Discrete Math Day
April 23, 2022

Wrap-Up

Thank you, everybody, for helping to make this DMD a great one! We could not have done it without you. For those of you who were unable to make it, we hope to see you in the near future!

About

Discrete Math Days in the Northeast is a series of one- or two-day conferences having the goal of strengthening the discrete mathematics community in the northeast, encouraging collaboration across institutions and disciplines. These conferences strive for a welcoming environment for discrete mathematicians at all stages of their careers, including graduate students and undergraduates.

There are two DMDs per year: one in the fall and one in the spring. During the summer there is also Summer Combo. These meetings are hosted at different institutions across the northeast, helping to keep those with higher teaching loads to remain at the forefront of research in their field.

Location and Date

Colgate University, Hamilton NY
Saturday, April 23, 2022

All talks will take place in Robert H. N. Ho Science Center, Room 101. For those preferring to attend remotely, these talks will be streamed in a Zoom webinar format (link forthcoming).

Local Organizers

Rob Davis, Gabe Sosa Castillo, and Aaron Robertson. If you have questions, please contact Rob Davis at rdavis [a-in-circle] colgate [dot] edu.

Plenary Speakers

Titles, Abstracts, and Slides

Marcelo Aguiar: Descents for cones of a real hyperplane arrangement (slides)

Reading a permutation from left to right and recording the positions where the values decrease yields the descent set of the permutation. More generally, descents may be attached to elements of a finite Coxeter group. In this talk we will discuss a different extension of the notion of descents, to the setting of real hyperplane arrangements. The notion on which this is based is the Tits product, an operation that exists on the set of faces of such an arrangement. The talk will center around this fundamental notion and include a discussion of faces, cones, and descents, together with lots of illustrations. No previous familiarity with hyperplane arrangements will be assumed.

Annie Raymond: Tropicalization of graph profiles (slides)

The number of homomorphisms from a graph H to a graph G, denoted by hom(H;G), is the number of maps from V(H) to V(G) that yield a graph homomorphism, i.e., that map every edge of H to an edge of G. Given a fixed collection of finite simple graphs {H_1, ..., H_s}, the graph profile is the set of all vectors (hom(H_1; G), ..., hom(H_s; G)) as G varies over all graphs. Graph profiles essentially allow us to understand all polynomial inequalities in homomorphism numbers that are valid on all graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known. To simplify these objects, we introduce their tropicalization which we show is a closed convex cone that still captures interesting combinatorial information. We explicitly compute these tropicalizations for some sets of graphs, and relate the results to some questions in extremal graph theory.

Anne Shiu: Which neural codes are convex? (slides)

This talk focuses on combinatorial problems motivated by neuroscience. Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? This talk describes how to use tools from combinatorics and commutative algebra to uncover a variety of signatures of convex and non-convex codes.

Shanise Walker: Zero Forcing, Power Domination, and Product Power Throttling of a Graph (slides)

Zero forcing and power domination are coloring processes on a graph based on color change rules. Zero forcing and power domination are related in that power domination of a graph is the process of using a domination step and then the zero forcing process to observe vertices in the graph. The study of power domination in graphs results from the analysis of electrical network monitoring. The product power throttling number of a graph studies product throttling for power domination. We define the product power throttling number of a graph to be the minimum product of the size of a power dominating set of a graph and the number of steps it takes for all the vertices in the graph to be observed. In this talk, we briefly introduce zero forcing and power domination. We provide bounds for the product throttling number and discuss certain families of graphs for which the product throttling number is known.

Poster Session

Following lunch there will be a poster session for participants to present their own work. All are welcome to contribute to the session! If you are interested in contributing a poster, please indicate so when registering for the conference.

Registration and Funding for Participants

Registration is now closed.

There will be a very limited amount of funding available to participants to help offset travel costs. If you would like to request funding, please indicate this when you register. There will be no need to collect or submit receipts.

Accommodations

Most participants drive to and from the conference on the same day. For those arriving from further way, we recommend reserving your room at Hampton Inn & Suites - Cazenovia, although space is limited.

Additional recommended local options are:

• Colgate Inn

• Hamilton Inn

• Brewster Inn (in Cazenovia)

• Brae Loch Inn (in Cazenovia)

Many more options, including various B&Bs within Hamilton, are listed here. Participants will be responsible for payment.

Schedule

8:45 am -- 9:30 am: Registration and light refreshments, first floor of Ho Science Center

9:30 am -- 9:45 am: Opening remarks

9:45 am -- 10:45 am: Marcelo Aguiar

10:45 am -- 11:00 am: Break

11:00 am -- 12:00 pm: Annie Raymond

12:00 pm -- 1:30 pm: Lunch

1:30 pm -- 2:30 pm: Poster session, held in the 2nd floor atrium

2:30 pm -- 3:30 pm: Shanise Walker

3:30 pm -- 3:45 pm: Break

3:45 pm -- 4:45 pm: Anne Shiu

Campus Map and Parking information

All talks and the poster presentation will take place in the Robert H. N. Ho Science Center. Below are different maps to help orient you with buildings and parking lots. Colgate has a very walkable campus; even many of the lots that are "farther" from Ho Science Center are less than a 15-minute walk away.

Participants may park in any lot, and there is no cost for parking, but please make sure to fill out this form ahead of time to obtain your temporary parking permit. All you need to do is provide your email address and indicate that you are taking part in an "Adult conference or seminar" (i.e. click the top button, not the bottom button).

• Map of the entire campus, with buildings and parking lots labeled. Ho Science Center is on the east edge of G6.

• Map of the portion of campus near to the location of the conference, with buildings and parking lots labeled

• Interactive map

Funding Acknowledgements

This conference is organized by the Northeast Combinatorics Network and is funded in part by the National Science Foundation and Colgate University.

Land Acknowledgement

Colgate University is situated on the traditional territories of the Oneida peoples. Before Payne's Farm describes the history of how this came to be.