Andy Zucker, Department of Pure Mathematics, University of Waterloo
"NIP"
We continue to read through Pierre Simon's a Guide to NIP Theories.
MC 5403
Yuming Zhao, Department of Pure Mathematics, University of Waterloo
"Positivity and sum of squares in quantum information"
A multivariate polynomial is said to be positive if it takes only non-negative values over reals. Hilbert's 17th problem concerns whether every positive polynomial can be expressed as a sum of squares of other polynomials. Many problems in math and computer science are closely connected to deciding whether a given polynomial is positive and finding certificates (e.g., sum-of-squares) of positivity. In quantum information, we are interested in noncommutative polynomials in *-algebras. A well-known theorem of Helton states that an element of a free *-algebra is positive in all *-representations if and only if it is a sum of squares. The theorem provides an effective way to determine if a given element is positive, by searching through sums of squares decompositions. In this talk, I'll present joint work with Arthur Mehta and William Slofstra in which we show that no such procedure exists for the tensor product of two free *-algebras: determining whether an element of such an algebra is positive is coRE-hard. Consequently, tensor products of free *-algebras contain elements which are positive but not sums of squares. I will also discuss the connetions to quantum information theory.
This seminar will be held both online and in person:
Cynthia Dai, Department of Pure Mathematics, University of Waterloo
"Combinatorial aspects of Schubert calculus on Grassmannian"
We will explain how to take intersection products of two Schubert classes using combinatorics. If time permits, we will define Schubert polynomials.
This seminar will be held both online and in person:
Freid Tong, Harvard University
"On complete Calabi-Yau metrics and a free-boundary Monge-Ampere equation"
Calabi-Yau metrics are Ricci-flat, Kähler metrics, and they are central objects in Kähler geometry. The existence problem for Calabi-Yau metrics on compact manifolds was answered by Yau in his solution of the Calabi conjecture. The situation in the non-compact setting is much more delicate, and many questions related to the existence and uniqueness of non-compact Calabi-Yau metrics remain unanswered. A major difficulty lies in the lack of suitable model metrics that model the asymptotics of the Calabi-Yau metric at spatial infinity. In this talk, I will give an introduction to this subject and discuss some joint work with T. Collins and S.-T. Yau, on a new relationship between non-compact Calabi-Yau metrics and a free-boundary Monge-Ampere equation, which allows us to resolve this problem of the lack of model metrics.
MC 5501
Huixi Li, Nankai University
"On Covering Systems of Polynomial Rings Over Finite Fields"
In 1950, Erd\H{o}s posed a question known as the minimum modulus problem on covering systems for $\mathbb{Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough in 2015. In this presentation, we will discuss the analogous minimum modulus problem for $\mathbb{F}_q[x]$. We proof that the smallest degree of the moduli in any covering system for $\mathbb{F}_q[x]$ of multiplicity $s$ is bounded by a constant depending only on $s$ and $q$. This is a joint work with Shaoyun Yi, Biao Wang, and Chunlin Wang.
Zoom: https://uwaterloo.zoom.us/j/98950813087?pwd=SEl1NlNqNHl0QzlYNGJzeDVla204QT09
Adam Jelinsky, University of Waterloo
"Properties of the Pseudo-randomness of a³ mod p²"
In this talk, I will be discussing methods and tactics used to quantify and understand the apparent pseudo-random distribution of a³ mod p², the evidence we have to fit the random model proposed, and the required steps in order to formally prove it.
MC 5403
Chatchai Noytaptim, Department of Pure Mathematics, University of Waterloo
"The Laplacian on the Berkovich projective line"
We discuss various examples in computing the Laplacian of functions of bounded differential variation on the Berkovich projective line. We follow closely Chapter 5 in “Potential Theory and Dynamics on the Berkovich Projective Line” by Baker and Rumely.
MC 5479
Anne Dranowski, University of Southern California
"Some spaces associated to KLR/W algebras"
We discuss two projects related to the KLR algebra which is a diagrammatic algebra categorifying representations of the quantum group U_q(sl_n). In one project, joint with Guo, Lauda and Manion, we *adapt* a spectral refinement of Khovanov homology to a foamy version thereof, and *extend* it to KLR-mod. In another project, joint with Leroux-Lapierre, we study an equivalence relating KLR-mod and quantizations of slices in the affine Grassmannian using canonical bases and characters. We hope to be as self-contained as possible up to a first course in algebra or topology.
M3 3127
Anwesh Ray, Chennai Mathematical Institute
"Diophantine stability for elliptic curves on average"
Let K be a number field and ℓ≥ 5 be a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety X/K at a prime ℓ. Under the hypothesis that all elliptic curves E/ℚ have finite Tate-Shafarevich group, we show that there is a positive density set of elliptic curves E/ℚ of rank 1, such that E/K is diophantine stable at ℓ. This result has implications to Hilbert's tenth problem for number rings. This is joint work with Tom Weston.
Zoom link: https://uwaterloo.zoom.us/j/2433704471?pwd=aXJoSDh0NDF0aFREbkthSnFBOUI4UT09
Andy Zucker, Department of Pure Mathematics, University of Waterloo
"Ultracoproducts of G-flows"
Given a topological group G, a G-flow is a continuous action of G on a compact Hausdorff space X. This talk will discuss a notion of ultracoproduct for G-flows, which arise from considering ultraproducts of commutative G-C*-algebras by Gelfand duality. We apply the construction to develop an understanding of the properties of various classes of subflows of a flow, i.e. minimal, topologically transitive, etc. For groups which are locally Roelcke precompact, ultracoproducts of G-flows lead to a well-behaved notion of weak containment for a wide class of G-flows, and in particular for all G-flows when G is locally compact. In joint work with Gianluca Basso, we apply ultracoproducts of G-flows to achieve a new characterization of those Polish groups G with the property that every minimal flow has a comeager orbit.
MC 5479