Topological quantum field theories (TQFTs), inspired by theoretical physics, produce manifold invariants behaving well under gluing. For every discrete group G, homotopy quantum field theories (HQFTs) are G-equivariant versions of TQFTs. In this talk we define and classify 2-dimensional extended HQFTs by generalizing methods introduced for TQFTs by Chris Schommer-Pries in 2009. We list generators and relations for the extended G-equivariant bordism bicategory and use them to classify 2-dimensional extended HQFTs.

Title: Educational Professionalism within the Fifth Estate: Networks of Influence Within Social Media and Education

Date: 10/03/2018

Time: 12:00 PM - 1:00 PM

Place: B310 Wells Hall

Dr. Kaitlin Torphy will speak about an emergent phenomenon, social media in education. She will present the notion of a Fifth Estate within the digital age, redefining network influence (Dutton, 2009). Dr. Torphy will review research regarding teachers’ engagement within Pinterest, a prevalent social media platform amongst teachers nationwide. In related work, she will explore how teachers are turning to social media (Pinterest) to connect with instructional resources and one another as they work to support the academic needs of their students and respond to education reforms. Dr. Torphy will provide a first look at characterizing the quality and standards alignment of over 5000 mathematics tasks within Pinterest. For more information on the work or the Teachers in Social Media project, visit www.TeachersInSocialMedia.org.

This talk will be an introduction to Riemann surfaces, including branched covering and monodromy in this setting. I will prove Riemann's existence theorem of branched covers, illustrate this using algebraic curves, and finish with Riemann-Hurwitz.

In 1948-1950, V.Ya.Kozlov (1914-2007) stated a series of
interesting geometric properties of dilated systems D(f)= {f(kx): k=
1,2,...} in the spaces L^p(0,1). Since that, no proofs were published.
In particular, for a Rademacher-Haar-Walsh type generator f=
2-periodic odd extension of the indicator function of (0,a), 0<a<1,
the system D(f) was claimed to be complete/incomplete for many
particular values of a. We prove all Kozlov's statements and several
new, as well as discuss other geometric properties of D(f).

Title: Unitary paths and quantum computational supremacy: A proof of average-case hardness of Random Circuit Sampling

Date: 10/04/2018

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

Demonstration of computational advantages of Noisy Intermediate-Scale Quantum (NISQ) computers over classical computers is an imperative near-term goal, especially with the exuberant experimental frontier in academia and industry. Because of a large industrial push (e.g., from IBM and Google), NISQ computers with hundred(s) of qubits are at the brink of existence with the promise of outperforming any classical computer.
A goal-post is to demonstrate the so called {\it quantum computational supremacy}, which is to show that a NISQ computer can perform a computational task that is tremendously difficult for any classical (super-)computer. The foremost candidate problem to show quantum supremacy is Random Circuit Sampling (RCS), which is the task of sampling from the output distribution of a random circuit. For example, this is Google's primary current objective, whose delivery is promised within the next few months.
In this work, we first develop a mathematical framework for and prove various useful facts applicable to random circuits such as construction of rational function valued unitary paths that interpolate between two arbitrary unitaries, an extension of Berlekamp-Welch algorithm that efficiently and exactly interpolates rational functions, and construction of probability distributions over unitaries that are arbitrarily close to the Haar measure. Lastly, we then prove that the exact sampling from the output distribution of random circuits is $\#P$-Hard on {\it average}; we also prove that this is necessary for proving the quantum supremacy conjecture.

Title: Khovanov homology and Bar-Natan's deformation via immersed curves in the 4-punctured sphere.

Date: 10/04/2018

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

We will describe a geometric interpretation of Khovanov homology and its deformation due to Bar-Natan as Lagrangian Floer homology of two immersed curves in the 4-punctured 2-sphere S^2 \ 4pt. We will first start with a certain cobordism theoretic algebra H, where elements are all cobordisms between two trivial tangles )( and = up to certain relations. The central point then will be the observation that this algebra is isomorphic to an algebra B = Fuk(a0, a1), whose elements are generators of wrapped Lagrangian Floer complexes between two arcs a0 and a1 inside S^2 \ 4pt. The results will follow because D structures over H give Khovanov/Bar-Natan invariants for 4-ended tangles, and D structures over B give curves in S^2 \ 4pt (due to [Haiden, Katzarkov, Kontsevich]).
The construction is originally inspired by a result of [Hedden, Herald, Hogancamp, Kirk], which embeds 4-ended reduced Khovanov arc algebra (or, equivalently, Bar-Natan dotted cobordism algebra) into the Fukaya category of the 4-punctured sphere. This is joint work with Liam Watson and Claudius Zibrowius.

Title: Cell Decompositions for Rank Two Quiver Grassmannians

Date: 10/04/2018

Time: 3:00 PM - 4:00 PM

Place: C117 Wells Hall

A quiver Grassmannian is a variety parametrizing subrepresentations of a given quiver representation. Reineke has shown that all projective varieties can be realized as quiver Grassmannians. In this talk, I will study a class of smooth projective varieties arising as quiver Grassmannians for (truncated) preprojective representations of an n-Kronecker quiver, i.e. a quiver with two vertices and n parallel arrows between them. The main result I will present is a recursive construction of cell decompositions for these quiver Grassmannians. If there is time I will discuss a combinatorial labeling of the cells by which their dimensions may conjecturally be directly computed. This is a report on joint work with Thorsten Weist.

Title: Some inverse source and coefficient problems for the wave operators (special colloquium)

Date: 10/04/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Inverse problems seek to infer causal factor from the resulting observation, and waves are among the most prevalent and significant observations in nature. In this talk, we will discuss two inverse problems for the acoustic wave equation and its generalizations. The first is an inverse source problem where one attempts to determine an instantaneous source from the boundary Dirichlet data. We give sharp conditions on unique and stable determination, and derive an explicit reconstruction formula for the source. The second is an inverse coefficient problem on a cylinder-like Lorentzian manifold (M,g) for the Lorentzian wave operator perturbed by a vector field A and a function q. We show that local knowledge of the Dirichlet-to-Neumann map (DN-map) stably determines the jets of (g,A,q) up to gauge transformations, and global knowledge of the DN-map stably determines the lens relation of g as well as the light ray transforms of A and q. This is based on joint work with P. Stefanov.

Prime numbers are the building blocks of arithmetic. Starting with Euclid's classical proof that there are infinitely many primes, I will discuss various approaches to thinking about the infinitude of primes, culminating with Dirichlet's theorem on primes in arithmetic progression.