User:Robert Abramovic

Robert Abramovic 410 Yale Avenue Swarthmore, PA 19081 rabramo1@jhu.edu Home (610) 541-0368 Cell (610) 316-8265 __________________

Research Topics

Differential Geometry and its application in physics, especially as related to gravitation - general relativity/quantum gravity. Mathematical Physics. Geometric Flows. Partial Differential Equations. Lie Groups.

Summary of Achievements

Young Scholar 2005 – 2007, University of Pennsylvania, Philadelphia, PA

Dean’s List: Spring 2008, Spring 2009, and Fall 2009 Johns Hopkins University, Baltimore, MD

Nomination for Best Tutor Fall 2009

Membership in Sigma Pi Sigma Physics Honors Society Spring 2010

Seven Nominations for a Teaching Award from Students in my Recitation Section Fall 2010 – I served as a TA for section 5 of 110.106 Calculus I for Biology and Medicine

Teaching Experience

Tutoring:

Through the Learning Den, Office of Academic Advising Johns Hopkins University, Baltimore MD

Courses Tutored:

Linear Algebra, Spring 2008 – Present

Calculus III, Fall 2007 – Present

Ordinary Differential Equations, Fall 2008 – Present

Teaching Assistant (TA): Fall 2010 – Present Through the Johns Hopkins University Mathematics Department Course: 110.106 Calculus I for Biology and Medicine In addition to holding a recitation section, my work involves holding office hours, working at the Mathematics Help Room, proctoring and grading exams, grading homework, and constructing, giving, and grading quizzes.

Schools Attended

2003 – 2006 Strath Haven High School, High School Diploma Honors

Fall 2004 – Summer 2005 Temple University: Continuing Education

Fall 2006 – Spring 2007 Temple University: Sub-matriculation from High School

Fall 2005 – Spring 2007: University of Pennsylvania Mathematics Classes through the Young Scholars Program for High School Students

Currently pursuing MA/BA in Math and a BA in Physics at the Johns Hopkins University

Seminars Attended

I have consistently attended the following lecture series:

Spring 2003 – Spring 2006 (As a High School Student) Distressing Mathematics Collective at Bryn Mawr College. Topics ranged from number theory to knot theory.

Spring 2006 (as a High School Student) Geometry and Topology Seminar at the University of Pennsylvania Topics included four dimensional manifolds and Kirby calculus along with group theory.

Fall 2009: (As a Johns Hopkins University Junior) Analysis Seminar organized by Professor Chrisopher Sogge

Fall 2007, Fall 2008, and Fall 2010 (As a Johns Hopkins University Freshman, Sophomore, and Senior) Attended training for and participated in both the Virginia Tech Regional Mathematics Contest (Fall 2007, Fall 2008) and William Lowell Putnam Examination (Fall 2007, Fall 2008, and Fall 2010)

Community Service:

Fall 2007: Habitat for Humanity Pre-Orientation Program 7 hours a day for 4 days in August 2007 Reference: Ariana Barkley, E-mail: abarkle2@jhu.edu

B-More Program social service; 2 hours a day; last week for intercession 2008 Reference: Adam Richard, E-mail: aricha30@jhu.edu

Relay for Life All Night, April 16, 2010 Reference: Ms. Novic, e-mail: tutoring@jhu.edu

Fall 2010: Member of Circle K Community Service Group 2-5 Hours a Week for duration of Semester Refernce: Laritza Mendoza, E-mail: laritza.mdza@gmail.com

Additional Work Experience Fall 2003- Spring 2007 Go Vertical Rock Climbing Gym Belayer Job Description: I held the ropes for the climbing customers and fitted them in their harnesses Employer: Howard

Research Experience

Summer 2010 – Present REU – Geometric Differential Equations with Professor Xiaodong Cao Cornell University, Ithaca, New York Research Included: Ricci flow on warped products, differential Harnack estimates, References Texts Included: Collected Papers on Ricci Flow, HD Cao, B Chow, SC Chu and ST Yau eds, Series in Geometry and Topology 37 (2003) 163-165, International Press.

Summer 2009 - Present REU- Geometry and Physics on Graphs with Professor Stratos Prassidis – Canisius College, Buffalo, NY Research included: Harmonic Maps on Cayley Graphs (relationship with group theory), Compactification , Floyd Boundaries, the Cayley Graph of a Heisenberg Group over a Ring, and Heat Kernel on Higher Dimensional Lattice Graphs.

Summer 2008 – Summer 2009 
Research on Theoretical Astrophysics with Mr. Scott Noble
-Johns Hopkins, Baltimore, MD 

Research Included: Numerical Solutions of the Geodesic Equation using Java and C++, Black Holes, Spins, Orbits, and Plots using Mathematica Reference Textbook: Robert M. Wald’s General Relativity

Presentations of REU Research Results:

           Harmonic Maps and the Dirichlet Problem 
           Canisius-RIT Joint Conference, Rochester Institute of Technology, July 2009 
            
           Harmonic Maps on Cayley Graphs and Compactification 
           Young Mathematicians Conference, Ohio State University, August 2009

           Harmonic Maps on Cayley Graphs and Compactification
           Shenandoah Undergraduate Mathematics and Statistics Conference 
           (SUMS Conference) at James Madison University, October 2009 
            I also served as a panelist for REU programs. 

           Ricci flow on Warped Product Manifolds and Pinching Estimates 
           Young Mathematicians Conference, Ohio State University, August 2010 

Works in Progress : Harmonic Maps on Cayley Graphs and Compactification Professors Stratos Prassidis and BJ Kahng Canisius College

Harnack Estimates and Ricci Flow on Warped Products Professor Xiaodong Cao Cornell University

Essay on REU Research and Research Experience:

As an undergraduate, I have worked independently on three research projects. My first experience involved taking the initiative to become involved in theoretical astrophysics with a scientist in the physics department. Two subsequent projects were completed during summer REU, research experience for undergraduate, programs, at Canisius College and at Cornell University respectively. During both of the REU programs, I collected my results in two papers entitled Harmonic Maps on Cayley Graphs and Compactification and Harnack Estimates and Ricci Flow on Warped Products, at least one of which will be considered for publication. The summer after my freshmen year, I became involved in a project with Mr. Scott Noble, then an associate research scientist at Johns Hopkins University. Our research aimed to investigate the behavior holes interact is a groundbreaking and vital topic in astrophysics. How black holes interact is a groundbreaking and vital topic in astrophysics. In order to observe and analyze their behavior in the cosmos, scientists need to first understand how general relativity theory will predict the interaction of the black holes. Using the Java computer program, I wrote a code for numerically solving ordinary differential equations, with both Runge-Kutta and Euler’s method, for the geodesics in the Schwarzchild spacetime metric arising in general relativity of an array, or surface, of light in the vicinity of two rotating black holes. At first, I worked with the metric for a non-rotating black hole, entering different initial conditions for the four-momentum of the photon to test the existence and radii of various circular orbits. Next, I moved on to testing different spins. Finally, I produced a variety of plots of the orbits using Mathematica, and compared my results to predictions based on intuition from general relativity.

	At Canisius, I first worked with Professor Stratos Prassidis on finding the heat kernel on a two-dimensional infinite lattice graph. These findings were inconclusive. Next I studied a paper on discrete analogues of the Laplacian, combinatorial Laplacian, and harmonic maps on finite graphs. From this article, I investigated to what extent these same results held for infinite graphs. Additionally, I discovered a basic relationship between homomorphisms on groups and harmonic maps on their Cayley graphs. This relationship lead to a conjecture on how the nature of harmonic maps on Cayley graphs determine algebraic group structure.  In particular we determined whether or not the generating group was cyclic. Since certain groups appeared to give counter-examples, I reduced the case to one of free groups or abelian groups. After working with some effort for both cases, I proved the contrapositive of my conjecture using induction and some basic results from group theory. Professor Prassidis also asked me to investigate information on Floyd boundaries and compactification of metric spaces. Using a variety of papers on the subject as a basis, I constructed and proved two propositions about what I termed “harmonic isometries,” harmonic maps that also preserve metric properties, and the Cayley graphs on which they are defined. Metric properties and differential operators on graphs have important applications to studying the quantization of position, energy, and spacetime postulated by quantum physics. The Laplacian operator in particular plays an important role in thermodynamics and electromagnetism, both essential physical theories in the development of technology. 

My most recent experience involved studying Ricci flow with Professor Xiaodong Cao at Cornell University. Ricci flow has become an increasingly noteworthy topic in pure mathematics and the differential geometry of higher-dimensional manifolds since its use in Perelman’s proof of the Poincare conjecture. Before attending the program, I researched a few of Professor Cao’s and Professor Richard Hamilton’s papers on the subject of Ricci flow on warped product metrics and isoperimetric estimates. The papers used isoperimetric estimates to exclude the possibility of the cigar soliton as a dilation limit. Since Perelman has already used gradient estimates to prove a more general result, employing further applications of the isoperimetric estimate did not seem necessary in approaching similar but more general problems. After discussing the topic with Professor Cao, we came up with a new conjecture, that the Ricci flow on a warped product of a two-dimensional manifold with constant sectional curvature, the sphere, the hyperbolic plane, or torus, will approach the direct product metric under Ricci flow. We began considering the case of a compact manifold and looked into a paper written by Professor Cao and Professor Hamilton on differential Harnack estimates. Using the technique outlined in this paper, I proved a lemma asserting our original conjecture in the case of a two-dimensional, compact base manifold, assuming long-time existence of the flow. To move to the case of a higher-dimensional base manifold, I reviewed three papers under Professor Cao’s direction that dealt with finding the evolution equations for the Hessian and the gradient of the function responsible for warping the metric. I generalized one of Professor Cao’s results about finding a lower bound for the Ricci curvature tensor restricted to the circle (non-base) manifold, comparing the Laplacian of the function to the total scalar curvature of the entire manifold. We are currently exploring the evolution of the Hessian and trying to find a similar estimate, or lower bound, for the Ricci curvature tensor restricted to a three-dimensional base manifold. Our findings are important because, for manifolds of dimension less than three, nonnegative Ricci curvature is always preserved under Ricci flow. However, this property does not hold for four manifolds. Finding a lower bound for the Ricci curvature of a four manifold (three-dimensional base manifold) therefore has important geometric implications in the theory of four-manifolds and its application in analyzing the curved spacetime of general relativity.