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Kieran Mastel, Department of Pure Mathematics, University of Waterloo

"Introduction to the PCP theorem"

This is the first meeting of a working seminar that will take place every Wednesday 1:30PM-3:00PM in QNC 1201. The PCP theorem from computational complexity theory was a important part of the groundbreaking MIP*=RE result, and has very strong connections to stability problems for representations of algebras and groups. We plan to study this theorem, the quantum PCP conjecture, and their connections to (approximate) representations and stability.  The first meeting will be organizational and will include an introductory talk by Kieran Mastel on the PCP theorem.

QNC 1201

Panagiotis Dimakis, Université du Québec à Montréal, CIRGET

"The moduli space of solutions to the dimensionally reduced Kapustin-Witten equations on $\Sigma\times\mathbb{R}_+$"

Since their introduction in 2006, the Kapustin-Witten (KW) equations have become the subject of a number of conjectures. Given a knot $K$ embedded in a closed $3$-manifold $Y$, the most prominent conjecture predicts that the number of solutions to the KW equations on $Y\times\mathbb{R}_+$ with boundary conditions determined by the embedding and with fixed topological charge, is a topological invariant of the knot. A major obstacle with this conjecture is the difficulty of constructing solutions satisfying these boundary conditions. In this talk we assume $Y\cong \Sigma\times\mathbb{R}_+$ and study solutions to the dimensionally reduced KW equations with the required boundary conditions. We prove that the moduli spaces are diffeomorphic to certain holomorphic lagrangian sub-manifolds inside the moduli of Higgs bundles associated to $\Sigma$. Time permitting, we explain how one could use this result to construct knot invariants.

MC 5417

Kiseok Yeon, Purdue University

"The Hasse principle for random homogeneous polynomials in thin sets"

In this talk, we introduce a framework via the circle method in order to confirm the Hasse principle for random homogeneous polynomials in thin sets. We first give a motivation for developing this framework by providing an overall history of the problems of confirming the Hasse principle for homogeneous polynomials. Next, we provide a sketch of the proof of our main result and show a part of the estimates used in the proof. Furthermore, by using our recent joint work with H. Lee and S. Lee, we discuss the global solubility for random homogeneous polynomials in thin sets.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09

Amy Huang, Auburn University

"Matrix Multiplication Complexity: Tensor Geometry and Commutative Algebra"

Tensors are just multi-dimensional arrays. Tensor decomposition also has a lot of applications in data analysis, physics, and other areas of science. I will survey my recent two results about matrix multiplication complexity and classification of special tensors. The first result computes the border rank of 3 X 3 permanent, which is important in the theory of matrix multiplication complexity. The second result classifies linear spaces of matrices of bounded rank 4, making progress on an old problem that has been open for decades in linear algebra society. I will also briefly discuss how the role of commutative algebra, algebraic geometry, and representation theory comes into the picture. 

Zoom link: https://uwaterloo.zoom.us/j/2433704471?pwd=aXJoSDh0NDF0aFREbkthSnFBOUI4UT09

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory II"

We will continue our discussion of forcing in computable structure theory.

MC 5479

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory I"

In this talk we give a basic introduction to computable structure theory. 

MC 5479

Changho Han, Department of Pure Mathematics, University of Waterloo

"Extending the torelli map to alternative compactifications of the moduli space of curves"

It is well-known that the Torelli map, that turns a smooth curve of genus g into its Jacobian (a principally polarized abelian variety of dimension g), extends to a map from the Deligne—Mumford moduli of stable curves to the moduli of semi-abelic varieties by Alexeev. Moreover, it is also known that the Torelli map does not extend over the alternative compactifications of the moduli of curves as described by the Hassett—Keel program, including the moduli of pseudostable curves (can have nodes and cusps but not elliptic tails). But it is not yet known whether the Torelli map extends over alternative compactifications of the moduli of curves described by Smyth; what about the moduli of curves of genus g with rational m-fold singularities, where m is a positive integer bounded above? As a joint work in progress with Jesse Kass and Matthew Satriano, I will describe moduli spaces of curves with m-fold singularities (with topological constraints) and describe how far the Torelli map extends over such spaces into the Alexeev compactifications.

MC 5417

AJ Fong, Department of Pure Mathematics, University of Waterloo

"Affine Schemes"

We will introduce affine schemes, the building blocks of schemes and a generalisation of affine varieties, and discuss the interesting and nontrivial geometry that can happen in them. We will briefly describe some sheaf theory in the process. This talk closely follows section I.1 of Eisenbud-Harris.

MC 5417

Benoit Charbonneau, Department of Pure Mathematics, University of Waterloo

"Deformed Hermitian-Yang-Mills equation"

The Deformed Hermitian-Yang-Mills equation has been an intense topic of study in the recent past. I will describe the equation, the concept of central charge pertinent in this story, and various conjectures and progress that has been made.

MC 5403

Jason Bell, Department of Pure Mathematics, University of Waterloo

"Sparse subsets of the reals"

We look at the first-order theory of the real numbers augmented by a predicate X that is in some natural sense self-similar with respect to a positive integer base. We show that there is a dichotomy: either we can define a Cantor set in our structure or our expansion of the reals is interdefinable with the real numbers augmented by a set of the form {1/r, 1/r^2, 1/r^3, …} for some integer r>=2.  In the latter case, this is equivalent to the structure having NIP and NTP_2.  This is joint work with Alexi Block Gorman.

MC 5479