let $C$ be a convex caustic. Must $C$ be symmetric about $l$? Pr...

Suppose a bounded strictly convex planar billiard has two nested closed caustics such that the smaller caustic is itself a caustic for the billiard in...

Let $C\subset\mathbb{R}^2$ be a curve and translate it through an $\varepsilon$-square lattice, recording a click whenever it meets a lattice point. C...

For an oval $\gamma$, the area spectrum of its outer billiard is the set of areas of the circumscribed polygons formed by periodic outer-billiard traj...

Can polynomial tunings be constructed by quasiconformal surgery?...

For a fixed polynomial $P_1$, does the polynomial obtained by tuning $P_1$ with $P_2$ vary continuously with $P_2$?...

Among polynomials $P_1$ of degree greater than two with a superstable orbit of fixed period, does the tuning with a fixed $P_2$ vary continuously with...

Let $P_{1,k}$ have a superstable orbit whose period tends to infinity and suppose $P_{1,k}\to P_{1,\infty}$. Do the tunings with a fixed polynomial $P...

When does the topological intertwining construction for two polynomial dynamical planes yield a branched map conjugate to a polynomial?...

Can polynomial intertwinings be constructed by quasiconformal surgery?...

For a fixed first polynomial $P_1$, does the polynomial obtained by intertwining $P_1$ with $P_2$ vary continuously with $P_2$?...

Is the Mandelbrot set locally connected? Equivalently, for the quadratic family $z\mapsto z^2+\lambda$, is the boundary of the unbounded component of ...

For a degree-$n$ Blaschke product $A$, let $B(A)$ be the rational maps obtained by mating $A$ with a varying Blaschke product, and let $F:B(z^n)\to B(...

Quotient the boundary of $B(A)$ by quasiconformal conjugacy, writing the quotient as $\partial(A)$. Prove that the natural isomorphism $F:B(z^n)\to B(...

Give a combinatorial description, possibly by laminations, of the quotient boundary space $\partial(z^n)$ obtained from the boundary of $B(z^n)$ by id...

Is $B(z^n)$ a domain of holomorphy? More generally, is every component of the space of expanding rational maps, or of expanding polynomials, a domain ...