Events

Utkarsh Bajaj

An introduction to the Mckay correspondence

The McKay correspondence is a bijection between the finite subgroups of SL(2,C) and the Dynkin diagrams of the type  A_r, D_r, E_6, E_7, E_8. One bijection takes a subgroup G, constructs the orbit space C^2/G, resolves the singularities by inserting Riemann spheres multiple times, sees how the spheres intersect, and then constructs a graph to represent this information. Another bijection constructs irreducible representations of G. We will see how these bijections are related.

MC 5417

Filip Milidrag

The relation between the Wythoff construction and abstract polytopes

In this talk we will use the Wythoff construction of a geometric polytope to describe its face lattice and then use this to make the connection between geometric polytopes and the notion of an abstract polytope. We will then go on to speak a bit about abstract polytopes and some related definitions.

MC 5417

Aiden Suter, Department of Pure Mathematics, University of Waterloo 

“Mathematical aspects of Higgs and Coulomb branches” 

3d mirror symmetry is a duality between topological twists of 3 dimensional quantum field theories (QFTs) with N=4 supersymmetry. One of the most salient features of this duality is the symplectic duality between the branches of the moduli space of vacua of the full physical theory know as the “Higgs” and “Coulomb” branches. These branches are singular hyperkahler varieties that are interchanged under the duality. In this talk, I will primarily discuss two results regarding these varieties:

The first utilises a construction due to Costello and Gaiotto allowing one to associate a vertex operator algebra (VOA) to certain boundary conditions of these twisted QFTs and it has been conjectured that the associated variety of this VOA is the Higgs branch of the theory. In this talk I will outline a proof of this conjecture in the case of U(1) gauge theory acting on n>3 hypermultiplets, building on prior work of Beam and Ferrari who conjectured that the boundary VOA is the simple quotient of the psl(n|n) affine VOA.

In the second part of this talk I will outline a construction of a tilting generator for the derived category of sheaves on T*Gr(2,4). This space is the Coulomb branch for a certain quiver gauge theory and the construction is a realisation of a result due to Webster who proved the existence of such a tilting generator. In the case of quiver gauge theories such as this, the Coulomb branch algebra can be described in terms of a cyldrinical KLRW algebra, a type of diagrammatic algebra. Using these diagrammatic methods we explicitly describe the tilting generator and find that it differs to those previously known in the literature.

Remote - contact Ben Webster for the Zoom link. 

Aiden Suter

Mathematical aspects of Higgs and Coulomb branches

3d mirror symmetry is a duality between topological twists of 3 dimensional quantum field theories (QFTs) with N=4 supersymmetry. One of the most salient features of this duality is the symplectic duality between the branches of the moduli space of vacua of the full physical theory know as the “Higgs” and “Coulomb” branches. These branches are singular hyperkahler varieties that are interchanged under the duality. In this talk, I will primarily discuss two results regarding these varieties:

The first utilises a construction due to Costello and Gaiotto allowing one to associate a vertex operator algebra (VOA) to certain boundary conditions of these twisted QFTs and it has been conjectured that the associated variety of this VOA is the Higgs branch of the theory. In this talk I will outline a proof of this conjecture in the case of U(1) gauge theory acting on n>3 hypermultiplets, building on prior work of Beam and Ferrari who conjectured that the boundary VOA is the simple quotient of the psl(n|n) affine VOA.

In the second part of this talk I will outline a construction of a tilting generator for the derived category of sheaves on T*Gr(2,4). This space is the Coulomb branch for a certain quiver gauge theory and the construction is a realisation of a result due to Webster who proved the existence of such a tilting generator. In the case of quiver gauge theories such as this, the Coulomb branch algebra can be described in terms of a cyldrinical KLRW algebra, a type of diagrammatic algebra. Using these diagrammatic methods we explicitly describe the tilting generator and find that it differs to those previously known in the literature.

Join online at: https://pitp.zoom.us/j/97622507197

Yuming Zhao, Department of Pure Mathematics, University of Waterloo 

“Tsirelson's Bound and Beyond: Verifiability and Complexity in Quantum Systems” 

Suppose we have a physical system consisting of two separate labs, each can mark several measurements. If the two labs are entangled, then their measurement statistics can be correlated in surprising ways. In general, we do not directly see the entangled state and measurement operators, only the resulting correlations. There are typically many different models achieving a given correlation, hence it is remarkable that some correlations have a unique quantum model. A correlation with this property is called a self-test. In the first part of this thesis, we give a new definition of self-testing in terms of states on C*-algebras. We show that this operator-algebraic definition of self-testing is equivalent to the standard one and naturally extends to the commuting operator framework for nonlocal correlations. We also give an operator-algebraic formulation of robust self-testing in terms of tracial states on C*-algebras.

Self-testing provides a powerful tool for verifying quantum computations. In the second part of this thesis, we propose a new model of delegated quantum computation where the client trusts only its classical processing and can verify the server's quantum computation, and the server can conceal the inner workings of their quantum devices. This delegation protocol also yields the first two-prover one-round zero-knowledge proof systems of QMA.

Mathematically, bipartite measurements can be modeled by the tensor product of free *-algebras. Many problems for nonlocal correlations are closely related to deciding whether an element of these algebras is positive and finding certificates of positivity. In the third part of this thesis, we show that it is undecidable (coRE-hard) to determine whether a noncommutative polynomial of the tensor product of free *-algebras is positive.

QNC 2101 

Colin Jahel (TU Dresden)

When invariance implies exchangeability (and what it means for invariant Keisler measures)

Let M be a countable model-theoretic structure. We study the actions of Aut(M) on spaces of expansions of M and more precisely, the invariant probability measures under this action. In particular we are interested in understanding when Aut(M)-invariance actually implies S_\infty invariance. We obtain a nice classification for many classical structures. Finally we connect this to invariant Keisler measures, showing how for many structures, they must be S_\infty-invariant. We use this fact to illustrate the difference between two notion of smallness for formulas, forking and universal measure zero.

MC 5403

Faisal Romshoo

Perspectives on the moduli space of torsion-free $\textrm{G}_2$-structures

Joyce showed that the moduli space of torsion-free $\textrm{G}_2$-structures for a compact $7$-manifold forms a non-singular smooth manifold. In this talk, we consider the action of gauge transformations of the form $e^{tA}$ where $A$ is a $2$-tensor, on the space of torsion-free $\textrm{G}_2$-structures. This gives us a new framework to study the moduli space. 

We will see that a $\textrm{G}_2$-structure $\widetilde{\varphi} = P^*\varphi$ acted upon by a gauge transformation $P = e^{tA}$ is infinitesimally torsion-free if and only if  $A \diamond \varphi$ is harmonic and if $A$ satisfies a ``gauge-fixing" condition, where $A \diamond \varphi$ is a special type of $3$-form. This may be the first step in giving an alternate proof of the fact that the moduli space forms a manifold in our framework of gauge transformations.

MC 5501

Spencer Whitehead (Duke University)

An introduction to the Nahm transform and construction of instantons on tori

A Nahm transform recognizes the moduli space of instantons in some setting as an isometric 'dual space'. In this sense the Nahm transform is a 'nonlinear Fourier transform'. In this talk, I will give an introduction to Nahm transforms, sketching from two different points of view the classical Nahm transform of hermitian bundles over 4-tori. Along the way, we will develop a zoo of instanton examples in all ranks using constructions from differential and complex geometry.

MC 5417

Lucia Martin Merchan

Closed G2 manifolds with finite fundamental group

In this talk, we construct a compact closed G2 manifold with b1=0 using orbifold resolution techniques. Then, we study some of its topological properties: fundamental group, cohomology algebra, and formality.

MC 5417

Jacques Van Wyk

An Introduction to Generalised Geometry

Generalised geometry is a field in differential and complex geometry in which one views the direct sum TM ⊕ T*M instead of TM as the bundle associated to a manifold M. Generalised geometry has seen great success in acting as a unifying framework in which structures defined on TM and T*M can be viewed as specific instances of structures defined on TM ⊕ T*M. For example, almost complex structures and pre-symplectic structures can both be viewed as generalised almost complex structures, a certain kind of automorphism of TM ⊕ T*M.

In this talk, I will give an introduction to generalised geometry. I will show TM ⊕ T*M comes with an intrinsic non-degenerate bilinear form. I will introduce the Dorfman bracket on Γ(TM ⊕ T*M), an analogue of the Lie bracket, which together with the aforementioned bilinear form gives TM ⊕ T*M the structure of a Courant algebroid. I will define generalised almost complex structures in this setting, and show how almost complex structures and pre-symplectic structures can be viewed as generalised almost complex structures. I will introduce generalised metrics and generalised connections, and if time permits, I will discuss integrability of generalised almost complex structures in terms of generalised connections, and/or discuss the analogue of the Levi-Civita connection and what complications it comes with.

MC 5417