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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

Ty Ghaswala

More on automatic continuity

I will talk about some specific Polish groups arising naturally (for some definition of natural) from low-dimensional topology and dynamical systems. The talk will then continue, all on its own, to a discussion about automatic continuity.

MC 5403

Michael Xu

Applications of large sieve in variance analysis

Due to its analytic nature, large sieve is considered one of the popular sieves in many number theory estimates, free from many combinatorial constraints. Thanks to the same reason, large sieve is also considered one of the difficult sieves as it lacks combinatorial nature and a proper visualization of sieving.

In this talk, I will attempt to showcase both characteristics of this sieve with applications, demonstrating its capability, in process of proving results about distribution of primes in arithmetic progression, namely Barban-Davenport-Halberstam bound.

MC 5403

Jason Fang

Large sieve inequality and classical large sieve

As its name suggests, the large sieve inequality has many applications in sieve theory, but it is not easy to construct the idea about the techniques involved. In this talk I will prove the large sieve inequality and classical large sieve result using tricks such as Parseval identity and Cauchy-Schwarz Inequality.

MC 5403

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

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

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

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 

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. 

Speaker: Cynthia Dai

"Elliptic curves"

We will study elliptic curves, derive Weistrass short form, and study the law of addition.

You can access the handout at Height Study Seminar | Doing Meow-thematics (cynthiaddai.github.io)

MC 5417