UROP Research Mentor Project Submission Portal: Submission #1117

Research Mentor Information
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Primary Research Mentor Name: Alexander Reznikov
Research Mentor Preferred Pronouns: He/His
When potential research assistants are reaching out via email, what is your preferred honorific?: Dr. 
Contact Email (FSU Email if affiliated): areznikov@fsu.edu
Position Title: Faculty
FSU College (if applicable): Arts and Sciences
FSU Department or Non-FSU Organization Affiliation: Mathematics
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Additional Research Mentor(s)
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Research Assistant Supervisor (if different from above): {Empty}
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Research Assistant Supervisor Preferred Honorific?: {Empty}
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Overall Project Details
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Title of the Project: *** Gradient descent method for potential energy of four equal charges
Project Keywords: Potential theory, real analysis, optimization
Are you currently looking for research assistants?: Yes
Number of Research Assistants Needed : 1
Relevant Research Assistant Major(s):
Mathematics, statistics, physics, computer science, engineering

Project Location:: On FSU Main Campus
If the project location is off campus, does the research assistant(s) need to provide their own transportation?: {Empty}
Please select the choice that most accurately describes your expectations for the research assistant(s):: Partially Remote
Approximately how many hours a week would the research assistant(s) need to work?: 7
Roughly what time frame do you expect research assistant(s) to work?: Flexible schedule (Combination of business and outside of business. TBD between student and research mentor.)
Overall Research Project Description:
Imagine placing four electrons on the sphere of radius 1 (a common beast in a Calculus class). They will repel each other, and therefore move along the sphere. They will stop moving when they reach an "equilibrium state"; mathematically speaking, when their potential energy is minimized. Unfortunately, the word "minimized" has different meanings in pure math vs the real life. In reality, this stable state can be reached at a critical point of the potential energy (the difference between a critical point, a local minimum and a global minimum is a common Calculus 3 problem). 

For the project, I propose to collect numerical evidence for the following conjecture: unless the four electrons were placed in a very unfortunate locations, they will reach the stable state at the global minimum of the potential energy. Specifically, they will arrange themselves in the vertices of a regular pyramid. 

Research Tasks:
The main task will be to develop an efficient script implementing the gradient descent method for a specific "potential energy" function. 

The efficiency of the script can be evaluated as follows: the same question can be asked in a more general setting (for example, in higher dimensions), when the number of variables grows. The script is expected to be adaptable to those settings.

Skills that research assistant(s) may need::
Required: good knowledge of the first half of Calculus 3
Recommended: knowledge of a common programming language, such as C or Python. 

The project can be viewed as a learning opportunity to better learn such a language.

Mentoring Philosophy:
The main point in my mentoring philosophy is fostering independent work of a student and teaching the student to come up with ideas, problems, and their solutions on their own.

Please provide a link to your publications, a video clip, or a website for your research project (if applicable):: https://scholar.google.com/citations?user=x5-srkIAAAAJ&hl=en&oi=sra
Please add any additional information here (if applicable)::
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Are you interested in participating in the UROP Research Mentor Roundtable?: No
Roundtable times and Zoom links:
N/A

UROP Program Elements
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Mentor Handbook, FAQs, and Communication: Yes
UROP Performance Evaluation: Yes
Materials Grant: Yes
UROP Poster Presentation: Yes

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Year: 2025
update url: https://cre.fsu.edu/urop-research-mentor-project-submission-portal?token=91ayh55k6hUAgb7uGp4sioFocPyLCV7b3u0_eibv3H8