I'm a fourth-year graduate student in UC Davis working with Anne Schilling in the field of algebraic combinatorics and symmetric functions. I'm also interested in enumerative combinatorics, where I'm researching different statistics on Dyck paths.

More recently my interests are shifting towards Data Science. In particular, I'm currently looking into topological properties of the image data corresponding to different types of motion.

I also enjoy teaching mathematics. As an Associative Instructor in UC Davis, I've taught Calculus courses and a Number Theory course, and I've been a Teaching Assistant for many more courses.

I received my undergraduate degree with honors in Moscow State University in 2013, where I specialized in stochastic processes.

Topological Analysis of Image Data

We consider a simple angular motion of two beams with fixed center (e.g. two hands of a clock). We randomly generate 50x50-images corresponding to different states of that motion. Here, we want to choose an appropriate fading to make the eucledian distances between close images relatively small.

Next, we represent 2000 samples of the images as vectors in 2500-dimensional space and put them together in a dataset. In order reduce the noise in the topology of data, we apply the diffusion map. A scatter plot of the 'diffused' dataset is to the right.

Finally, we use persistent homology utilities of the rTDA package to determine topological structure of the dataset. Unfortunately, memory limitations only allow us to get a persistence diagram for dimension 1 of the homology (to the left). The diagram tells us that Betty number of dimension 1 is equal to 1 (corresponds to the red triangle on top of the diagram), which implies that the surface is either a real projective plane or a Klein bottle.

One of the future research directions is to optimize the algorithm of persistent homology in rTDA and apply it to the other types of motion.

Bigraded Fibonacci Numbers

It was discovered that the set of rational (a,b)-Dyck paths are in bijection with certain partitions called (a,b)-cores. Thus, the number of (a,b)-cores is equal to the rational Catalan number C(a,b). Here, we count only (a,b)-cores with distinct parts, which turn out to be the generalization of Fibonacci numbers F(a,b). We use the Anderson's bijection between cores and abaci diagrams (to the right), and introduce a natural grading on that set.

Here is the poster with more detailed explanation.

Summer 2016 - MAT115A - Number Theory (Final Exam)

Fall 2015 - MAT21C - Calculus (Final Exam)

Summer 2016 - MAT16C - Calculus (Final Exam)