Events

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

Faisal Romshoo, University of Waterloo

Constructing calibrated submanifolds through evolution equations

I will talk about how we can construct examples of calibrated submanifolds using the techniques of evolution equations. We will begin by defining the ideas involved in coming up with these evolution equations and then look at some of the examples of calibrated submanifolds that are constructed this way, following arXiv:math/0008021, arXiv:math/0008155, arXiv:math/0010036 and arXiv:math/0401123.

MC 5403

Xuemiao Chen, University of Waterloo

On the space of lines

I will make a story about the space of oriented lines in the three dimensional Euclidean space.

MC 5403

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

Jiahao Hu, Yau Mathematical Sciences Center, Tsinghua University

Homotopy theoretical holomorphic invariants of complex manifolds

In this talk, I will begin by presenting a method for extracting new holomorphic invariants of a complex manifold from its de Rham algebra of complex-valued differential forms. These invariants can be seen as refined versions of the complexified homotopy groups. I will then explore their potential connections with Hermitian geometry. Specifically: (1) the non-abelian part (refined fundamental group) should be related to a generalization of Higgs bundle; (2) the abelian part (refined higher homotopy) may provide new tools for studying the geometry of holomorphic mappings.

MC 5417

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

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

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

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

Joey Lakerdas-Gayle, University of Waterloo

Effective Algebra 6

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

MC 5417