Ross Willard, Pure Mathematics, University of Waterloo
"Natural dualities for finitely generated quasi-varieties - definitions and first results"
Rahim Moosa, Pure Mathematics, University of Waterloo
"Pseudo-finite sets and dimension, Part 2"
Having discussed the normalised counting measure on pseudo-finite sets and its application to Szemeredi Regularity, we now introduce the fine and coarse pseudo-finite dimensions for these sets, with an eye toward the Szemerdi-Trotter theorem.
Fei Hu, University of British Columbia
"Jordan property for algebraic groups in arbitrary characteristic"
Kübra Benli, University of Georgia
"On the number of small prime power residues"
Dan Ursu, Department of Pure Mathematics, University of Waterloo
"Relative C* - simplicity"
Catherine St-Pierre, University of Waterloo
"Borel fixed Ideals (in Krull dimension 0)"
Amanda Petcu, University of Waterloo
Some results on hypersymplectic structures
A conjecture of Simon Donaldson is that on a compact 4-manifold X^4 one can flow from a hypersymplecticstructure to a hyperkahler structure while remaining in the same cohomology class. To this end thehypersymplectic flow was introduced by Fine-Yao. In this thesis the notion of a positive triple on X^4 is used todefine a hypersymplectic and hyperkahler structure. Given a closed positive triple one can define either a closedG2 structure or a coclosed G2 structure on T^3 x X^4. The coclosed G2 structure is evolved under the G2Laplacian coflow. This descends to a flow of the positive triple on X^4, which is again the Fine-Yaohypersymplectic flow. In the second part of this thesis we let X^4 = R^4 \0 with a particular cohomogeneity oneaction. A hypersymplectic structure invariant under this action is introduced. The Riemann and Ricci curvaturetensors are computed and we verify in a particular case that this hypersymplectic structure can be transformed toa hyperkahler structure. The notion of a soliton for the hypersymplectic flow in this particular case is introducedand it is found that steady solitons give rise to hypersymplectic structures that can be transformed to hyperkahlerstructures. Some other soliton solutions are also discussed.
MC 5403
William Dan, University of Waterloo
A Characterization of Random, Left C.E. Reals
An immediate property of the halting probability of a prefix-free machine is that it is a left c.e. real. An easycorollary of the Kraft-Chaitin theorem is that the converse is true: any left c.e. real is the halting probability ofsome prefix-free machine. The most common example of a random real is Chaitin's omega, the haltingprobability of a universal prefix-free machine. In fact, it is a random left c.e. real. It is natural then to ask if theconverse holds in this case as well: that any random left c.e. real is the halting probability of some universalprefix-free machine. As it turns out, this is the case, and in this talk I will explain the concept used to solve thisquestion, Solovay reducibility, then prove the theorems demonstrating the converse. This talk follows sections9.1 and 9.2 of the Downey and Hirschfeldt book.
MC 5403
Rahim Moosa, University of Waterloo
Definable groups in CCM
I will survey what is known about the structure of definable groups in both the standard and nonstandard models of CCM.
MC 5479
Faisal Romshoo, University of Waterloo
Deformations of calibrations
We will look at a criterion for unobstructedness for calibrations and see when the corresponding moduli spacesform smooth manifolds, following the approach by Goto in https://arxiv.org/abs/math/0112197
MC 5403