Schedule, Notes/Handouts & Videos

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We discuss an effective equidistribution theorem for semisimple periodic orbits on compact adelic homogeneous spaces. The obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. We apply this equidistribution theorem to the problem of establishing a local-global principle for representations of integral quadratic forms, improving the codimension assumptions and providing effective bounds in a theorem of Ellenberg and Venkatesh. This is joint work with Manfred Einsiedler, Elon Lindenstrauss, and Amir Mohammadi.

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A Besicovitch set is a compact subset of R^n that contains a unit line segment pointing in every direction. The Kakeya set conjecture asserts that every Besicovitch set in R^n has Minkowski and Hausdorff dimension n. I will discuss some recent progress on this conjecture, leading to the resolution of the Kakeya set conjecture in three dimensions. This is joint work with Hong Wang.

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I will discuss a version of Host's equidistribution theorem on multidimensional tori, giving an essentially optimal result that removed some unnecessary restrictions that appeared in Host's work on the subject. In the commuting case this gives a new proof and an extension of the measure classification theorem of Einsiedler and Lindenstrauss. We also obtain equidistribution results for points drawn from various fractal-type measures on the torus.

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I will discuss upper bounds for the dimensions of the affine and smooth slices of Cartesian products of Cantor sets invariant under multiplication by p_i on the circle. These upper bounds generalize the case of linear slices of products of two invariant Cantor sets, which is the original Furstenberg slicing problem. The higher rank version requires several new tools and ideas. Joint work with Emilio Corso.

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I will talk about recent developments on establishing effective versions of Ratner’s equidistribution theorem. I will explain the connection between quantitative behavior of unipotent orbits and problems from harmonic analysis on restricted projections. Then I will explain proofs for some important cases relying on arguments from incidence geometry. Based on joint work with Elon Lindenstrauss, Amir Mohammadi, and Zhiren Wang.

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We review the rudiments of a diophantine theory of affine Markoff cubics. These enjoy an action of the mapping class group on p-adic integral points thanks to their realization as the character variety of the once punctured torus. This provides a powerful tool making them one of the few families of affine log K_3's whose integral points can be studied. We highlight recent advances and some applications.

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I will explain how methods from analytic number theory and automorphic forms can be used to prove versions of simultaneous equidistribution and the mixing conjecture on the upper half plane or more generally quotients of quaternion algebras.

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In 1975, Szemer\'edi proved that any subset of the natural numbers with positive upper density must contain arbitrarily long finite arithmetic progressions. Szemer\'edi's original argument was purely combinatorial, and then Furstenberg gave an alternative proof using ergodic theory a couple of years later. Objects called "nonconventional ergodic averages" appeared for the first time in Furstenberg's proof, and understanding the limiting behavior of very general such averages became an important problem in ergodic theory. After breakthrough work of Bourgain in the late 1980s and early 1990s, no further progress had been made on proving pointwise almost everywhere convergence of nonconventional ergodic averages until recently. I will report on this progress, along with some of the key inputs from additive combinatorics.

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We will discuss the reflection groups associated to hyperbolic triangles from a variety of perspectives:  the arithmetic of
their cusps; totally geodesic curves on Hilbert modular surfaces; and Minkowski's question mark function.  The latter
arises when triangle groups are mated with polynomials, to produce fractals and dynamical systems on the Riemann sphere.

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They say the only normal people are the ones you don't know very well. But what about numbers? Which ones are normal, and how well do we really know them? A real number x is normal in base b if every block of digits of the same length appears in the b-adic digit expansion of x with equal limiting frequency. While the concept is easy to define, proving that specific numbers are normal often requires deep and intricate methods. The study of normal numbers connects diverse areas of mathematics, including harmonic analysis, ergodic theory, Diophantine approximation, and fractal geometry.

In this talk, I will survey key results and techniques in the study of normal numbers, explore the challenges posed by long-standing open problems, and present the recent resolution of a conjecture by Kahane and Salem regarding the properties of partially normal numbers. This work draws on methods from Fourier analysis, geometric measure theory, and probabilistic number theory. Joint work with Junqiang Zhang.