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Robert Harris, University of Waterloo

Exotic constructions on covers branched over hyperplane arrangements

As a consequence of embedded surfaces and codimension two submanifolds coinciding in dimension four, many of the tools that are used in higher dimensions fail or are underwhelming when applied to 4-manifolds. For this reason, the development and advancement of techniques that are applicable to 4-manifolds are of particular interest and importance to low dimensional topologists. The general techniques of interest are those that either construct a 4-manifold in a novel way or those that provide ample control over the geometric data of the resulting 4-manifold.

In this talk, I will discuss my thesis, in which we investigate ways to construct 4-manifolds with positive signature. We also describe a construction that can guarantee the existence of algebraically interesting embedded symplectic submanifolds.

Specifically, we discuss how the combinatorial data of line arrangements and the algebraic data of their complements in rational complex surfaces can be utilized to construct symplectic 4-manifolds with arbitrarily large signatures through the method of branched coverings. In general, we not only show that these line arrangements can be used to provide asymptotic bounds for the existence of symplectic 4-manifolds but we also show that for any line arrangement, there exists symplectic branched covers with sufficiently nice geometric and topological properties. Namely, we show they contain embedded symplectic Riemann surfaces which carry their fundamental group.

Online presentation: contact r26harri@uwaterloo.ca for details on how to attend

Amanda Petcu, University of Waterloo

Cohomogeneity one solitons of the hypersymplectic flow

Given a hypersymplectic manifold X^4, one can give a flow of hypersymplectic structures. In this talk, we let X^4 be R^4 with an SO(4) action, and the hypersymplectic triple depend on three different functions h_k that depend solely on the radial coordinate. We will examine how the triple evolves under the hypersymplectic flow and given this initial structure, determine all possible solitons of the flow.

MC 5403

Larissa Kroell, University of Waterloo

Partial C*-dynamical systems and the ideal structure of partial reduced crossed products

We study C*-algebras stemming from partial C*-dynamical systems. We develop equivariant injective envelopes associated to such systems, which allow us to obtain canonical connections to enveloping actions as well as results on the ideal structure of partial crossed products.

We extend the theory of equivariant injective envelopes pioneered by Hamana in the 1980s to partial C*-dynamical systems. To do so, we introduce the category of generalized unital partial actions by allowing for *-automorphisms acting on families of special hereditary subalgebras. Utilizing properties of injective envelopes and the notion of an injective unitization of partial C*-dynamical systems, we argue that it suffices to consider unital objects in our category. This also allows us to connect our theory to Abadie's notion of enveloping actions leading to a canonical relationship of their G-injective envelopes.

Utilizing properties of injective envelopes, we introduce novel non-triviality conditions for partial *-automorphisms inspired by global C*-dynamics. We contrast this notion with existing conditions in the literature. Lastly, we study a non-commutative generalization of stabilizer subgroups for pseudo-Glimm ideals. In particular, we show that for Glimm ideals in the C*-injective envelope, these stabilizer subgroups are in fact amenable --- a result which is crucial for our main theorems regarding the ideal structure of partial reduced crossed products.

Finally, our main application of the theory of G-injective envelopes is a characterization of the ideal intersection property for partial C*-dynamical systems subject to a cohomological condition as a generalization of the result for global group actions. To state this generalization, we utilize the dynamical conditions introduced previously and generalize the notion of equivariant pseudo-expectations to partial C*-dynamical systems. We also give a sufficient intrinsic condition in terms non-commutative uniformly recurrent partial subsystems utilizing pseudo-Glimm ideals. As a consequence of our results, we obtain a full characterization of the ideal intersection property for partial actions on commutative C*-algebras in terms of freeness of the partial action on the spectrum of the G-injective envelope.

MC5403

Joey Lakerdas-Gayle, University of Waterloo

Effective Algebra 6

We will prove Dobritsa's Theorem following the monograph by Downey and Melnikov.

MC 5417

Kieran Mastel, University of Waterloo

The complexity of constraint system games with entanglement

Constraint satisfaction problems (CSPs) are a natural class of decision problems where one must decide whether there is an assignment to variables that satisfies a given formula. Schaefer's dichotomy theorem and its extension to all alphabets due to Bulatov and Zhuk, shows that CSP languages are either efficiently decidable or NP-complete. It is possible to extend CSP languages to quantum assignments using the formalism of nonlocal games. The recent MIP*=RE theorem of Ji, Natarajan, Vidick, Wright, and Yuen shows that the complexity class MIP* of multiprover proof systems with entangled provers contains all recursively enumerable languages. As a consequence, general succinctly presented CSPs are RE-complete. We show that a wide range of NP-complete CSPs become RE-complete when the players are allowed entanglement, including all boolean CSPs, such as 3SAT and 3-colouring. This implies that these CSP languages remain undecidable even when not succinctly presented.

Prior work of Grilo, Slofstra, and Yuen shows (via a technique called simulatable codes) that every language in MIP* has a perfect zero knowledge (PZK) MIP* protocol. The MIP*=RE theorem uses two-prover one-round proof systems. Hence, such systems are complete for MIP*. However, the construction in Grilo, Slofstra, and Yuen uses six provers, and there is no obvious way to get perfect zero knowledge with two provers via simulatable codes. This leads to a natural question: are there two-prover PZK-MIP* protocols for all of MIP*?

In this work, we show that every language in MIP* has a two-prover one-round PZK-MIP* protocol, answering the question in the affirmative. For the proof, we use a new method based on a key consequence of the MIP*=RE theorem, which is that every MIP* protocol can be turned into a family of boolean constraint system (BCS) nonlocal games. This makes it possible to work with MIP* protocols as boolean constraint systems. In particular, it allows us to use a variant of a CSP due to Dwork, Feige, Kilian, Naor, and Safra that gives a classical MIP protocol for 3SAT with perfect zero knowledge.

To prove our results, we develop a toolkit for analyzing the quantum soundness of reductions between constraint system (CS) games, which we expect to be useful more broadly. In this formalism, synchronous strategies for a nonlocal game correspond to tracial states on an algebra. We equip the algebra with a finitely supported weight that allows us to gauge the players' performance in the corresponding game using a weighted sum of squares. The soundness of our reductions hinges on guaranteeing that specific measurements for the players are close to commuting when their strategy performs well. To this end, we construct commutativity gadgets for all boolean CSPs and show that the commutativity gadget for graph 3-colouring due to Ji is sound against entangled provers. We define a broad class of CSPs that have simple commutativity gadgets. We show a variety of relations between the different ways of presenting CSPs as games. This toolkit also applies to commuting operator strategies, and our argument shows that every language with a commuting operator BCS protocol has a two prover PZK commuting operator protocol.

QNC 2101

9:30am -12:30pm

Kaleb Ruscitti, University of Waterloo

Embedding a family of moduli spaces of SL(2,C) bundles into projective spaces

The moduli space of polystable degree-0 SL(2,C) bundles on a compact connected Riemann surface of genus g>=2 is a Kähler manifold, and an open subset of the moduli space of semi-stable bundles, which is a projective variety of dimension 3g-3. Biswas and Hurtubise constructed a toric degeneration of this moduli space, meaning a family of moduli spaces over C whose fiber over 0 is a toric variety. The toric variety has a moduli interpretation as a space of framed parabolic bundles.

In this talk, I will describe the family and then describe how one can embed the entire family into P^N x C. This is the key step in a current project I am working on, about relating different geometric quantizations of the moduli space of SL(2,C) bundles.

MC 5403

Paul Cusson, University of Waterloo

Vector bundles over a complex torus

We will cover basic results about the complex geometry of a complex torus X, followed by a discussion of holomorphic vector bundles over X. An immediate result due to Hodge theory is the existence of complex bundles that don't admit holomorphic structures when the complex dimension of X is at least 2. We will thus focus on bundles whose Chern classes lie in the diagonal of the Hodge diamond and ask which ones can be holomorphic.

MC 5403

Joey Lakerdas-Gayle, University of Waterloo

Effective Algebra 5

We will continue learning about computable Abelian groups following the monograph by Downey and Melnikov.

MC 5501

Rachael Alvir, University of Waterloo

More Torsion-Free Abelian Groups

We will continue learning about torsion-free Abelian Groups following the Monograph by Downey and Melnikov.

MC 5417

Sean Lee, University of Waterloo

Dynamics of the Rado graph II

We continue our discussion about the topological dynamics of the automorphism group of the Rado graph.

MC 5417