Tuesday January 5 |
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| 3:00-3:50pm Philip T. Gressman (UPenn) Radon-like Transforms, Geometric Measures, and Invariant Theory, slides Fourier restriction, Radon-like operators, and decoupling theory are three active areas of harmonic analysis which involve submanifolds of Euclidean space in a fundamental way. In each case, the mapping properties of the objects of study depend in a fundamental way on the "non-flatness" of the submanifold, but with the exception of certain extreme cases (primarily curves and hypersurfaces), it is not clear exactly how to quantify the geometry in an analytically meaningful way. In this talk, I will discuss a series of recent results which shed light on this situation using tools from an unusually broad range of mathematical sources. |
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| 4:00-4:50pm Alexander Dobner (UCLA) Extreme values of the argument of the zeta function, slides Let S(t) = 1/(π Im log ζ(1/2+it)). The behavior of this function is intimately connected to irregularities in the locations of the zeros of the zeta function. In particular S(t) measures the difference between the "expected" number of zeta zeros up to height t and the actual number of such zeros. I will discuss what is known about the distribution of S(t) and prove a new unconditional lower bound on how often S(t) achieves large values. |
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Tuesday January 19 |
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| 3:00-3:50pm Adi Glücksam (U. Toronto) Stationary random entire functions and related questions, slides The complex plane acts on the space of entire function by translations, taking f(z) to f(z+w). B. Weiss showed in `97 that there exists a probability measure defined on the space of entire functions, which is invariant under this action. In this talk I will present optimal bounds on the minimal possible growth of functions in the support of such measures and discuss other growth-related problems inspired by this work. In particular, I will focus on the question of minimal possible growth-rate of frequently oscillating subharmonic functions. The talk is partly based on a joint work with L. Buhovsky, A. Logunov, and M. Sodin. |
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| 4:00-4:50pm Yufei Zhao (MIT) Joints of varieties, slides We generalize the Guth-Katz joints theorem from lines to varieties. A special case of our result says that N planes (2-flats) in 6 dimensions (over any field) have O(N3/2) joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery's conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects$ Joint work with Jonathan Tidor and Hung-Hsun Hans Yu. |
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Tuesday February 2 |
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| 3:00-3:50pm Sigurd Angenent (UW-Madison) Nonunique evolution through cones in Mean Curvature Flow and Ricci Flow, slides For any integer k>1 there exist smooth solutions Mt (t<0) of MCF that form a one-point singularity at time t=0, after which there exist at least 2k forward evolutions Mt1, ..., Mtk, Nt1, ... , Ntk (t>0) by the flow. The solutions Mtj and Ntj are topologically distinct. The analogous statement for Ricci Flow also holds, and I will explain both.
Building on these self similar solutions to MCF, I will also describe non-self similar solutions that have a given
cone as their initial data. One conclusion is that for any k>1 there is a smooth self similar solution to MCF that
forms a one point singularity, and for which the set of possible smooth forward evolutions contains a k-dimensional
continuum. Another conclusion is that the set of smooth solutions to MCF whose initial condition is one of the
stationary cones in ℝn (n∈{4, 5, 6, 7}) is infinite dimensional.
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| 4:00-4:50pm Polona Durcik (Chapman) Multilinear singular and oscillatory integrals and applications, slides We give an overview of some recent results in the area of multilinear singular and oscillatory integrals. We discuss their connection with certain questions about point configurations in subsets of the Euclidean space and convergence of some ergodic averages. Based on joint works with Michael Christ, Vjekoslav Kovac, and Joris Roos. |
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Tuesday February 16 |
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| 10:00-10:50am Tsviqa Lakrec (U. Jerusalem) Equidistribution of affine random walks on some nilmanifolds, slides We consider the action of the group of affine transformations on a nilmanifold. Given a probability measure on this group and a starting point x, a random walk on the nilmanifold is defined. We study quantitative equidistribution in law of such affine random walks on nilmanifolds. Under certain assumptions, we show that a failure to have fast equidistribution on a nilmanifold is due to a failure on some factor nilmanifold. Combined with equidistribution results on the torus, this leads to an equidistribution statement on some nilmanifolds, such as Heisenberg nilmanifolds.
This talk is based on joint works with Weikun He and Elon Lindenstrauss.
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| 11:00-11:50am Felipe Gonçalves (Bonn) Sign Uncertainty, slides I will talk about some of the recent developments of the sign uncertainty principle and its relation with sphere packings and modular forms. |
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Tuesday March 2 |
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| 10:00-10:50am Cyrill Muratov (New Jersey Institute of Technology) Magnetic skyrmions in the conformal limit, slides We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate existence of minimizers for an explicit range of the model parameters when the energy is dominated by the exchange energy. We then investigate the conformal limit, in which only the exchange energy survives and identify the asymptotic profiles of the skyrmions as degree 1 harmonic maps from the plane to the sphere, together with their radii, angles and energies. A byproduct of our analysis is a quantitative rigidity result for degree ±1 harmonic maps from the two-dimensional sphere to itself. |
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| 11:00-11:50am Betsy Stovall (UW-Madison) Existence of extremizers for Fourier restriction operators We learn in first year graduate analysis that an operator from one Banach space to another is continuous if and only if the image of the unit ball is a bounded set. In this talk, we will discuss the question of whether this image has a point of maximal norm, in the specific context of certain Fourier restriction operators. |
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Last update: March 2, 2021 | |