Title: The homology polynomial and pseudo-Anosov braids

Date: 10/17/2018

Time: 4:10 PM - 5:00 PM

Place: A202 Wells Hall

Every orientation preserving homeomorphism of a compact, connected, orientable surface S is isotopic to a representative that is periodic, reducible, or pseudo-Anosov (pA). In the last case, the representative is neither periodic nor reducible and the surface admits two (singular) transverse measured foliations. The pA representative "stretches" with respect to one of these measures by a number called the stretch factor.
The homology polynomial, introduced by Birman, Brinkmann, and Kawamuro, is an invariant of the isotopy class and contains the stretch factor as it's largest real root. It can also distinguish some distinct pA maps with the same stretch factor. In this talk I will discuss the ideas behind the homology polynomial and how it is obtained. As time permits I will discuss some examples involving pA braids and touch on a connection with the Burau representation.

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.

ABSTRACT. Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type,
$$
\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[b(u)+ \sigma(u)\stackrel{\cdot}{F}(t,x)]
$$
in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$, $\alpha\in (0,2]$. The operator $\partial^\beta_t$ is the Caputo fractional derivative while $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic $\alpha$-stable L\'evy process and $I^{1-\beta}_t$ is the Riesz fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. These equations might be used as a model for materials with random thermal memory. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on $b$, $\sigma$ and the initial condition. Our results complement those of P. Chow in ``P.-L. Chow. Unbounded positive solutions of nonlinear parabolic It$\hat{o}$ equations. Commun. Stoch. Anal., 3(2)(2009), 211--222.'' and ``P.-L. Chow. Explosive solutions of stochastic reaction-diffusion equations in mean $l_{p}$-norm. J. Differential Equations, 250(5) (2011), 2567--2580.'' and Foondun and Parshad ``M. Foondun and R. Parshad, On non-existence of global solutions to a class of stochastic heat equations. Proc. Amer. Math. Soc. 143 (2015), no. 9, 4085--4094'', among others. The results presented are our recent joint work with Sunday Asogwa, Mohammud Foondun, Wei Liu, and Jebessa Mijena.

Title: A family of freely slice good boundary links

Date: 10/18/2018

Time: 3:10 PM - 4:00 PM

Place: C304 Wells Hall

The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links. This is joint work with Jae Choon Cha and Mark Powell.

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.

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.

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.