The theory of q-analogues is important in both combinatorics and the study of hypergeometric series. Roughly speaking, the q-analogue of a mathematical object (which could be a number or a theorem or ...) is another object depending on a parameter q which reduces to the original object when q=1. This talk will be a gentle introduction to q-analogues. No background will be assumed.

Title: Proof of average-case #P- hardness of random circuit sampling with some robustness, and a protocol for blind quantum computation

Date: 09/05/2019

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

A one-parameter unitary-valued interpolation between any two unitary matrices (e.g., quantum gates) is constructed based on the Cayley transformation. We prove that this path induces probability measures that are arbitrarily close to the Haar measure and prove the simplest known average-case # P -hardness of random circuit sampling (RCS). RCS is the task of sampling from the output distribution of a quantum circuit whose local gates are random Haar unitaries, and is the lead candidate for demonstrating quantum supremacy in the "noisy intermediate scale quantum (NISQ)" computing era. Here we also prove exp(-Θ(n^4 )) robustness with respect to additive error. This overcomes issues that arise for extrapolations based on the truncations of the power series representation of the exponential function. (Dis)Proving the quantum supremacy conjecture requires an extension of this analysis to noise that is polynomially small in the system's size. This remains an open problem. Lastly, an efficient and private protocol for blind quantum computation is proposed that uses the Cayley deformations proposed herein for encryption. This is an efficient protocol that only requires classical communication between Alice and Bob.
** The talk is self-contained and does not require any pre-req beyond basic linear algebra (e.g, knowing what a unitary matrix is).

I'll discuss a forthcoming paper studying families of $G_2$-instantons on $S^7$, focusing on those which are obtained by pulling back asd instantons on $S^4 $ via the quaternionic Hopf fibration. In the charge-1 case this yields a smooth and complete 15-dimensional family. The situation for higher charge is more complicated, but we are able to compute all the infinitesimal deformations.

Title: Localization of Gaussian disordered systems at low temperature

Date: 09/05/2019

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

The fundamental premise of statistical mechanics is that a physical system's state is random according to some probability measure, which is determined by the various forces of interaction between the system's constituent particles. In the ``disordered" setting, these interactions are also random (meant to capture the effect of a random medium), meaning the probability measure is itself a random object. This setting includes several of the models most widely studied by mathematical physicists, such as the Random Energy Model, the Sherrington--Kirkpatrick spin glass, and directed polymers. The most intriguing part of their phase diagrams occurs at low temperature, when the measure concentrates, or "freezes", on energetically favorable states. In general, quantifying this phenomenon is especially challenging, in large part due to the extra layer of randomness created by the disorder. This talk will describe recent progress on this question, leading us to some conjectures on further open problems. (Joint work with Sourav Chatterjee)

Title: Applications of Constructible Sheaves to Symplectic Topology

Date: 09/05/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

My goal is to explain a few applications of constructible sheaves to symplectic topology through examples that we can calculate together on the board.
In particular, I would like to explain how sheaves relate to: 1) Legendrian knot invariants, 2) cluster varieties, 3) nonfillability results for Legendrian surfaces.

Come learn what AMS is all about, what events are scheduled for this year, and meet your student community! This event is for ALL members, new and returning.
We'd love it if you could bring a snack or dish to share if you're able to.
We are also looking to fill two eboard positions: secretary and treasurer! We will discuss more about these positions on Tuesday and would love to hear from you if you're interested.

Title: Gröbner basis and the Ideal Membership problem

Date: 09/09/2019

Time: 4:30 PM - 5:30 PM

Place: C304 Wells Hall

We know from the Hilbert Basis Theorem that any ideal in a polynomial ring over a field is finitely generated. However, there remains question as to the best generators to choose to describe the ideal. Are there generators for a polynomial ideal $I$ that make it easy to see if a given polynomial $f$ belongs to $I$? For instance, does $2x^2z^2+2xyz^2+2xz^3+z^3-1$ belong to $I=(x+y+z, xy+xz+yz, xyz−1)$? Deciding if a polynomial is in an ideal is called the Ideal Membership Problem. In polynomial rings of one variable, we use long division of polynomials to solve this problem. There is a corresponding algorithm for $K[x_1,\ldots, x_n]$, but because there are multiple variables and multiple divisors, the remainder of the division is not unique. Hence a remainder of $0$ is a sufficient condition, but not a necessary condition, to determine ideal membership. However, if we choose the correct divisors, then the remainder is unique regardless of the order of the divisors. These divisors are called a Gröbner basis. In our talk we will define the Gröbner basis and see how it solves the Ideal Membership Problem.

Title: Introduction to Free Products of von Neumann Algebras

Date: 09/10/2019

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

In this learning seminar, I will give an introduction to the free product construction for von Neumann algebras, which is the direct analogue of a free product for groups. Moreover, it defines the non-commutative independence relation most frequently used in free probability. No prior knowledge of von Neumann algebras will be necessary.

Title: Combinatorial interpretations of Lucas analogues

Date: 09/11/2019

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

The Lucas sequence is a sequence of polynomials in $s,t$ defined recursively by $\{0\}=0$, $\{1\}=1$, and $\{n\}=s\{n-1\}+t\{n-2\}$ for $n\ge2$. On specialization of $s$ and $t$ one can recover the Fibonacci numbers, the nonnegative integers, and the $q$-integers $[n]_q$. Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of $n$ in the expression with $\{n\}$. It is then natural to ask if the resulting rational function is actually a polynomial in $s$ and $t$ and, if so, what it counts. Using lattice paths, we give combinatorial models for Lucas analogues of binomial coefficients. We also consider Catalan numbers and their relatives, such as those for finite Coxeter groups. This is joint work with Curtis Bennett, Juan Carrillo, and John Machacek.

Title: Rough solutions to the three-dimensional compressible Euler equations with vorticity and entropy

Date: 09/11/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We prove a series of intimately related results tied to the regularity and geometry of solutions to the three-dimensional compressible Euler equations.
The solutions are allowed to have nontrivial vorticity and entropy, and an arbitrary equation of state with positive sound speed. The central theme is that under low regularity assumptions on the initial data, it is possible to avoid, at least for short times, the formation of shocks. Our main result is that the time of classical existence can be controlled under low regularity assumptions on the part of the initial data associated with propagation of sound waves in the fluid. Such low regularity assumptions are in fact optimal. To implement our approach, we derive several results of independent interest: (i) sharp estimates for the acoustic geometry, which in particular capture how the vorticity and entropy interact with the sound waves; (ii) Strichartz estimates for quasilinear sound waves coupled to vorticity and entropy; (iii) Schauder estimates for the transport-div-curl-part of the systems. Compared to previous works on low regularity, the main new feature of our result is that the quasilinear PDE system under study exhibit multiple speeds of propagation. In fact, this is the first result of its kind for a system with multiple characteristic speeds. An interesting feature of our proof is the use of techniques that originated in the study of the vacuum Einstein equations in general relativity.

Title: Free products of finite-dimensional von Neumann algebras

Date: 09/12/2019

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

I will present joint work with Brent Nelson, where we classify the structure of free products of von Neumann algebras equipped with arbitrary states. Our techniques use our other joint work of assigning a von Neumann algebra associated to a weighted graph. I will discuss this work and how it leads to computing finite-dimensional free products.

We study two different invariants of framed oriented links. Augmentations are rank one representations of a non-commutative algebra, whose definition is motivated by Floer homology. Sheaves in microlocal theory can be thought of as generalizations of link group representations. We will demonstrate two constructions going back and forth between these invariants. We will also tell a motivating story behind the scene, using SFT and microlocalization correspondence in symplectic topology.

Speaker: Jiangguo (James) Liu, Colorado State University

Title: Developing Finite Element Solvers for Poroelasticity in the Two-field Approach

Date: 09/13/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

This talk presents results from our recent efforts for reviving the 2-field approach (fluid pressure and solid displacement) for numerically solving poroelasticity problems. We choose quadrilateral and hexahedral meshes for spatial discretization since they are equally flexible in accommodating complicated domain geometry but involve less unknowns, compared to simplicial meshes. The Darcy equation is solved for fluid pressure by the novel weak Galerkin finite element methods, which establish the discrete weak gradient and numerical velocity in the Arbogast-Correa spaces. The elasticity equation is solved for solid displacement by the enriched Lagrangian elements, which were motivated by the Bernardi-Raugel elements for Stokes flow. These two types of finite elements are coupled through the implicit Euler temporal discretization to solve poroelasticity. Numerical experiments on benchmarks will be presented to show that the new solvers are locking-free. Implementation on deal.II will be discussed also. This talk is based on a series of joint work with several collaborators.

In this talk we will define P.D. rings, which are triples consisting a ring, an ideal of the ring and a map on an ideal mimicking $x^n/n!$. We will give some examples of P.D. rings and discuss their properties. Then we will use the P.D. structures to define the crystalline site of schemes and crystals. If time admits we will talk about some examples of crystals and explain why we care about them.

Title: Legendrian knots and augmentation varieties

Date: 09/17/2019

Time: 3:00 PM - 4:00 PM

Place: C204A Wells Hall

We begin with a gentle introduction to Legendrian knot and its invariant theory. We will define the Chekanov-Eliashberg different graded algebra and augmentations associated to the dga. We also present an example where the augmentation variety is a cluster variety.

In this talk we will introduce perfectoid fields and tilting. Perfectoid fields provide the the correct base scheme for perfectoid spaces. Tilting is a fundamental tool that will let us lift characteristic $0$ results to characteristic $p$ results. For example, if $K$ is a characteristic $0$ perfectoid field and $K^{\flat}$ is a tilt of $K$ then $K^{\flat}$ is a characteristic $p$ field; $K^{\circ}/K^{\circ\circ}\cong K^{\flat \circ}/K^{\flat\circ\circ}$; if $[L:K]$ is finite then $[L^{\flat}:K^{\flat}]=[L:K]$ (in particular, $L$ is perfectoid); and there is an equivalence of categories between finite étale covers of $K$ and finite étale covers of $K^{\flat}$ via $L\mapsto L^{\flat}$.
This talk will not require any material beyond first-year graduate algebra. However, the sophistication required may be higher. To make this talk as accessible as possible, we will include numerous examples.

Speaker: Scott Atkinson, University of California, Riverside

Title: Tracial stability and related topics in operator algebras

Date: 09/19/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

We will discuss the notion of tracial stability for operator algebras. Morally, an algebra A is tracially stable if approximate homomorphisms on A are near honest homomorphisms on A. We will discuss several examples and non-examples of tracially stable algebras including certain graph products (simultaneous generalization of free and tensor products) of C*-algebras. We will also discuss properties closely related to tracial stability that provide new characterizations of amenability. Parts of this talk are based on joint work with Srivatsav Kunnawalkam Elayavalli.

Title: Introduction to the Energy Identity for Yang-Mills

Date: 09/19/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In this talk we give an introduction to the analysis of the Yang-Mills equation in higher dimensions. In particular, when studying sequences of solutions we will study the manner in which blow up can occur, and how this blow up may be understood through the classical notions of the defect measure and bubbles. The energy identity is an explicit conjectural relationship relating the energy density of the defect measure at a point to the bubbles which occur at that point. This talk is introductory and we will spend most of our time understanding the words of this abstract. If time permits we will briefly discuss the ideas needed to prove this conjecture and the related $W^{2,1}$-conjecture. The work is joint with Daniele Valtorta.

Title: Splitting Criteria for Vector Bundles on $\mathbb{P}^n$

Date: 09/23/2019

Time: 4:30 PM - 5:30 PM

Place: C304 Wells Hall

Grothendieck's Theorem says that any vector bundle on $\mathbb{P}^1$ can be decomposed as a finite sum of line bundles. In this talk, we will discuss a generalization of this theorem: Horrocks Splitting Criterion. This criterion completely describes when a vector bundle on $\mathbb{P}^n$ splits as a sum of line bundles. We will then discuss an open conjecture of Hartshorne. If time permits, we will also consider the similar question of classifying when a vector bundle on $\mathbb{P}^n$ decompose as line bundles and twists of the tangent bundle.

Title: The higher dimensional algebra of matrix product operators and quantum spin chains

Date: 09/26/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

In the context of 1D quantum spin chains, matrix product operators provide a way to study non-local operators such as translation in terms of quasi-local information. They have been used to describe a generalized form of symmetry for 1D systems on the boundary of 2D topological phases. In this talk, we will introduce some concepts of higher dimensional algebra, and a broad hypotheses about higher categories and spatially extended quantum systems. We will then explain how the collection of matrix product operators assembles into a higher (symmetric monoidal 2-) category, and discuss some implications of this. Based on joint work with David Penneys.

Title: Holonomy perturbations of the Chern-Simons functional for lens spaces

Date: 09/26/2019

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

We describe a scheme for constructing generating sets for Kronheimer and Mrowka's singular instanton knot homology for the case of knots in lens spaces. The scheme involves Heegaard-splitting a lens space containing a knot into two solid tori. One solid torus contains a portion of the knot consisting of an unknotted arc, as well as holonomy perturbations of the Chern-Simons functional used to define the homology theory. The other solid torus contains the remainder of the knot. The Heegaard splitting yields a pair of Lagrangians in the traceless $SU(2)$-character variety of the twice-punctured torus, and the intersection points of these Lagrangians comprise the generating set that we seek. We illustrate the scheme by constructing generating sets for several example knots. Our scheme is a direct generalization of a scheme introduced by Hedden, Herald, and Kirk for describing generating sets for knots in $S^3$ in terms of Lagrangian intersections in the traceless $SU(2)$-character variety for the 2-sphere with four punctures.

Title: Root systems - a powerful tool for classification

Date: 09/30/2019

Time: 4:30 PM - 5:30 PM

Place: C304 Wells Hall

Root systems arose historically as a tool for classifying semisimple Lie algebras, but they can also be understood without that context. I will describe several concrete examples of root systems, with plenty of pictures. I will describe how to associate a special graph called a Dynkin diagram to a root system, and briefly describe the classification of root systems. If time allows, I will describe some of the applications to classifying semisimple Lie algebras and reductive algebraic groups. All you need to know to understand my talk is how to compute dot products on $\mathbb{R}^n$.