Math and Cookies is an undergraduate research seminar. Starting from Spring 2023, Math and Cookies will be combined with the course Seminar Talk Experience.

The following is the list of tentative speakers for the seminar for Spring 2023. We meet on Tuesdays 12:30 - 1:30 

• January 24

Informal meeting with students and Faculty members (Moshe Cohen, Cheyne Glass, Hunter Park, Jeungeun Park)

This is the first meeting of the semester. Several faculties will join to discuss what mathematics research means to each of us. 

• January 31

Speaker: Jaiung Jun (SUNY New Paltz)

Title: Introduction to chip-firing games

Abstract:  A chip-firing game is a combinatorial game which one can play on a finite graph. Chip firing games arise from various areas of mathematics such as algebraic geometry, combinatorics, and number theory.  In this talk, I will introduce basic notions of chip-firing games. Then I will explain several results on chip-firing games which I proved with my former students at New Paltz, Alessandro Chilelli and Matthew Pisano. 

• February 7

Speaker: Matthew Pisano (Rensselaer Polytechnic institute)

Title: Directed chip-firing games and their undirected couterparts. 

Abstract: We explore a combinatorial game on finite graphs, called Chip-Firing Games, which has various connections to other areas, such as algebraic geometry, number theory and economics. To play the game, one first puts an integer amount of chips at each vertex. Then, each vertex is allowed to borrow or lend chips from its neighbors, equally, as the game progresses. One can study chip-firing games on a graph G through the Picard group Pic(G), a finitely generated abelian group, and its torsion subgroup, the Jacobian Jac(G). These can be computed by using the Laplacian matrix of G. When a graph G is directed, one may define Pic(G) and Jac(G) as in the case of undirected graphs by using Laplacian matrices, but computations become much more complicated in this case. These graphs, and their relations to their undirected counterparts is what we focus on here. In our project, we study Picard groups and Jacobians for directed trees, cycles, pseudotrees, wheel graphs, and multipartite graphs. Even in these seemingly simple cases, we find some new phenomenon. For example, we can observe that some orientations of wheel graphs, closely match their undirected counterparts. We can also see how, by closely examining trees and cycles how Picard groups and Jacobians change with (suitably defined) vertex and edge gluing, we obtain several results for pseudotrees. This is joint work with Jaiung Jun and Youngsu Kim. 

• February 14

Speaker: David Hobby (SUNY New Paltz)

Title: Vector proofs in geometry.

Abstract: Fix an origin O in n-dimensional space. We consider a point P to be the vector from O to P. Then the midpoint of the segment AB corresponds to the vector (1/2)(A + B). We can then use a vector calculation to show that the medians of a triangle intersect at one point. Extending these ideas, we can investigate midpoints of sides of other polygons. 

• February 21

Speaker:  Alessandro Chilelli (University at Albany)

Title: Graph Jacobians, Gluing, and Tutte's Rotor Construction 

Abstract:  Every finite connected graph has a finite abelian group associated with it called the Jacobian which translates graph-theoretic properties into algebraic properties. Due to joint work published with Jaiung Jun, we studied these objects and asked how the Jacobian of a graph can be predicted from the Jacobians of smaller graphs glued together. In this talk, I’ll discuss the main results of our paper including our most useful techniques. Then I will finish with an application to Tutte’s Rotor Construction which was the main motivation of our work. 

• February 28

Speaker: The talk has been cancelled due to the snowstorm. 

• March 7

Speaker: Vincent Martinez (CUNY Hunter College)

Title: Non-differentiable solutions of differential equations? 

Abstract: Recall that a differential equation is an equation that involves a function and it derivatives. How is it then possible to solve a differential equation if the function isn’t differentiable? This talk will introduce one of the most fundamental ideas in the modern study of differential equations in the concept of “weak solution.” We discuss this notion in the particular context of a simple model for fluid motion called Burgers’ equation, for which some non-differentiable solutions actually appear in nature in a dramatic way! 

• March 14

No meeting. Spring break!

• March 21

No meeting. Thursday schedule!

• March 28

Speaker: Youngsu Kim (California State University San Bernardino)

Title: Sandpile Groups 

Abstract: In this talk, we introduce abelian sandpile groups and their connections to graph theory and algebra. Sandpile models were invented by Bak, Tang, and Wiesenfeld to study self-organized criticality in dynamical systems. Its algebraic interpretation, called critical or Picard groups, can be defined by chip-firing games. We provide explicit forms of critical groups for several classes of directed graphs. This is joint work with J. Jun and M. Pisano.  

• April 18

Speaker: Anthony Cooper (SUNY New Paltz)

Title: Synchronization and clustering in complex quadratic networks 

Abstract: Many systems in the life sciences are organized as networks of dynamic, interacting units. One extremely studied example is the brain. In neuroscience, a lot of effort has been invested recently in understanding how brain connectivity patterns affect the ensemble dynamics of the system and, subsequently, brain function and behavior. In our work, we place this question in a canonical framework using model networks with complex quadratic nodes. Using an extension of the traditional Mandelbrot set for our complex quadratic networks (CQNs), we define a concept of "synchronization" (based on behavior of individual nodes), then we investigate the mechanisms that lead to nodes belonging to the same synchronization cluster. We show that clustering is strongly determined by the network connectivity patterns, with the geometry of the clusters further controlled by the connection weights.  Finally, we illustrate the concept of synchronization in an existing data set of whole brain, tractography-based networks obtained from 197 human subjects. This example showed us how our approach can help understand and contextualize existing open questions in neuroscience, such as the role of weak connections between brain regions.  As an undergraduate student working on this project throughout a five week long summer research experience, this was my introduction to how mathematics can be applied to the life sciences. I was able to gain a deeper understanding of both the mathematical content, and on how mathematical modeling can increase our understanding of cognition and behavior.

• May 2 (Last meeting)

Informal meeting and discussion with students

• Note: Dr. Sistko's talk is rescheduled to Fall 2023. 

Speaker: Alex Sistko (Manhattan College)

Title: Linear Recurrences and the F1-Representation Theory of Quivers 

Abstract: Representation theory is the study of abstract mathematical objects via linear algebra. Emerging over a century ago, this theory has become an indispensable tool within modern mathematics, having its impact felt in disciplines as varied as number theory, physics and data science. The representation theory of quivers (i.e. directed graphs) is of particular interest in this field. In this talk, we discuss recent advances in the related combinatorial theory of quiver representations over F1, the so-called "field with one element." More explicitly, we show how linear recurrences can be used to describe the asymptotic growth of F1-representations of proper pseudotrees. No background knowledge will be assumed in this talk, and all students with an interest in contemporary mathematical research are encouraged to attend. This is joint work with Drs. Jaiung Jun and Jaehoon Kim.