PhD Candidate: Alessandro Marinelli
Mathematics, UBC

Wed 21 Mar 2018, 12:30pm
SPECIAL
Room 203, Graduate Student Centre, UBC

PhD Exam: The Unboundedness of the Maximal Directional Hilbert Transform

Room 203, Graduate Student Centre, UBC
Wed 21 Mar 2018, 12:30pm2:30pm
Details
Abstract:
In this dissertation we study the maximal directional Hilbert transform operator associated with a set U of directions in the ndimensional Euclidean space. This operator shall be denoted by H U. We discuss in detail the proof of the (p,p)weak unboundedness of H U in all dimensions n ≥ 2 and all Lebesgue exponents 1 < p < +∞ if U contains infinitely many directions in IR^n.
This unboundedness result for H U is an immediate consequence of a lower estimate for the (p,p) norm of the operatorH U that we prove if the cardinality of U (denoted by #U) is finite. In this case, we prove that the aforementioned operator norm is bounded from below by the square root of log(#U) up to a positive constant depending only on p and n, for any exponent p in the range 1 < p < +∞ and any n ≥ 2.
These results were first proved by G. A. Karagulyan in the case n = p = 2. The structure of our argument follows Karagulyan’s, but includes the results that are necessary for the extension of the lower estimate to the case 1 < p < +∞ and to all dimensions n ≥ 2.
Finally, a review of the scientific literature on H U and related topics is also included.
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PhD Candidate: Niki Myrto Mavraki
Mathematics, UBC

Wed 21 Mar 2018, 12:30pm
SPECIAL
Room 200, Graduate Student Centre, UBC

PhD Exam: Unlikely intersections and Equidistribution with a Dynamical Perspective

Room 200, Graduate Student Centre, UBC
Wed 21 Mar 2018, 12:30pm2:30pm
Details
Abstract:
In this thesis we investigate generalizations of a theorem by Masser and Zannier concerning torsion specializations of sections in a fibered product of two elliptic surfaces.
We consider the Weierstrass family of elliptic curves 𝐸𝐸𝑡𝑡∶𝑦𝑦2=𝑥𝑥3+𝑡𝑡 and points 𝑃𝑃𝑡𝑡(𝑎𝑎)=(𝑎𝑎,√𝑎𝑎3+𝑡𝑡)in 𝐸𝐸𝑡𝑡parametrized by nonzero 𝑡𝑡.
Given 𝛼𝛼,𝛽𝛽algebraic over 𝑄𝑄2 with rational ratio, we provide an explicit description for the set of parameters 𝑡𝑡=𝜆𝜆 such that 𝑃𝑃𝜆𝜆(𝛼𝛼) and 𝑃𝑃𝜆𝜆(𝛽𝛽) are simultaneously torsion for 𝐸𝐸𝜆𝜆. In particular, we prove that the aforementioned set is empty unless 𝛼𝛼/𝛽𝛽∈{−2,−1/2}. Furthermore, we show that this set is empty even when 𝛼𝛼/𝛽𝛽∉𝑄𝑄 provided that 𝛼𝛼 and 𝛽𝛽 have distinct 2adic absolute values and the ramification index of 𝛼𝛼/𝛽𝛽 over 𝑄𝑄2is coprime with 6.
Our methods are dynamical. Using our techniques, we derive a recent result of Stoll concerning the Legendre family of elliptic curves 𝐸𝐸𝑡𝑡:𝑦𝑦2=𝑥𝑥(𝑥𝑥−1)(𝑥𝑥−𝑡𝑡), which itself strengthened earlier work of Masser and Zannier by establishing, as a special case, that there is no complex parameter 𝑡𝑡=𝜆𝜆∉{0,1} such that the points with xcoordinates 𝑎𝑎 and 𝑏𝑏 are both torsion in 𝐸𝐸𝜆𝜆, provided 𝑎𝑎,𝑏𝑏 have distinct reduction modulo 2.
We also consider an extension of Masser and Zannier's theorem in the spirit of Bogomolov's conjecture.
Let 𝐸𝐸→𝐵𝐵 be an elliptic surface defined over a number field 𝐾𝐾, where 𝐵𝐵 is a smooth projective curve, and let 𝑃𝑃:𝐵𝐵→𝐸𝐸 be a section defined over 𝐾𝐾 with nonzero canonical height. We use Silverman's results concerning the variation of the NeronTate height in elliptic surfaces, together with complexdynamical arguments to show that the function 𝑡𝑡→ℎ𝐸𝐸𝑡𝑡(𝑃𝑃𝑡𝑡) satisfies the hypothesis of Thuillier and Yuan's equidistribution theorems. Thus, we obtain the equidistribution of points 𝑡𝑡∈𝐵𝐵 where 𝑃𝑃𝑡𝑡 is torsion. Finally, combined with Masser and Zannier's theorems, we prove the Bogomolovtype extension of their theorem. More precisely, we show that there is a positive lower bound on the height ℎ𝐴𝐴𝑡𝑡(𝑃𝑃𝑡𝑡), after excluding finitely many points 𝑡𝑡∈𝐵𝐵, for any `non special' section 𝑃𝑃 of a family of abelian varieties 𝐴𝐴→𝐵𝐵 that split as a product of elliptic curves.
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