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Mario Garcia Armas
Mathematics, UBC
Thu 19 Mar 2015, 12:30pm SPECIAL
One Time Event
Room 203 of the Graduate Student Centre (6371 Crescent Rd.), UBC
Doctoral Exam: Group Actions on Curves over Arbitrary Fields
Room 203 of the Graduate Student Centre (6371 Crescent Rd.), UBC
Thu 19 Mar 2015, 12:30pm-2:30pm

Details

This thesis consists of three parts. The common theme is finite group actions on algebraic curves defined over an arbitrary field k.

In Part I we classify finite group actions on irreducible conic curves defined over k. Equivalently, we classify finite (constant) subgroups of SO(q) up to conjugacy, where q is a nondegenerate quadratic form of rank 3 defined over k. In the case where k is the field of complex numbers, these groups were classified by F. Klein at the end of the 19th century. In recent papers of A. Beauville and X. Faber, this classification is extended to the case where k is arbitrary, but q is split. We further extend their results by classifying finite subgroups of SO(q) for any base field k of characteristic not 2 and any nondegenerate ternary quadratic form q.

In Part II we address the Hyperelliptic Lifting Problem (or HLP): Given a faithful G-action on the projective line defined over k and a double cover H of a finite group G, determine the conditions for the existence of a hyperelliptic curve C/k endowed with a faithful H-action that lifts the prescribed G-action on the projective line. In this thesis, we find a complete solution to the HLP in characteristic 0 for every faithful group action on the projective line and every exact sequence as above.

In Part III we determine whether, given a finite group G and a base field k of characteristic 0, there exists a strongly incompressible G-curve defined over k. Recall that a G-curve is an algebraic curve endowed with the action of a finite group G. A faithful G-curve C is called strongly incompressible if every dominant G-equivariant rational map of C onto a faithful G-variety is birational. We prove that strongly incompressible G-curves exist if G cannot act faithfully on the projective line over k. On the other hand, if G does embed into PGL(2,k), we show that the existence of strongly incompressible G-curves depends on finer arithmetic properties of k.
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Ed Kroc
Mathematics, UBC
Fri 20 Mar 2015, 4:00pm SPECIAL
One Time Event
Room 203 of the Graduate Student Centre (6371 Crescent Rd), UBC
Doctoral Exam: Kakeya-type Sets, Lacunarity, and Directional Maximal Operators in Euclidean Space
Room 203 of the Graduate Student Centre (6371 Crescent Rd), UBC
Fri 20 Mar 2015, 4:00pm-6:00pm

Details

Given a Cantor-type subset Ω of a smooth curve in d-dimensional Euclidean space, we construct random examples of Euclidean sets that contain unit line segments with directions from Ω and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. We also develop a notion of finite order lacunarity for direction sets in arbitrary d-dimensional space, and use it to extend our construction to direction sets Ω that are sublacunary according to this definition. This generalizes to higher dimensions a pair of planar results due to Bateman and Katz. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set Ω is unbounded on the Lebesgue spaces of finite exponent.
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Nishant Chandgotia
Mathematics, UBC
Tue 24 Mar 2015, 12:00pm SPECIAL
One Time Event
Room 126 of the Mathematics Bldg.
Doctoral Exams
Room 126 of the Mathematics Bldg.
Tue 24 Mar 2015, 12:00pm-2:00pm

Details

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Carmen Bruni
Mathematics, UBC
Mon 30 Mar 2015, 9:00am SPECIAL
One Time Event
Room 203 of the Graduate Student Centre (6371 Crescent Rd), UBC
Doctoral Exams
Room 203 of the Graduate Student Centre (6371 Crescent Rd), UBC
Mon 30 Mar 2015, 9:00am-11:00am

Details


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Felipe Garcia Ramos Aguilar
Mathematics, UBC
Tue 31 Mar 2015, 12:30pm SPECIAL
One Time Event
Room 203 of the Graduate Student Centre (6371 Crescent Road), UBC
Doctoral Exam: Randomness and Structure in Dynamical Systems: Different Forms of Sensitivity and Equicontinuity
Room 203 of the Graduate Student Centre (6371 Crescent Road), UBC
Tue 31 Mar 2015, 12:30pm-2:30pm

Details

In this thesis we study topological (continuous map on a compact metric space) and measure theoretical (measure preserving map on a probability space) dynamical systems.

Dynamical systems range from chaotic (random) to predictable (high structure). Structure and randomness can be represented with different forms of equicontinuity and sensitivity to initial conditions (sensitivity).

Inspired by the classical dichotomy between sensitivity and equicontinuity we define weak forms of topological and measure theoretical equicontinuity and strong forms of sensitivity for dynamical systems, and we study their relationships with spectral properties and sequence entropy. We also prove results of how measure theoretically equicontinuous cellular automata (a particular class of topological systems with close connections to computer science) behave in the long term.

The work of this thesis answers questions from - B. Scarpellini. Stability properties of flows with pure point spectrum. Journal of the London Mathematical Society, 2(3):451–464, 1982. - F. Blanchard and P. Tisseur. Some properties of cellular automata with equicontinuity points. Annales de l’Institut Henri Poincare (B) Probability and Statistics, 36(5):569 – 582, 2000.
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