webnotice

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

Speaker: Benoit Charbonneau

"Maple for differential geometry"

While we are certainly competent to do with pen and paper the myriad of computations required by our research, refereeing and our supervision work, I find that using tools can improve speed and accuracy and reduce frustration. I will share some principles and illustrate using Maple, including packages useful for differential geometry: difforms, DifferentialGeometry, and Clifford. Code displayed for this presentation can be found at https://git.uwaterloo.ca/bcharbon/maple-demos

MC 5417

Research Area: Algebraic Geometry/Number Theory

Speaker: Cynthia Dai

"Group Varieties"

Abstract: 

We will define group varieties and study their basic properties. 

You can access the handout at https://cynthiaddai.github.io/2024/05/17/HeightSeminar/

MC 5417

Speaker: Cynthia Dai

"Global Heights"

In this talk, we will wrap up local heights with respect to a presentation of a line bundle, then define global heights. If time permits, we will also talk about Weil heights.

MC 5417

Speaker: Utkarsh Bajaj

"Klein's icosahedral function"

Can we define a rational function on the sphere? Sure we can. Can we define a rational function on the sphere so that it is invariant under the rotational symmetries under the icosahedron? Yes - by embedding the icosahedron in the Riemann sphere (and then doing some algebra). We then show how this beautiful function reveals connections between the symmetries of the icosahedron and the E8 lattice  - the lattice that gives the most efficient packing of spheres in 8 dimensions!

MC 5501