Colloquium, Department of Mathematics and Statistics
Colloquium, Department of Mathematics and Statistics
Colloquium Lectures
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Duan Chen

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  • Fall 2022
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  • Spring 2022

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Friday, April 22, 2022, 2:00-3:00, via Zoom

April 18, 2022 by William Brian
Categories: Spring 2022

Speaker: Dr. Elizabeth Bumgardner, Chair of Mathematics at the North Carolina School of Science and Mathematics – Morganton (invited by Adalira Saenz-Ludlow)

Title: Active Learning and What It Entails: From Curriculum, to Intellectual Environment, to Physical Environment

Abstract:  Educators often discuss the importance of active learning in the classroom to deepen students’ understanding of mathematical concepts and help students take ownership of their own learning.  Active learning tasks such as think-pair-share, reading for comprehension, chunked problem analysis, and muddiest point are readily offered as possible activities for instructors interested in implementing active learning within their own classroom. Why do some instructors succeed in implementing such tasks while others struggle? What factors contribute to the success of active learning in a mathematics classroom? In this colloquium talk, we will discuss the role of curriculum design in a successful active learning classroom and the importance of a curriculum focused on making conceptual connections and establishing habits of thinking.  We will look at the process of preparing and aligning tasks that allow for hypothesizing, time for thinking, active engagement with content, and reflection.  In addition, we will consider the intellectual environment and the physical environment conducive to active learning.

Thursday, April 21, 2022, 11:30-12:30, in person

April 18, 2022 by William Brian
Categories: Spring 2022

Speaker: Dr. Domynikas Norgilas from the University of Michigan (invited by Adriana Ocejo Monge)

Title: The Monge-Kantorovich mass transport with supermartingale constraints.

Abstract:  Given two measures μ,ν on R with μ(R) ≤ ν(R), and such that μ is smaller than ν in positive convex-decreasing order (i.e., μ ≤pcd ν), there exists a two-period supermartingale S = (S₁,S₂) that transports μ to ν. For each such supermartingale, S₁ ~ μ, but there are many possible choices for the law of S₂. In this talk we study two canonical choices (the minimal and the maximal measures) with respect to convex-decreasing order. We show how these measures give rise to the so-called supermartingale shadow couplings of S₁ and S₂.

Friday, April 8, 2022, 11:00-12:00, via Zoom

April 04, 2022 by William Brian
Categories: Spring 2022

Speaker: Dr. Jonathan Meddaugh from Baylor University (invited by Will Brian)

Title: Shadowing in Dynamical Systems: When are approximate orbits meaningful?

Abstract:  In a discrete dynamical system f: X → X, the orbit of a point x ϵ X is the sequence x, f(x), f²(x), . . . . In even modestly complex systems, small errors compound rapidly in computations of orbits, leading to true orbits and computed orbits that are wildly different. Surprisingly, however, there is a large class of systems for which these approximate orbits can yield useful qualitative information about the dynamics of the system—systems with shadowing. Informally, a dynamical system f: X → X has the shadowing property provided that for any specified level of tolerance, there is a level of precision that guarantees that any approximate orbit which is computed with this level of precision is itself approximated (shadowed) within the specified tolerance by a true orbit for the system. In this talk, we will discuss the shadowing property with a focus on recent results concerning characterizations of this property and its relative prevalence in certain classes of dynamical systems.

Friday, April 1, 2022, 11:00-12:00, via Zoom

March 28, 2022 by William Brian
Categories: Spring 2022

Speaker: Dr. Akshaa Vatwani from IIT Gandhinagar (invited by Arindam Roy)

Title: Limitations to equidistribution in arithmetic progressions

Abstract: We will discuss the distribution of prime numbers lying in an arithmetic progression a modulo q with a coprime to q. Dirichlet observed that any such sequence contains infinitely many primes. Moreover, for a fixed q, the proportion of primes in each such progression is the same. As q varies, more questions can be asked regarding how far such equidistribution persists. We will explore this theme and discuss some variants and applications. This talk is based on joint work with Aditi Savalia.

Friday, March 25, 2022, 11:00-12:00, in person

March 21, 2022 by William Brian
Categories: Spring 2022

Speaker: Dr. Qiliang Wu from Ohio University (invited by Helen Li)

Title: Pearling and Localized Undulation of Bilayers in Amphiphilic Morphology

Abstract: Amphiphiles, such as lipids and functionalized polymers, plays a central
role in the self-assembly of solvent accessible, intricately structured
nano-scaled network structures, which are vital in cell functionality and
offer wide applications to drug delivery, detergent production, emulsion
stabilization and energy conversion devices. We study amphiphilic
morphology in the framework of the functionalized Cahn-Hilliard (FCH)
energy. The FCH is a continuum model accommodating various co-dimensional
structures such as bilayers (co-dim 1), filaments (co-dim 2) and micelles
(co-dim 3). We focus on defect structures that break the dimensional
reduction and include endcaps that terminate filaments or bilayers and Y
junctions. More specifically, we show the existence of pearled bilayer
solutions via a spatial-dynamics formulation, in combination with center
manifold reduction and a fixed-point argument. In addition, we also show
via a functional analytic framework that in the presence of spatial
inhomogeneity, localized undulation appears under proper functionalization
terms. More interestingly, both the pearling and localized undulation are
shown to be a manifestation of a degenerate 1:1 resonance Hopf
bifurcation encoded in a reduced ODE system from the FCH energy.

Friday, March 18, 2022, 11:00-12:00 in person

March 16, 2022 by William Brian
Categories: Spring 2022

Speaker: Dr. Daniel Massatt rom the University of Chicago (invited by Helen Li)

Date and Time: Friday, March 18, 2022, 11:00-12:00 in person. The talk will also be broadcast online via Zoom for those unable to attend in person. Please contact Will Brian to obtain the Zoom link.

Title: Electronic Structure of Incommensurate 2D Heterostructures with Mechanical Relaxation

Abstract: 

Since their discovery by Geim, 2D heterostructures have become a hotbed of research due to their novel structure. Stacking various materials on top of each other allows for a vast range of possible materials and corresponding properties. However, this stacking typically leads to an aperiodic, or incommensurate, structure due to lattice mismatch or rotational misalignment. This incommensuration increases the potential for tunability of electronic and mechanical properties, leading for example to the famous discovery of unconventional superconductivity of twisted bilayer graphene at the magic angle.

Experiments are vastly more expensive than numerical simulations, so much work is needed on building analysis and algorithms for understanding and computing efficiently for appropriate models. In this work, we discuss configuration and momentum space methodologies to build algorithms to compute observables of incommensurate heterostructures with mechanical relaxation in a tight-binding framework. We exploit the ergodicity of the misalignment, and for appropriate materials (such as those with conic or parabolic bands) we use carefully selected perturbative expansions in momentum space.

Friday, March 4, 2022, 11:00-12:00 in person

February 28, 2022 by William Brian
Categories: Spring 2022

Speaker: Dr Hung Tran from the University of Wisconsin (invited by Loc Nguyen)

Date and Time: Friday, March 4, 2022, 11:00-12:00 in person. The talk will also be broadcast online via Zoom for those unable to attend in person. Please contact Will Brian to obtain the Zoom link.

Title: Periodic homogenization of Hamilton-Jacobi equations: optimal rate and finer properties

Abstract:  I will describe some recent progress in periodic homogenization of Hamilton-Jacobi equations. First, we show that the optimal rate of convergence is O(ε) in the convex setting. We then give a minimalistic explanation that the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts in two dimensions. This is joint work with Wenjia Jing and Yifeng Yu.

Friday, February 25, 2022, 11:00-12:00 in person

February 21, 2022 by William Brian
Categories: Spring 2022

Speaker: Dr. Lynne Yengulalp from Wake Forest University (invited by Will Brian)

Date and Time: Friday, February 25, 2022, 11:00-12:00 in person. The talk will also be broadcast online via Zoom for those unable to attend in person. Please contact Will Brian to obtain the Zoom link.

Title: Completeness in topology

Abstract:  A metric space is complete if every Cauchy sequence converges (to a point in the space). The real line is complete, but the open unit interval (which is topologically the same, i.e. homeomorphic to the real line) is not complete. In this talk, we will survey some topological notions of completeness. One strong notion of completeness is complete metrizability; a space X is completely metrizable if it is homeomorphic to a complete metric space. On the weaker end, a space is called Baire if the intersection of countably many dense open sets is dense. There is an interesting spectrum of topological properties in between. Such properties arise from generalizing convergence of sequences, from topological games, and from domain theory. 

Friday, February 18, 2022, 2:00-3:00 via Zoom

February 14, 2022 by William Brian
Categories: Spring 2022

Speaker: Dr. Tracie McLemore Salinas from Appalachian State University (invited by Adalira Saenz-Ludlow)

Date and Time: Friday, February 18, 2022, 2:00-3:00 via Zoom. Please contact Will Brian to obtain the Zoom link.

Title: STEM Education: Integrated Perspectives and a Focus on Mathematics

Abstract:  Policy-makers and practitioners have latched onto the idea of STEM (Science-Technology-Engineering-Mathematics) in many ways, often by focusing on engineering and technology as driving components and by invoking an obligation for workforce development. Historically, STEM as an integration of constituent disciplines is more nuanced, informed by the inherent interdependence of and combination of habits of mind within and across the disciplines.  In this colloquium talk, I will share a historical look at STEM and suggest a practices-based framework for considering its teaching and learning.  In addition, we will consider mathematics-specific questions related to the teaching and learning of STEM.

Friday, February 11, 2022, 9:00-10:00 via Zoom

February 08, 2022 by William Brian
Categories: Spring 2022

Speaker: Dr. Luo Ye from Xiamen University

Date and Time: Friday, February 11, 2022, 9:00-10:00 via Zoom. Please contact Will Brian to obtain the Zoom link.

Title: Tropical convexity analysis and some applications

Abstract:  The tropical semiring is an idempotent semiring where  the usual operations of addition and multiplication are replaced by operations of minimum/maximum and addition respectively. Tropical geometry is a theory of geometry over the tropical semiring which has rich combinatorial features and can be described as a degenerated version of algebraic geometry over the field of complex numbers under Maslov dequantization or over a non-archimedean field under the valuation map. The  features of “linear combinations” in tropical geometry can be captured by the notion of tropical convexity.   In this talk, I will introduce a general theory of tropical convexity analysis based on the so-called “B-pseudonorms” on tropical projective spaces, and show some subsequent results, e.g., a tropical version of Mazur’s Theorem on closed tropical convex hulls and a fixed point theorem for tropical projections. Two applications will also be presented. The first is to establish a connection between tropical projections and  reduced divisors on (metric) graphs, and the second is to construct  min-max-plus neural networks, a new type of artificial neural networks.





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