**Lidia Mrad**
Portrait of Lidia Mrad

Assistant Professor of Mathematics

Department of Mathematics and Statistics

Mount Holyoke College

lmrad [at] mtholyoke.edu

My research focuses on solving problems, analytically and/or computationally, arising from physical setups such as those describing liquid crystals, population growth patterns, or epidemiology. The techniques I use belong to the areas of partial differential equations, the calculus of variations, agent-based modeling, and numerical simulations. Teaching mathematics has been an integral part of my professional journey, starting as a high school teacher, then throughout my graduate and postdoctoral years, and now as faculty. I enjoy teaching such a rich topic to students from a variety of backgrounds. I am always looking for outreach opportunities beyond the classroom as well as ways to support students from under-represented identities in their pursuit of mathematics. Before coming to Mount Holyoke College, I was a Postdoctoral Research Associate in the Department of Mathematics at the University of Arizona where I was mentored by Joceline Lega. I received my Ph.D. in mathematics from Purdue University under the supervision of Daniel Phillips. Research ============================================================== My work is situated along a spectrum between theoretical and applied mathematics. The motivation for the problems I work on is always rooted in answering questions which interest scientists, specifically experimental physicists or public health specialists. My technical approaches utilize both analytical and computational tools. My focus has recently been more on the computational aspects, but I still aim for analytical results to complement them. **************************************************************************************************** *.------------------------------..-----------------------..----------------..---------------------.* *|partial differential equations|| calculus of variations|| liquid crystals||agent-based modeling |* *'------------------------------''-----------------------''----------------''---------------------'* **************************************************************************************************** From a mathematical perspective, the methods I use fall under the areas of *partial differential equations* and the *calculus of variations*. I also build tools for agent-based modeling or utilize existing *computational methods* for my work. Themes --------------------------------------------------------------- **_Liquid Crystals_** One of my main application areas is that of liquid crystals, which are intermediate phases of matter between liquids and crystalline solids. Molecules of these phases possess an orientational order (molecules tend to align with each other), with some phases having positional order as well (molecules aggregate in planes in addition to aligning). Such characteristics endow liquid crystals with unique physical properties, rendering them useful in a variety of applications as illustrated by their widespread use in optical displays (LCDs). Their applicability, however, is not limited to optical devices and it has been extended over the years to biological and pharmaceutical applications, among others. My research targets the optical as well as the biological utilizations of liquid crystals. There are many phases of liquid crystals that range from liquid-like to solid-like, each phase requiring different modeling approaches, and I consider a variety of phases in my work. **_Population Dynamics_** Another line of applications, which I started during my postdoctoral training, concerns spatial population dispersion, specifically of mosquitoes for the purpose of disease control. I address questions linked to the spread of *aedes aegypti*, the yellow fever mosquito that can cause dengue fever and other diseases. My work focuses on simulating the flight of mosquitoes over heterogeneous spatial grids with the goal of producing efficient intervention suggestions. In addition to creating agent-based modeling tools for mosquito populations, I am also interested in the more abstract question of population dynamics and how different species can interact in ways that affect their coexistence. Publications --------------------------------------------------------------- - A multi-faceted study of nematic order reconstruction in microfluidic channels, J. Dalby, Y. Han, A. Majumdar, **L. Mrad**, SIAM Journal on Applied Mathematics , 83(6), p.2284-2309 (2023) [doi] [arxiv]
Abstract We study order reconstruction (OR) solutions in the Beris-Edwards framework for nematodynamics, for both passive and active nematic flows in a microfluidic channel. OR solutions exhibit polydomains and domain walls, and as such, are of physical interest. We show that OR solutions exist for passive flows with constant velocity and pressure, but only for specific boundary conditions. We prove the existence of unique, symmetric and non-singular nematic profiles, for boundary conditions that do not allow for OR solutions. We compute asymptotic expansions for OR-type solutions for passive flows with non-constant velocity and pressure, and active flows, which shed light on the internal structure of domain walls. The asymptotics are complemented by numerical studies that demonstrate the universality of OR-type structures in static and dynamic scenarios.
- Dynamics of a linearly-perturbed May-Leonard competition model, G. Jaramillo, **L. Mrad**, T. L. Stepien, *Chaos*, 33, 063121 (2023) [doi] [arxiv]
Abstract The May–Leonard model was introduced to examine the behavior of three competing populations where rich dynamics, such as limit cycles and nonperiodic cyclic solutions, arise. In this work, we perturb the system by adding the capability of global mutations, allowing one species to evolve to the other two in a linear manner. We find that for small mutation rates, the perturbed system not only retains some of the dynamics seen in the classical model, such as the three-species equal-population equilibrium bifurcating to a limit cycle, but also exhibits new behavior. For instance, we capture curves of fold bifurcations where pairs of equilibria emerge and then coalesce. As a result, we uncover parameter regimes with new types of stable fixed points that are distinct from the single- and dual-population equilibria characteristic of the original model. On the contrary, the linearly perturbed system fails to maintain heteroclinic connections that exist in the original system. In short, a linear perturbation proves to be significant enough to substantially influence the dynamics, even with small mutation rates.
- Aggregation phenomena in lyotropic chromonic liquid crystals, **L. Mrad**, L. Zhao, M. I. Español, L. Xu, M.C. Calderer, *Communications in Nonlinear Science and Numerical Simulations*, 120, 107139 (2023) [doi] [arxiv]
Abstract We study the aggregation phenomenon in lyotropic chromonic liquid crystals as the molecular concentration changes and condensing agents are added into the system. Using properties of the critical points of the Oseen-Frank energy of a nematic liquid crystal, combined with the geometric constraints of the hexagonal columnar chromonic phases, we show that the minimizers of the total energy are topologically equivalent to tori, in agreement with available experimental evidence on chromonic liquid crystals and DNA condensates, in viral capsids as well as in free solution. We model the system as bi-phasic, consisting of liquid crystal molecules and water, and postulate the total energy as the sum of the Flory-Huggins energy of mixing together with the bending and surface tension contributions of the liquid crystal. Two types of problems are considered, one related to finding the optimal shape of a torus, once the phase separation has occurred, and the second one that models the conditions leading to molecular aggregation. This work follows recent experimental investigations, but without addressing the topological properties of the toroidal nuclei observed and focusing on how the liquid crystal order competes with the aggregation phenomenon.
- Dynamic analysis of chevron structures in liquid crystal cells, **L. Mrad**, D. Phillips, *Molecular Crystals and Liquid Crystals*, 647(1), p.66-91 (2017) [doi] [arxiv]
Abstract If a surface stabilized ferroelectric liquid crystal cell is cooled from the smectic-A to the smectic-C phase, its layers thin causing V-shaped (chevron like) defects to form. These create an energy barrier that can prevent switching between equilibrium patterns. We examine a gradient flow for a mesoscopic Chen-Lubensky energy $F(\Psi,\mathbf{n})$ that allows the order parameter to vanish, so that the energy barrier does not diverge if the layer thickness becomes small. The liquid crystal can evolve during switching in such a way that the layers are allowed to melt and heal near the chevron tip in the process.
**_In preparation_** - Spatial dispersal of _aedes aegypti_: Modeling mosquito flight using mark-release-recapture data, **L. Mrad**, J. Lega, H. Brown. - Switching dynamics in polar columnar phases of bent-core liquid crystals, **L. Mrad**. Talks --------------------------------------------------------------- Being a mathematician has allowed me to experience the joy of travel while meeting other mathematicians/scientists and discussing research. - Invited talks at institutions - Biomathematics Seminar, *University of Florida* (Apr. 2023) - Analysis and PDE Seminar, *Worcester Polytechnic Institute* (Feb. 2023) - Lunch Talk Series, *Smith College* (Oct. 2022) - Applied Mathematics Seminar, *University of Massachusetts at Amherst* (Nov. 2021) - Graduate Mathematics Seminar, *California State University Channel Islands*, virtual (Apr. 2021) - Math and Physics Seminar, *University of New Haven* (Feb. 2020) - Soft Materials Research Center Seminar, *University of Colorado Boulder* (Jan. 2020) - Condensed Matter Seminar, *University of Massachusetts Amherst* (Oct. 2019) - Applied Math Seminar, *Brigham Young University* (Apr. 2018) - Departmental Colloquium, *New Mexico State University* (Oct. 2016) - Talks at conferences - Special Session on Connections between Theoretical and Applied Dynamical Systems, *AMS Sectional Meeting, Amherst* (Oct. 2022) - Joint Dynamics and PDE Seminar, *University of Massachusetts at Amherst* (Nov. 2021) - Variational Models: Theory, Computations, and Applications to Materials, *SIAM Conference on Mathematical Aspects of Materials Science, Bilbao, Spain*, virtual (May 2021) - Numerical Methods and New Perspectives for Extended Liquid Crystalline Systems, ICERM, Providence (Dec. 2019) - New Trends in the Variational Modeling and Simulation of Liquid Crystals, *Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria* (Dec. 2019) - Spatiotemporally Complex Patterns minisymposium, *SIAM Conference on Dynamical Systems, Snowbird* (May 2019) - Optimal Design of Soft Matter workshop with a celebration of Women in Materials, *Isaac Newton Institute for Mathematical Sciences, Cambridge, UK* (May 2019) - Women in Mathematics of Materials (WiMM) session, *AWM Symposium, Houston* (Apr. 2019) - MUVE Symposium: Predicting Vector-Borne Disease Spread in Changing Landscapes, *Entomology Joint Annual Meeting (ESA, ESC, ESBC), Vancouver, Canada* (Nov. 2018) - Variational Problems from Materials Science minisymposium, *SIAM Conference on Mathematical Aspects of Materials Science, Portland* (Jul. 2018) - Modeling and Analysis of Condensed Matter Systems minisymposium, *SIAM Conference on Analysis of PDE, Baltimore* (Dec. 2017) - AMS Special Session on Analysis and Numerics on Liquid Crystals and Soft Matter, *Hunter College, City University of New York* (May 2017) - AMS Special Session on Problems in Geometry and Design of Materials, *Joint Mathematics Meetings, Seattle* (Jan. 2016) - Poster presentations - Learning from Nature to Transform Technology through Liquid Crystal Science, *Gordon Research Conference, Southern New Hampshire University* (June 2023) - Women in Math of Materials Workshop, *Michigan Center for Applied and Interdisciplinary Mathematics, Ann Arbor* (May 2018) - Liquid Crystals, Soft-matter Packing, and Active Systems Workshop, *IMA, Minneapolis* (Jan. 2018) - Rocky Mountain Partial Differential Equations Conference, *Brigham Young University, Provo City* (May 2017) - Other conferences attended - Optimal Transport in Data Science, *ICERM, Providence* (May 2023) - The 94th New England Complex Fluids workshop, *University of Rhode Island* (Mar. 2023) - Joint Mathematics Meetings, *Boston* (Jan. 2023) - SIAM Conference on Mathematics of Data Science, *San Diego*, virtually (Sept. 2022) - Spring School on the Mathematical Design of Materials, *Isaac Newton Institute for Mathematical Sciences, Cambridge, UK* (Mar. 2019) **_Research and Travel Awards_** - PUI Faculty Travel Grant, *Joint Mathematics Meetings* (2023) - SIAM Early Career Travel Award, *SIAM Conference on Mathematical Aspects of Materials Science* (2018 and 2021) - Bio5 Institute Team Scholars Program Award (one course release), PI’s: Joceline Lega and Heidi Brown, *Bio5 Institute, University of Arizona* (2018-2019) - Certificate of Excellence in Research Award, *Office of Interdisciplinary Graduate Programs, Purdue University* (2015) Teaching ============================================================== My teaching is sustained by my passion for mathematics and my evolving practice teaching it. Over time, I came to believe that a student-centered classroom provides an opportunity for students to fully participate and do mathematics together. In this environment, my role is to promote this participation by instructing with questions rather than answers and giving students appropriate time, space, and resources to engage these questions. My hope is to make learning mathematics both fun and appropriately challenging to promote community and growth. Furthermore, my grading evolved to be based on student reflection on my feedback and opportunities for showing progress with less time restrictions than the traditional system provides. My pedagogical methods are a fusion of ideas from inquiry approaches to learning and alternative grading. In 2016-2017, I received the Teaching and Service Prize for non-tenure faculty from the Mathematics Department at the University of Arizona. In 2012-2013, I received the Excellence in Teaching Award from the Department of Mathematics at Purdue University. The same year, I received the CETA Excellence in Teaching Award from the Committee for the Education of Teaching Assistants at Purdue University. Courses --------------------------------------------------------------- - Spring 2024 courses - Introduction to Proofs through Analysis, Math 206 - Linear Algebra, Math 211 Over the years, I've taught courses ranging from introductory to advanced undergraduate levels. I also coordinated a seminar for teaching assistants. During my graduate studies, I was a recitation instructor for the first 3 semesters and the main instructor afterwards. After receiving my Maîtrise (Bachelor of Science and Teaching Diploma) in Mathematics, I worked as a full-time high school teacher for 3 years. During that time, I earned a professional degree modeled after the French CAPES, a certificate for teaching mathematics in secondary education. Below is a list of courses I have taught. - At Mount Holyoke College - Calculus I, Math 101 (2 semesters) - Calculus III, Math 203 (3 semesters) - Linear Algebra, Math 211 (1 semester) - Introduction to Proofs through Analysis, Math 206 (4 semesters) - Real Analysis, Math 301 (2 semesters) - Introduction to Partial Differential Equations, Math 339PD (1 semester) - At the University of Arizona - Calculus I, Math 125 (1 semester) - Vector Calculus, Math 223 (1 semester) - Real Analysis of Several Variables, Math 425/525B (1 semester) - Real Analysis of One Variable, Math 425A (1 semester) - Complex Analysis, Math 424/524 (1 semester) - Undergraduate Teaching Assistants seminar, Math 491 (4 semesters) - At Purdue University - Calculus I - life sciences focus, MA23100 (1 semester) - Calculus II - business focus, MA22400 (3 semesters) - As a high school teacher: grades 9 to 12, mathematics and SAT prep. Professional Development --------------------------------------------------------------- I strive to keep sharpening my teaching skills, so I participate in workshops and reading groups regularly. I've also been part of long-term programs focused on growing as a faculty member and making the mathematics field more inclusive. Below is a list of programs I have participated in. - Cultural Competence in Computing (3C) Fellows Program, Cohort 2, *Alliance for Identity-Inclusive Computing Education (AiiCE)* (2021-2022) - Faculty Success Program (12 weeks), *National Center for Faculty Development and Diversity (NCFDD)* (Fall 2022) - Project NExT (New Experiences in Teaching), *Silver '19* (2019-2020) - Named one of two inaugural SIAM Project NExT fellows **_Professional development workshops_** - Workshops I regularly participate in: - The Grading Conference, Higher Ed. STEM Focus - New England Community for Mathematics Inquiry in Teaching (NE-COMMIT) - Other workshops: - Broadening Participation: 2020 MPS Workshop, NSF, 2-day virtual workshop (Nov. 2020) - STEM Faculty Launch workshop, Worcester Polytechnic Institute (Oct. 2018) - Inquiry Based Learning workshop (certificate), DePaul University (June 2018) - Diversity in the Classroom workshop series (certificate), University of Arizona (Spring 2018) Outreach & Service ============================================================== **_Outreach_** I am passionate about making mathematics more accessible and enjoyable for a variety of groups I work with. Such groups include: high school teachers, teaching assistants, women in mathematics/statistics, and middle school students. - Association for Women in Mathematics Chapter Advisor, *Department of Mathematics and Statistics, Mount Holyoke College* (2020-present) - Outreach one-week course instructor - Mathematical Modeling in Ecology: Population Dynamics, for early college students (specifically first-generation students), *Jumpstart Workshop* (Aug. 2023) - Single Variable Optimization, for middle school students, *MathPath* (July 2021, 2022) - Tucson Math Teachers' Circle Facilitator, *Center for Recruitment and Retention of Mathematics Teachers, University of Arizona* (2016-2018) - President and Founding Member, *Purdue Chapter of the Association for Women in Mathematics* (2011-2013) - Chapter received the AWM Professional Development award in the 2018 MAA MathFest, in part due to ongoing programs started by our cabinet **_Service_** Below you can find my service to the profession and the mathematical community. - Undergraduate Teaching Assistantship Program Organizer, *Math Center, University of Arizona* (2017-2019) - Initiator and Organizer of Basic Skills Workshop, *Purdue Chapter of the Association for Women in Mathematics* (2012-2016) - Organized workshops to help graduate students with career planning, teaching, using technological tools, preparing for job applications, etc. - Still run by current graduate students and now counts as a 1-credit seminar - Seminar/Minisymposium Co-organizer - Developing Computational and Applied Mathematics Courses Minisymposium, *Project NExT session, MAA Math Fest* (July 2020) - Analysis, Dynamics, and Applications Seminar, *Department of Mathematics, University of Arizona* (2017- 2018) - PDE Models for Pattern Forming Systems Minisymposium, *SIAM Conference on Analysis of Partial Differential Equations, Baltimore* (Dec. 2017) - Computational Science and Engineering Student Conference, *Purdue Chapter of the Society for Industrial and Applied Mathematics* (2015-2016) - NSF reviewer (2022, 2023)

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