Maciej Piwowarczyk
Doctoral Student in Mathematics @ UNL
What am I Currently Working On?
Who Do I Work With?
My advisor is Dr. Christine Kelley (her UNL website).
What do I Work On?
My current research interest is DNA Coding Theory!
I'm Sorry, What?
DNA Coding Theory! Let's break it down:
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Coding Theory
In very broad terms, Coding Theory is the study of transmitting data quickly, efficiently, and free of errors. Formally established in 1948 by Claude Shannon, the original focus was on the problem of how best to encode information being transmitted. Since then, the focus has split into several subfields, including source coding (how to take data and make it smaller), cryptographic coding (keeping data secret as it's transmitted), and channel coding (sending data efficiently and with the capacity to detect or correct errors).
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Here's a small example of channel coding;
I want to send the message "HI" to my friend, but every time I send a letter, there's a chance that the wrong letter is received. So, I protect the message by sending each letter three times.
Encode
"HI"
"HHH III"
Send
Decode
"HIH IIH"
"HI"
Even though two letters flipped (2nd "H" and 3rd "I"), my friend is still able to correctly decode the message based on the most common letters!
Okay, But What Was That About DNA?
DNA Coding Theory is all about storage, storage, storage. Humans produce a lot of information that we would like to keep around and modern storage mediums are good, but come with disadvantages that DNA can address.
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In short, DNA Coding Theory is all about creating a storage system that is dense, stable, and energy efficient. We hope to be able to store long-term rarely-accessed data, like government archives, in DNA. Initial experiments in 2012 by Church et al. as well as in 2013 by Goldman et al. showed the potential of DNA as a storage system. Research groups since then have continued developing new ideas to address current roadblocks, such as cost and time.
G
A
T
T
A
G
A
G
T
C
H
E
L
L
O
W
O
R
L
D
So What's Your Role In This?
Throughout the DNA Storage process, many errors will be introduced. Some letters will be omited, others will change to other letters, some might get moved around; All of these errors are possible. Currently my research is focused on designing codes that are resistant to these errors by detecting them quickly. I am using graphs to design my codes for now and our early results are looking promising!
NOTE: This is a VERY
inaccurate example.
Undergraduate Work
Senior Thesis
To finish off my undergraduate degree at DePaul University, I studied the irreducible representations of a very particular ring R. We construct R by first taking a finite abelian group G and scaling it with the complex numbers to form a group ring CG. We then consider the exterior algebra of a complex vector space V, denoted by Λ(V). Taking the semidirect product of these two objects gives us the ring
R = CG#Λ(C)
In the end, we found that all irreducible R-modules are one-dimensional and are spanned by a G-eigenvector.
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This research was done under the supervision of Dr. Christopher Drupieski.
Student Researcher
DePaul's Undergraduate Summer Research Program
In this project we studied the mathematical concept of the Frobenius number and some curious patterns that come with it. One common example of the Frobenius number is the Coin Problem: If handed two denominations of coins, say 4¢ and 5¢, and asked to create all possible values, we will eventually find ourselves in a position where we can make any value. With 4¢ and 5¢ coins, we can create any value above 11¢, but not 11¢ itself. So, that makes 11 the Frobenius number of 4 and 5. What we explore in this paper is a pattern we call Frobenius symmetry: when all non-negative integers below the Frobenius number can be paired up such that one number is attainable, and the other is not. We looked at sets of two and three numbers and arrived at results about both.
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This research was done under the supervision of Dr. David Sher.
Publications
[1] Piwowarczyk, Maciej (2019) "The Mystery of Frobenius Symmetry," DePaul Discoveries: Vol. 8 : Iss. 1, Article 5, https://via.library.depaul.edu/depaul-disc/vol8/iss1/5