Title: Localization via positivity and large deviations of the Lyapunov exponent

Date: 11/27/2018

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

In this talk, I will introduce the role of a dynamical object, the Lyapunov exponent, in
the spectral analysis of one-dimensional ergodic Schrodinger operators. I will show how to obtain
Anderson Localization via positivity and Large deviations of the Lyapunov exponent.
I will mainly focus on the one-dimensional Anderson model, where a relatively simple proof
of spectral localization and exponentially dynamical localization may be obtained.

Title: Active Learning 2.0: Being Intentionally Inclusive

Date: 11/27/2018

Time: 1:30 PM - 3:00 PM

Place: 252 EH

Active learning has many documented benefits both for students and instructors. Moreover, there is increasing evidence that it disproportionately benefits women, students of color, and students who were previously denied the same learning opportunities as others. However, the empirical evidence for this disproportionate benefit doesn't explain why it happens, nor does it guarantee that all students will benefit from active learning. In fact, my own experience with active learning is that it is difficult to do well and sometimes it can have detrimental effects on students if we're not careful. So, we should aim not just for active learning, but learning that is both active and inclusive. We'll discuss some principles and practical strategies for making active learning more inclusive.

Title: Improving L^1 Poincar\'e inequality on Hamming cube

Date: 11/28/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

L^1 Poincar\'e inequality on hypercube is related to many interesting questions in random graph theory (like Margoulis graph connectivity theorem, e.g.). The sharp constant is unknown, but I will show how to improve the previously known constant \pi/2 obtained by Ben Efraim and Lust--Piquard by using non-commutative harmonic analysis. The approach will be probabilistic and luckily commutative. For Gaussian space the constant is known, it is \sqrt{\pi/2}, and the short proof belong to Maurey--Pisier.

One of the nicer properties a knot can have is to be alternating. These knots tend to be easy to work with, and can give us several nice results about the whole class. Unfortunately, many knots are not alternating, causing general proofs about them to be difficult at best. In this talk, we will take a look at a couple of different ways we can broaden the class of alternating knots, and see what we can get from these different definitions of alternating.

Speaker: Dominic Culver, University of Illinois Urbana-Champaign

Title: Towards a counter example of the telescope conjecture

Date: 11/29/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

The telescope conjecture is the only remaining Ravenel conjecture to remain unsolved. In this talk, I will give a review of the chromatic perspective on stable homotopy theory and motivate the telescope conjecture. I will then proceed to sketch Mahowald’s proof of the height 1 telescope conjecture for 2-local spectra. Time permitting, I will describe work in progress with Beaudry, Behrens, Bhattacharya, and Xu on using the tmf-resolution of the Bhattacharya-Egger spectrum to find a counter example to the height 2 telescope conjecture.

Every cluster algebra has an associated 'scattering diagram': an affine space endowed with a (possibly very complicated) collection of 'walls'. The structure of this scattering diagram encodes essential information about the cluster algebra's exchange graph, Laurent coefficients, and theta functions. In this talk, I will discuss an ongoing project with Nathan Reading and Shira Viel to construct a scattering diagram associated to a triangulable marked surface. The affine space may be identified with the set of certain `measured laminations' on the surface, and the walls may be identified with certain forbidden subgraphs embedded in the surface, which we call `barricades'.

Title: Mean-field anticipated BSDEs driven by fractional Brownian motion and related stochastic control problem

Date: 11/29/2018

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

In this talk, we introduce a new type of BSDEs, we call it mean-field anticipated backward stochastic differential equations (MF-BSDEs, for short) driven by a fractional Brownian motion with Hurst parameter H>1/2. We will show that it's possible to prove the existence and uniqueness of this new type of BSDEs using two different approaches. Then, we will present a comparison theorem for such BSDEs. Finally, as an application of this type of equations, a related stochastic optimal control problem is studied.
This is a joint work with Yufeng Shi and Jiaqiang Wen : Institute for Financial Studies and School of Mathematics, Shandong University, Jinan 250100, China.

Title: Taking an Instructional Innovation to Scale: Characterizing, Supporting, and Evaluating Inquiry-Oriented Instruction

Date: 11/30/2018

Time: 10:20 AM - 11:10 AM

Place: C304 Wells Hall

Inquiry-oriented instruction has shown promise in regards to many features of student success, including conceptual understanding, affective gains, and persistence in STEM degrees. However, instructional change is difficult (especially at scale) and the research literature has documented a number of challenges instructors face when shifting their instructional practice. During this talk I will provide a characterization of inquiry-oriented instruction; discuss an instructional support model that was developed to support inquiry-oriented instruction in undergraduate mathematics courses; and present preliminary evaluation findings, drawing on a national sample of content assessment data, collected from 513 students at 46 different institutions. Analysis of this assessment data revealed no difference in the performance of men and women in the comparison sample; however, under the inquiry-oriented treatment, a gender performance difference was present – with men outperforming women. In an effort to understand this finding, I present related research literature on gendered experiences in collaborative settings and our preliminary analysis into the experiences of our students in these inquiry-oriented courses.

Reproducing kernel Hilbert spaces (RKHSs) are Hilbert spaces of functions on which point evaluation functionals are continuous. Thanks to the existence of an inner product, RKHSs are well-understood in functional analysis. Successful and important machine learning methods based on RKHSs include support vector machines, regularization networks and kernel-based approximation.
In the past decade, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBSs) for applied and theoretical purposes for instance sparse approximation. Recently, we propose a generic definition of RKBS and a framework of constructing RKBSs that unifies existing constructions in the literature, and leads further to new RKBSs. As a by-product, the space C([0,1]) of all continuous functions on the interval [0,1] is an RKBS.
Motivated by sparse multi-task learning, we constructed a class of vector-valued RKBSs with the l1 norm based on multi-task admissible kernels. The relaxed linear representer theorem holds for regularization networks in the obtained spaces if and only if the Lebesgue constant of kernels is uniformly bounded. A class of translation-invariant kernels of limited smoothness admissible for construction are given. Numerical experiments demonstrate the advantages of the proposed construction and regularization models.
This talk is based on two joint papers with Prof. Guohui Song (Clarkson University), Haizhang Zhang (Sun Yat-sen University), and Jun Zhang (University of Michigan).