Mar 03, 2025
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
- Supplements
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09:30 AM - 10:30 AM
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On values of quadratic forms and effective equidistribution of unipotent flows
Elon Lindenstrauss (The Hebrew University of Jerusalem)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Effective equidistribution of semisimple adelic periods and representations of quadratic forms
Andreas Wieser (Institute for Advanced Study)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
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.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Curved Kakeya and Nikodym problems
Shaoming Guo (Indiana University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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The Kakeya set conjecture in three dimensions
Joshua Zahl (University of British Columbia)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
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.
- Supplements
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Mar 04, 2025
Tuesday
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09:30 AM - 10:30 AM
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Subconvexity and equidistribution problems for higher-rank L-functions
Paul Nelson (Aarhus University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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11:00 AM - 12:00 PM
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L-functions and simultaneous equidistribution
Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
We will describe various techniques that occurred when dealing with the problem of evaluating certain moments of L-functions, which go beyond certain natural limits in harmonic analysis and involve simultaneous equidistribution results
(although not necessarily in the area of homogenous dynamics).
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Exponential Mixing Via Additive Combinatorics
Osama Khalil (University of Illinois, Chicago)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
We describe an approach to the problem of exponential mixing of geodesic flows by connecting it to the following general dichotomy: for a given measure on Euclidean space, either its Fourier transform decays polynomially outside a very sparse set of frequencies, or a large subset of the support concentrates near proper subspaces at many scales.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Finiteness of totally geodesic submanifolds
David Fisher (Rice University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
I will give a survey of recent results on finiteness of totally geodesic submanifolds in negatively curved manifolds, both locally symmetric and not. Some emphasis will be placed on open questions, particularly open questions about effective results. If time permits I may mention some related questions about rigidity of commensurators.
- Supplements
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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Mar 05, 2025
Wednesday
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09:30 AM - 10:30 AM
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Toral endomorphisms and equidistribution
Michael Hochman (The Hebrew University of Jerusalem)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
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.
- Supplements
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10:30 AM - 10:40 AM
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Group Picture
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- Location
- SLMath: Front Courtyard
- Video
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- Abstract
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- Supplements
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10:40 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Higher rank Furstenberg slicing
Pablo Shmerkin (University of British Columbia)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
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.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Effective equidistribution in homogeneous spaces and restricted projection theorems
Lei Yang (National University of Singapore)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
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.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
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- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Diophantine analysis of Markoff type surfaces
Peter Sarnak (Princeton University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
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.
- Supplements
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Mar 06, 2025
Thursday
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09:00 AM - 10:00 AM
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Complex continued fractions and beyond
Seonhee Lim (Seoul National University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
- Supplements
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10:00 AM - 10:15 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
Zoom Link
- Supplements
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10:15 AM - 11:15 AM
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Non-abelian Littlewood-Offord theorems
Emmanuel Breuillard (Université de Paris XI)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
We prove uniform exponential estimates for the probability that a product of independent random variables with values in a Lie group hits a point, a subgroup, a subvariety. Joint work with Oren Becker.
- Supplements
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11:30 AM - 12:30 PM
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Smallest denominators and extreme events
Jens Marklof (University of Bristol)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
I will review old and new results, by various authors, on the distribution of smallest denominators of rational numbers in a shrinking interval (as well as higher-dimensional variants) and explore the connection to extreme value laws in homogeneous dynamics, specifically horocycle flows. This lecture is in part based on joint work with Mark Pollicott.
- Supplements
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Notes
528 KB application/pdf
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12:30 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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The Kakeya set conjecture in three dimensions II
Hong Wang (New York University, Courant Institute)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Given any set of delta-tubes in R^3, we can factor tubes into clusters, such that within each cluster, the tubes are dense and fill out a convex set, and these convex sets are essentially disjoint. We discuss some new aspects of the argument, putting into a broader context of projection theory. This is joint work with Josh Zahl.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
- Supplements
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Mar 07, 2025
Friday
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09:30 AM - 10:30 AM
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Simultaneous equidistribution and L-functions
Valentin Blomer (Universität Bonn)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
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.
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Pointwise convergence of polynomial ergodic averages
Sarah Peluse (Stanford University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
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.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Triangle groups: From Hilbert modular varieties to complex dynamics
Curtis McMullen (Harvard University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
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.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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A Journey Through Normality
Malabika Pramanik (University of British Columbia)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
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.
- Supplements
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