We prove that if a Hamiltonian diffeomorphism of a closed monotone symplectic manifold with semisimple quantum homology has more contractible fixed points, counted homologically, than the total dimension of the homology of the manifold, then it must have an infinite number of contractible periodic points. This constitutes a higher-dimensional homological generalization of a celebrated result of Franks from 1992, as conjectured by Hofer and Zehnder in 1994.

For every abelian compact Lie group $A$, we prove that the homotopical $A$-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the $A$-equivariant Lazard ring, introduced by Cole--Greenlees--Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian $2$-groups. Our results generalize classical theorems of Quillen (1969) on the connection between non-equivariant bordism rings and formal group laws, and extend the case $A=C_2$ due to Hanke--Wiemeler (2018).

We work in the framework of global homotopy theory, which is essential for our proof. In addition to the statements for a fixed group $A$, we also prove a global algebraic universal property that characterizes the collection of all equivariant complex bordism rings simultaneously. We show that they form the universal contravariant functor from abelian compact Lie groups to commutative rings that is equipped with a coordinate; the coordinate is given by the universal Euler class at the circle group. More generally, the ring of $n$-fold cooperations of equivariant complex bordism is shown to be universal among functors equipped with a strict $n$-tuple of coordinates.

We give a construction that produces irreducible complex rigid local systems on $\mathbb{P}_C^1-\{p_1,\ldots ,p_s\}$ via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups $\mathrm{SU}(n)$ (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices of these polytopes give all possible unitary irreducible rigid local systems. As a consequence we obtain that the ranks of unitary irreducible rigid local systems, including those with finite global monodromy, on $\mathbb{P}^1-S$ are bounded above if we fix the cardinality of the set $S=\{p_1,\ldots ,p_s\}$ and require that the local monodromies have orders that divide $n$ for a fixed $n$. Answering a question of N.~Katz, we show that there are no irreducible rigid local systems of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing $n$, when $n$ is a prime number.

We also show that all unitary irreducible rigid local systems on $\mathbb{P}_C^1-S$ with finite local monodromies arise as solutions to the Knizhnik-Zamalodchikov equations on conformal blocks for the special linear group. Along the way, generalizing previous works of the author and J.~Kiers, we give an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for $\mathrm{SU}(n)$.

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages $$A_N(f,g)(x) := \frac{1}{N} \sum_{n=1}^{N} f(T^nx) g(T^{P(n)}x)$$as $N \to \infty$, where $T\colon X\to X$ is a measure-preserving transformation of a $\sigma$-finite measure space $(X,\mu), P(\mathrm{n})\in \mathbb{Z}[\mathrm{n}]$ is a polynomial of degree $d \geq 2$, and $f \in L^{p_1}(X)$, $g \in L^{p_2}(X)$ for some $p_1, p_2 > 1$ with $\frac{1}{p_1} + \frac{1}{p_2} \leq 1$. We also establish an $r$-variational inequality for these averages (at lacunary scales) in the optimal range $r > 2$. We are also able to ``break duality" by handling some ranges of exponents $p_1$, $p_2$ with $\frac{1}{p_1} + \frac{1}{p_2} > 1$, at the cost of increasing $r$ slightly.

This gives an affirmative answer to Problem 11 from Frantzikinakis' open problems survey for the Furstenberg--Weiss averages (with $P(\mathrm{n})=\mathrm{n}^2)$, which is a bilinear variant of Question 9 considered by Bergelson in his survey on Ergodic Ramsey Theory from 1996. This also gives a contribution to the Furstenberg--Bergelson--Leibman conjecture. Our methods combine techniques from harmonic analysis with the recent inverse theorems of Peluse and Prendiville in additive combinatorics. At large scales, the harmonic analysis of the adelic integers $\mathbb{A}_{\mathbb{Z}}$ also plays a role.

We give a negative answer to Ulam's Problem 19 from the Scottish Book asking is a solid of uniform density which will float in water in every position a sphere? Assuming that the density of water is $1$, we show that there exists a strictly convex body of revolution $K\subset \mathbb{R}^3$ of uniform density $\frac{1}{2}$, which is not a Euclidean ball, yet floats in equilibrium in every orientation. We prove an analogous result in all dimensions $d\ge 3$.