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.

Title: Étale Cohomology and its Applications to Curves

Date: 10/08/2018

Time: 1:00 PM - 1:50 PM

Place: C517 Wells Hall

Étale cohomology was originally introduced by Grothendieck in the 1960s as a tool for solving the Weil conjectures. Since then it has proven very useful in algebraic and arithmetic geometry. In my talk, I will introduce the notion of Grothendieck topology, étale site, and étale cohomology. I will then explore the cohomology of curves and briefly describe some applications. This talk will be accessible to all levels.

Title: A simple family of infinitely many absolutely exotic manifolds

Date: 10/08/2018

Time: 4:10 PM - 5:30 PM

Place: C304 Wells Hall

I wıll demonstrate a smooth 4-manifold M, obtained by attaching a 2-handle to B^4 along a certain knot K in S^3, which admits infinitely many absolutely exotic copies M_n, n=0,1,2.., such that each copy M_n is obtained by attaching 2-handle to a fixed compact smooth contractible manifold W along its boundary Y, along the iterates f^{n}(c) of a knot c in Y by a diffeomorphism f: Y---> Y. This generalizes the example I gave in “An exotic 4-manifold, Jour. of Diff. Geom. 33, (1991)” which corresponds to the n=1 case.

Geometric L^2(0,1) properties of dilated function systems D(f)= {f(kx): k= 1,2,...} are discussed, as completeness, Riesz basis property, etc. (Completeness of D(f) for f(x)= 1/x-[1/x] is equivalent to the Riemann Hypothesis). The Bohr's lift techniques permit to explain (all) known results and show some new, as well as to discuss open problems.

Title: Turaev-Viro invariants via quantum representations of the mapping class group

Date: 10/10/2018

Time: 4:10 PM - 5:00 PM

Place: A202 Wells Hall

The Turaev-Viro invariants are an infinite family of real valued 3-manifold invariants originally defined by state sums of a triangulation. Using SO(3)-TQFT, I will demonstrate an equivalent formulation in terms of traces of quantum representations and discuss its possible advantages in studying mapping tori of surfaces.

Title: Existence in the sense of sequences of stationary solutions for some non-Fredholm integro-differential equations

Date: 10/11/2018

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

We establish the existence in the sense of sequences of
stationary solutions for some reaction-diffusion type equations in
appropriate
H^2 spaces. It is shown that, under reasonable technical conditions, the
convergence in L^1 of the integral kernels implies the existence and
convergence in H^2 of solutions. The nonlocal elliptic equations involve
second order differential operators with and without the Fredholm property.

Title: An infinite-rank summand of the homology cobordism group

Date: 10/11/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

This talk explains a generalization of the techniques that Hom introduced to construct an infinite-rank summand of the topologically slice knot concordance group. We generalize Hom's epsilon-invariant to the involutive Heegaard Floer homology constructed by Hendricks-Manolescu. As an application, we see that there is an infinite-rank summand of the homology cobordism group, generated by Seifert spaces. The talk will contain a review of involutive Floer homology. This is joint work with Irving Dai, Jen Hom, and Linh Truong.

Title: Annulus SLE partition functions and martingale-observables

Date: 10/11/2018

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

In this talk, I will introduce a version of conformal ﬁeld theory (CFT) and explain its implementations to SLE theory in a doubly connected domain. The basic fields in these implementations are one-parameter family of Gaussian free fields whose boundary conditions are given by a weighted combination of Dirichlet boundary condition and excursion-reflected one. After explaining basic notions in CFT such as OPE families of central charge modiﬁcations of the Gaussian free ﬁeld and presenting certain equations including a version of Eguchi-Ooguri and Ward’s equations, I will outline the relation between CFT and SLE theory. As an application, I will explain how to apply the method of screening to find Euler integral type solutions to the parabolic partial differential equations for the annulus SLE partition functions introduced by Zhan and present a class of SLE martingale-observables associated with these solutions.
This is based on joint work with Nam-Gyu Kang and Hee-Joon Tak.

I will give basic definitions of scattering diagrams and wall-crossing automorphims, and finish by showing some examples related to rank-2 cluster algebras.

Speaker: Deanna Needell, University of California, Los Angeles

Title: Simple Classification from Binary Data

Date: 10/11/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Binary, or one-bit, representations of data arise naturally in many applications, and are appealing in both hardware implementations and algorithm design. In this talk, we provide a brief background to sparsity and 1-bit measurements, and then present new results on the problem of data classification from binary data that proposes a framework with low computation and resource costs. We illustrate the utility of the proposed approach through stylized and realistic numerical experiments, provide a theoretical analysis for a simple case, and discuss future directions.

Title: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture

Date: 10/18/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll use branched surfaces to build taut foliations for manifolds obtained by surgery on positive 3-braid closures. As an example, we'll construct taut foliations in every non-L-space obtained by surgery along the P(-2,3,7) pretzel knot. No background in Heegaard-Floer or foliation theories will be assumed.

Speaker: Dr. Kyeong Hah Roh, Arizona State University

Title: On the Teaching and Learning of Logic in Mathematical Contents

Date: 10/18/2018

Time: 2:30 PM - 4:00 PM

Place: 252 EH

Logical thinking plays a crucial role in generating valid arguments from the given information as well as in evaluating the validity of others’ arguments in workplaces. Training our students as logical thinkers has been a central component in mathematics education. By engaging in proving and validating activities in undergraduate mathematics, students are expected to enhance logical thinking and make sound decisions by deducing valid inferences from a tremendous amount of information and resources in their future workplaces. Many universities in the United States thus offer introductory proof courses, or so called transition-to-proof courses, to introduce logic and various proof structures for valid arguments in mathematical contents. This presentation will provide an overview of the empirical studies that I have been involved in relation to undergraduate students’ logic and logical thinking, instructional interventions that I have designed to enhance students’ logical thinking in mathematical contents, and some issues and challenges in the introductory proof courses in mathematics.

Title: Cluster Monomials and Theta Bases via Scattering Diagrams

Date: 10/18/2018

Time: 3:00 PM - 4:00 PM

Place: C117 Wells Hall

In this talk I will add to Nick’s presentation from last time by describing a portion of the scattering diagram using c-vectors and g-vectors. Then I will present some examples of computing cluster monomials using broken lines. If there is time I will compute an element of the theta basis which is not a cluster monomial.

The familiar double soap bubble is the least-area way to enclose and separate two given volumes in Euclidean space. What if you give space a density, such as r^2 or e^r^2 or e^-r^2? The talk will include recent results and open questions. Students welcome.

Speaker: Paul Bendich, Duke University and Geometric Data Analytics

Title: Topology and Geometry for Tracking and Sensor Fusion

Date: 10/19/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Many systems employ sensors to interpret the environment. The target-tracking task is to gather sensor data from the environment and then to partition these data into tracks that are produced by the same target. The goal of sensor fusion is to gather data from a heterogeneous collection of sensors (e.g, audio and video) and fuse them together in a way that enriches the performance of the sensor network at some task of interest.
This talk summarizes two recent efforts that incorporate mildly sophisticated mathematics into the general sensor arena.
First, a key problem in tracking is to 'connect the dots:' more precisely, to take a piece of sensor data at a given time and associate it with a previously-existing track (or to declare that this is a new object). We use topological data analysis (TDA) to form data-association likelihood scores, and integrate these scores into a well-respected algorithm called Multiple Hypothesis Tracking. Tests on simulated data show that the TDA adds significant value over baseline, especially in the context of noisy sensor data.
Second, we propose a very general and entirely unsupervised sensor fusion pipeline that uses recent techniques from diffusion geometry and wavelet theory to fuse time series of arbitrary dimension arising from disparate sensor modalities. The goal of the pipeline is to differentiate classes of time-ordered behavior sequences, and we demonstrate its performance on a well-studied digit sequence database.
This talk represents joint work with many people. including Chris Tralie, Nathan Borggren, Sang Chin, Jesse Clarke, Jonathan deSena, John Harer, Jay Hineman, Elizabeth Munch, Andrew Newman, Alex Pieloch, David Porter, David Rouse, Nate Strawn, Adam Watkins, Michael Williams, and Peter Zulch.

A regular hexagon is the least-perimeter unit-area tile of the Euclidean plane. What is the best pentagonal tile? What about the hyperbolic plane? What about higher dimensions? The talk will include open questions and recent results, some by undergraduates.