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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University of Washington

Wed 21 Mar 2018, 3:10pm
Probability Seminar
LSK 460

On Lambertian reflections and stirring coffee

LSK 460
Wed 21 Mar 2018, 3:10pm4:10pm
Abstract
The Lambertian distribution, also known as Knudsen's Law, is a model for random reflections of light or gas particles from rough surfaces. I will present a mathematical "justification" of the Lambertian distribution. Then I will discuss a deterministic model inspired by stirring coffee. The analysis of the model will be partly deterministic, and partly based on the Lambertian distribution.
Joint work with O. Angel, M. Duarte, C.E. Gauthier, J. San Martin, and S. Sheffield.
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University of Oregon

Wed 21 Mar 2018, 3:15pm
Topology and related seminars
ESB 4133

A structure theorem for RO(C_2)graded cohomology

ESB 4133
Wed 21 Mar 2018, 3:15pm4:15pm
Abstract
Computations in RO(G)graded Bredon cohomology can be challenging and are not well understood, even for G=C_2, the cyclic group of order two. In this talk I will present a structure theorem for RO(C_2)graded cohomology with constant Z/2 coefficients that substantially simplifies computations. The structure theorem says the cohomology of any finite C_2CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. I will give some examples and sketch the proof, which depends on a Toda bracket calculation.
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Colorado State

Wed 21 Mar 2018, 4:00pm
SPECIAL
Algebraic Geometry Seminar / Number Theory Seminar
MATH 126

Algebraic intermediate Jacobians are arithmetic

MATH 126
Wed 21 Mar 2018, 4:00pm5:00pm
Abstract
Consider a smooth projective variety over a number field. The image of the associated (complex) AbelJacobi map inside the (transcendental) intermediate Jacobian is an abelian variety. We show that this abelian variety admits a distinguished model over the number field. Among other applications, this tool allows us to answer a recent question of Mazur; recover an old result of Deligne; and give new constructions of period maps over arithmetic bases.
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UBC Math

Fri 23 Mar 2018, 3:00pm
Department Colloquium
ESB 2012

Graduate Research Award: Clustering: a common thread between superresolution image analysis and cancer

ESB 2012
Fri 23 Mar 2018, 3:00pm4:00pm
Abstract
Clustering appears in many guises, playing important roles in diverse areas of cell biology. One such guise is the spatial clustering of proteins on the membrane of a cell. The ability of cell membrane proteins to cluster in response to stimuli is important to the normal function of many cells, but spontaneous, uncontrolled clustering can lead to cancer. Biologists are therefore keen to analyse protein clustering to better understand how cells function and gain insight into related diseases. This quest is assisted by superresolution microscopy techniques that enable single molecules to be imaged down to nanoscale precision. In this talk, I will outline StormGraph, a graphbased clustering algorithm that I have developed for the analysis of protein clustering in superresolution microscopy data. Using simulated data, I have found StormGraph to recover groundtruth clusters more accurately than current leading algorithms, and I have demonstrated its use on superresolution microscopy data from normal and cancerous Bcells, our antibodyproducing immune cells.
I will also provide a brief overview of how I intend to use clustering in multidimensional proteomic space to potentially improve personalized cancer therapies in the future. Tumours are heterogeneous populations of cells, and the activity of various signalling proteins can differ between cells within the same tumour. This intratumour heterogeneity is a key driver of resistance to cancer therapies, and should therefore be considered if trying to develop effective personalized therapies. I am working to develop suitable experiments and computational analysis to analyse this heterogeneity in Bcell tumours.
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UC Riverside

Mon 26 Mar 2018, 4:00pm
Algebraic Geometry Seminar
MATH 126

TBA

MATH 126
Mon 26 Mar 2018, 4:00pm5:00pm
Abstract
TBA
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Queen's University

Tue 27 Mar 2018, 2:00pm
SPECIAL
Number Theory Seminar / PIMS Seminars and PDF Colloquiums
ESB 4127

Ramification Theory for Arbitrary Valuation Rings in Positive Characteristic (talk 1)

ESB 4127
Tue 27 Mar 2018, 2:00pm3:00pm
Abstract
In classical ramification theory, we consider extensions of complete discrete valuation rings with perfect residue fields. We would like to study arbitrary valuation rings with possibly imperfect residue fields and possibly nondiscrete valuations of rank ≥ 1, since many interesting complications arise for such rings. In particular, defect may occur (i.e. we can have a nontrivial extension, such that there is no extension of the residue field or the value group).
We present some new results for ArtinSchreier extensions of arbitrary valuation fields in positive characteristic p. These results relate the “higher ramification ideal” of the extension with the ideal generated by the inverses of ArtinSchreier generators via the norm map. We also introduce a generalization and further refinement of Kato’s refined Swan conductor in this case. Similar results are true in mixed characteristic (0, p).
This is talk 1 of 2 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
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Arizona State University

Tue 27 Mar 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

ESB 2012
Tue 27 Mar 2018, 3:30pm4:30pm
Abstract
I will discuss some recent joint work with Lu Wang in which we prove that a shrinking gradient Ricci soliton which agrees to infinite order at spatial infinity with a generalized cylinder along some end must be isometric to the cylinder on that end. When the shrinker is complete, it must be globally isometric to the cylinder or else to a Z_2quotient. This work belongs to a larger program aimed at obtaining a structural classification of complete noncompact shrinking solitons.
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Colorado State University

Tue 27 Mar 2018, 4:00pm
Discrete Math Seminar
ESB 4127

An efficient Markov chain sampler for plane curves

ESB 4127
Tue 27 Mar 2018, 4:00pm5:00am
Abstract
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Note for Attendees
Latecomers will not be admitted.