Distinct Distance Estimates and Low Degree Polynomial Partitioning

We give a shorter proof of a slightly weaker version of a theorem from Guth and Katz (Ann Math 181:155–190, 2015): we prove that if L is a set of L lines in R[superscript 3] with at most L[superscript 1/2] lines in any low degree algebraic surface, then the number of r-rich points of is L is ≲ L[superscript (3/2) + ε] r[superscript -2]. This result is one of the main ingredients in the proof of the distinct distance estimate in Guth and Katz (2015). With our slightly weaker theorem, we get a slightly weaker distinct distance estimate: any set of N points in R[superscript 2] c[subscript ε]N[superscript 1-ε] distinct distances

The joints problem for matroids

We prove that in a simple matroid, the maximal number of joints formed by L lines is o(L[superscript 2]) and Ω(L[superscript 2-ε]) for any ε > 0. Keywords: Matroids, The joints problem, Arithmetic progressionNational Science Foundation (U.S.) (Postdoctoral Fellowship

A family of maps with many small fibers

The waist inequality states that for a continuous map from S[superscript n] < to ℝ [superscript q], not all fibers can have small (n - q)-dimensional volume. We construct maps for which most fibers have small (n - q)-dimensional volume and all fibers have bounded (n - q)-dimensional volume. Keywords: waist inequality; isoperimetric inequalit

Polynomial Wolff axioms and Kakeya-type estimates in R4

We establish new linear and trilinear bounds for collections of tubes in R4 that satisfy the polynomial Wolff axioms. In brief, a collection of δ-tubes satisfies the Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a low degree algebraic variety. First, we prove that if a set of δ-3 tubes in R4 satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least δ1-1/28. We also prove a more technical statement which is analogous to a maximal function estimate at dimension 3+1/28. Second, we prove that if a collection of δ-3 tubes in R4 satisfies the polynomial Wolff axioms, and if most triples of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume at least δ3/4. Again, we also prove a slightly more technical statement which is analogous to a maximal function estimate at dimension 3+1/4. We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are unable to prove this. If our conjecture is correct, it implies a Kakeya maximal function estimate at dimension 3+1/28, and in particular this implies that every Kakeya set in R4 must have Hausdorff dimension at least 3+1/28. This would be an improvement over the current best bound of 3, which was established by Wolff in 1995

Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three

We prove the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves. Key words and phrases: discrete restriction estimates, Strichartz estimates, additive energy

Area-contracting maps between rectangles

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 207-208).In this thesis, I worked on estimating the smallest k-dilation of all diffeomorphisms between two n-dimensional rectangles R and S. I proved that for many rectangles there are highly non-linear diffeomorphisms with much smaller k-dilation than any linear diffeomorphism. When k is equal to n-l, I determined the smallest k-dilation up to a constant factor. For all values of k and n, I solved the following related problem up to a constant factor. Given n-dimensional rectangles R and S, decide if there is an embedding of S into R which maps each k-dimensional submanifold of S to an image with larger k-volume. I also applied the k-dilation techniques to two purely topological problems: estimating the Hopf invariant of a map from a 3-manifold to a high-genus surface, and determining whether there is a map of non-zero degree from a 3-manifold to a hyperbolic 3-manifold.by Lawrence Guth.Ph.D

Geometry and Destiny

The recognition that the cosmological constant may be non-zero forces us to re-evaluate standard notions about the connection between geometry and the fate of our Universe. An open Universe can recollapse, and a closed Universe can expand forever. As a corollary, we point out that there is no set of cosmological observations we can perform that will unambiguously allow us to determine what the ultimate destiny of the Universe will be.Comment: 7 pages, Gravity Research Foundation Essa

Magnetic Monopoles as the Highest Energy Cosmic Ray Primaries

We suggest that the highest energy \gsim 10^{20} eV cosmic ray primaries may be relativistic magnetic monopoles. Motivations for this hypothesis are that conventional primaries are problematic, while monopoles are naturally accelerated to $E \sim 10^{20} eV$ by galactic magnetic fields. By matching the cosmic monopole production mechanism to the observed highest energy cosmic ray flux we estimate the monopole mass to be $\sim 10^{10\pm1} GeV$.Comment: LaTex, 16 pages, no figure

Monopole annihilation at the electroweak scale---Not!

We examine the issue of monopole annihilation at the electroweak scale induced by flux tube confinement, concentrating first on the simplest possibility---one which requires no new physics beyond the standard model. Monopoles existing at the time of the electroweak phase transition may trigger $W$ condensation which can confine magnetic flux into flux tubes. However we show on very general grounds, using several independent estimates, that such a mechanism is impotent. We then present several general dynamical arguments constraining the possibility of monopole annihilation through any confining phase near the electroweak scale.Comment: 15 p