Julianne Vega

  • Graduate Student
  • Mathematics
722 Patterson Office Tower
859-257-6807
Research Interests:
Research
In 2017, I completed my Masters with Bert Guillou on Quillen's fiber lemma for posets. I am currently working towards my Doctorate in Topological Combinatorics with my advisor, Benjamin Braun. My research interests lie in Simplicial Complexes, Graphs, and Posets.  
 
In my current projects, my overall interests involve learning about properties of graphs. To do so, I create spaces, i.e. simplicial complexes, that encapsulate the properties that I am interested in and analyze the characteristics of the space. 
 
Within my dissertation research, I am currently finishing up my first project with my advisor, Ben Braun. Our project explores the topological effects of the Haj\'os merge and vertex identifications on the neighborhood complex of a graph. Since the connectivity of the neighborhood complex of a graph gives a topological lower bound for the chromatic number of the graph, we were curious about how  how the connectivity of the neighborhood complex changes under these operations; questioning how do highly connected neighborhood complexes arise through these operations? As this project comes to a close, I will begin a second project joint with Ben and start an independent project.
 
Outside of my dissertation work, I am involved in two collaborations that resulted from the Graduate Research Workshop in Combinatorics that I participated in at the beginning of last summer. One of the projects developed from a problem that I proposed that focuses on learning more about matching complexes for particular types of graphs. As the informal leader of this project, I have strengthened my skills to motivate a collaboration as well as maintain communication and focus. Two papers are resulting from this project. One that focuses on classifying the graphs that give rise to matching complexes isomorphic to spheres. The second develops tools to decompose graphs leading to a better understanding of homotopy type of the matching complexes. Determining the homotopy type is a particularly difficult question to answer and we have used our methods to understand the homotopy type of one-child binary trees and caterpillar graphs. My second collaboration is looking at a question that originates from Veit Elser's observations about a perturbation model in physics. We have been working to show that an alternating sum over vertex covers of graphs is non-negative. Since last summer, we have showed that the conjecture holds for particular types of graphs, constructed a chain complex whose Euler characteristic captures the sum, and used homological methods to understand the sum better. I have recently been awarded the Center for Graduate and Professional Diversity Initiatives Professional Development Grant through the university and I will use this money to host one of my collaborators in Spring 2019. During the course of her visit, we will continue to work on our collaboration and she will present in CATS seminar. This experience will help me build my network and strengthen my leadership skills. 
 
 
In addition to continuing my collaborations with my advisor and maintaining my collaborations with my GRWC groups, I am interested in trying to work on a research project independently and mentoring undergraduates in a research capacity. I am currently in the process of exploring topics for both of these aspirations and I expect to make continued progress toward these goals. 

 

Teaching
I am fortunate to have had by first teaching experience at Burgundy Farm Country Day School a progressive, independent school in Alexandria, VA. It was there that I developed as a teacher leader and learned the importance of working with colleagues to build the best possible learning community. I took on many leadership positions, both formal and informal, that placed me at the heart of what makes Burgundy beautiful. My classroom encouraged curiosity and invited students to actively participate in their learning.  
 
Moving into my current position as recitation leader and primary instructor has led me to reflect on how to adapt my teaching to elicit the same curiosity and engagement in college students. One way that I do this is by having students ask questions in groups and throughout class. I am currently teaching a medium-sized lecture of College Algebra (70 students). Through this experience I am learning how my teaching practices can be further adapted to reach more students at once. In the past, I could have class discussions on new topics, bringing discovery to the class as a whole, but in a larger lecture this is impractical, so I provide time for individual discovery at which time I walk around and work with small groups of students. My undergraduate assistant and I work as a seamless team, answering questions that arise during work sessions. 
 
Part of my professional development last year was to observe professors and post-docs at different types of departments. During my visit to Macalester College, after observing a professor's persistent enthusiasm in an early morning class, I concluded that I should always present my best self in every class, despite students' wavering enthusiasm. There are many outside factors that affect a student's engagement in a class and as a post-secondary educator flexibility and patience come in the form of understanding these outside factors and working with students to become dedicated and focused on their studies.  
Service
I strive to be a community leader. As such I believe in the importance of service. I am involved in several service commitments that I believe have a positive impact on the department and on the surrounding community. 
I am part of the Inclusive Pedagogies Graduate Learning Community, a collaborative effort for inclusion through the College of Arts and Sciences. Together, with graduate students from many departments, we challenge each other's views of what it means to build an inclusive space and are working towards creating a product that can be used university-wide to help faculty and graduate teaching assistants build more inclusive classrooms. This is a critical activity that will help recruit and retain underrepresented students. I am currently researching the question, how we can help marginalized students in a way that does not add to their burden? The questions that my cohort and I are trying to answer are not trivial and I have already experienced growth through our conversations and my reflections. 
 
In addition to participating in the Graduate Learning Community. I am also leading two efforts within our department, the Appalachian Initiative for Mathematics (AIM) and Inclusive Community Lunches (ICL). These two efforts work in tandem, through AIM we wish to help a broader group of undergraduate students view graduate mathematics as a viable next step. We want to encourage students to continue to explore their passions and help them find their way. As this is the first semester, our efforts are in building relationships. We have had two AIM visits and believe that we are making progress towards our goals. 
 
While recruitment is important, retention is vital. The purpose of Inclusive Community Lunches is to learn more from each other and to build an inclusive learning environment within the department. Thus far, we have had three meetings, each filled with open minds and open hearts. I am learning a lot from my colleagues and together we are pushing ourselves out of our comfort zones to learn more about our impact on building an inclusive space. Our hope is that we can help professors and other peers see the importance of reaching out, learning more about each other, and supporting each individual on their journey.  
 
Outside of my concerted efforts on diversity, I am also the AWM vice president of our current student chapter. Along with my AWM officers, we have developed committees to incorporate the input of our members into our event planning. A consequence of that is the formation of several committees and collaborations with other student organizations in STEM fields. I am currently leading the Research Workshop Committee. Our goal is create a math positive environment within the department by encouraging students to talk with each other and faculty about their math projects and to begin to see each other as colleagues and collaborators in addition to peers. 
 
I am also an active volunteer in Nerd Squad and Math Circle, two programs that reach out to elementary-high school students in the area to introduce them to math concepts. All of my service to the community evidences my desire to build a rich learning environment that encourages curiosity, passion, and life-long learning. 
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