The SUM Series features informal talks, panels, and discussions on mathematical topics.  All are welcome!  

**Join the mailing list to receive announcements before each event.  Add the SUM Series Google Calendar.**

Upcoming Events

5 Minute Math

Speakers: Many Undergraduate Students!

Thursday, April 23, 4:30-5:30 SAS Hall (pizza provided!)                   add to Google Calendar

Abstract: Join us for 5 Minute Math, a special event presented by UUG in collaboration with the SUM seminar series. Participants in UUG's Directed Reading Program share the results of a semester’s worth of research across diverse math disciplines, all in just 5 minutes each. Seven groups of talented students have been working hard all semester to explore advanced math topics with their graduate student mentors. This event is their chance to share that work with all of you!  Come ready to learn something new, celebrate student-driven projects, and support your peers. If you're interested in participating in the Directed Reading Program next year, this is a great opportunity to network, meet potential mentors/mentees, and see what kind of projects you could be working on.  (Questions? Contact Stephanie Stewart at smstewa7@ncsu.edu)  Be there or b^2.

Math Honors Presentations

Speakers: Nishant Ajitsaria, Mason McElroy, Noah Siekierski, Hutch Smith 

Tuesday, April 28, 4:30-5:30 SAS Hall (pizza provided!)                   add to Google Calendar

Note that this talk is on a TUESDAY.

Abstracts: 

Hutch Smith -  Conditions on the Expansion of Monomials as Summands of SEMs

This talk will introduce some combinatorial objects and concepts before diving into the algebra of symmetric polynomials. We will examine different bases for the symmetric polynomials and prove part of a conjecture on the appearance of a given monomial in the expansion of an Standard Elementary Monomial (SEM) using a variety of visual and combinatorial arguments. 

Nishant Ajitsaria - Maximal Convex Chains

Given a random set of points in a square, the problem of the longest increasing (monotone) path from the bottom-left to the top-right has been extensively studied. What happens if we add another geometric constraint to this? What if we require that the path not only be increasing, but convex (increasing slopes) too? We call this new problem the search for the Maximal Convex Chain.  We get to see some novel results due to this added constraint. As the number of points tends to infinity, we find that the optimal path converges asymptotically to a limiting curve. The number of points on the chain increases predictably. The path closes in on its true potential, the theoretical optimizer.  Previously, these chains had been studied under a uniform distribution. We extended this analysis to the non-uniform case, to study how the underlying distribution affects the shape and size of the optimal chain. Through this new perspective, we saw some phenomena that might have never been observed under a uniform distribution.

Mason McElroy - Avoidance Couplings: How to Mathematically Dodge Fate

A random walk is a process we can apply on any graph, where a “walker” starts at a single point and randomly chooses an edge to follow on each “turn.” A coupling of walkers then attempts to place two or more random walkers on a single graph. If the walkers are independent (on a finite graph), it’s well known that they will eventually intersect. However, by allowing them to be dependent on each other, we can construct avoidance couplings - pairs of walkers that do not intersect. Avoidance couplings are understood for certain classes of graphs, and in this paper we introduce a construction on a new category of graph - finite subsets of a square lattice. Further, two independent random walkers on the infinite square lattice are known to intersect each other infinitely often. However, we’ve found a construction that allows dependent walkers to not only avoid each other, but put an arbitrarily large distance between them. In fact, they manage to do this surprisingly quickly! 

Noah Siekierski - Approximate Message Passing for Quantum State Tomography 

Quantum state tomography (QST) is an indispensable tool for characterizing many-body quantum systems. However, due to the exponential scaling of the cost of the protocol with system size, many approaches have been developed for quantum states with specific structure, such as low-rank states. In this paper, we show how approximate message passing (AMP), an algorithmic framework for sparse signal recovery, can be used to perform low-rank QST. AMP provides asymptotically optimal performance guarantees for large sparse recovery problems, which suggests its utility for QST. We discuss the design challenges that come with applying AMP to QST, and show that by properly designing the AMP algorithm, we can reduce the reconstruction error by over an order of magnitude compared to existing approaches to low-rank QST. We also performed tomographic experiments on IBM Kingston and considered the effect of device noise on the reliability of the predicted fidelity of state preparation. Our work advances the state of low-rank QST and may be applicable to other quantum tomography protocols.