April 21, 2024

Undergraduate Network Meeting:
Research in Combinatorics

November 27
10:00 a.m. - 4:00 p.m.
at the University of Waterloo
Mathematics and Computer Science Building (MC)
Room MC1085

Balazs Szegedy (Toronto)
Levent Tuncel (Waterloo)
Dave Wagner (Waterloo)

Organizers: Richard Cerezo (rcerezo(at)fields.utoronto.ca), Sarah Sun, and Yifan Li (pmclub(at)gmail.com)
Faculty Advisor: Matthias Neufang

Undergraduate Network includes a series of mathematical talks aimed at undergraduates, and organized into a network involving the local universities. We will be stating with trial run of four events for next year with faculty members as consultants.

Levent Tuncel, University of Waterloo
A Guided tour of Mathematical Optimization

We will start the talk with linear optimization problems (optimizing a linear function subject to linear inequalities). This will take us to convex geometry, we will see an open problem related to the combinatorics of polytopes (and computational complexity) as well as some theorems of Caratheodory and Helly. Then, we will move on to hyperplane separation theorem and duality theory of convex optimization. We will introduce the semidefinite optimization problem (a class of convex optimization problems with matrices as the variables). Our path will lead us to optimization problems cast over multivariate polynomial inequalities and we will conclude by making a connection to real algebraic geometry and mentioning a new area of research ``convex algebraic geometry.'' Many open problems will be sprinkled throughout the presentation.

Dave Wagner, University of Waterloo
Integer flows in graphs and regular matroids

Imagine a graph G as a collection of pipes (the edges) and junctions (the vertices). Water currents can flow through these pipes at many different rates as long as mass is conserved at each junction. The set of all flows is in fact a real inner product space. The set of vectors with integer coordinates is the lattice ?(G) of integer flows of G.
If one forgets the coordinates of the integer flows and remembers only the geometric "shape" of the whole lattice ?(G), can one recover the original graph? There are a few obvious ambiguities, and in 1997 it was conjectured that these are the only ones. In 2008, together with my undergraduate research assistant Yi Su, we proved this conjecture.
I will sketch the main ideas of our proof, which is most naturally cast in terms of regular matroids. Of course, I will assume no knowledge of matroid theory, and will begin by explaining what regular matroids are and why they are as good as graphs in many ways

Balazs Szegedy, University of Toronto
Topics in Additive Combinatorics

Final List of Participants:

Full Name University/Affiliation
Aftab, Umar University of Waterloo
Ben-David, Shalev University of Waterloo
Bering, Edgar University of Waterloo
Boyko, Mariya University of Toronto (Mississauga)
Bradley, Nick Queen's University
Burton, Peter University of Toronto
Cerezo, Richard University of Toronto
Chammah, Tarek University of Waterloo
Chow, Kevin University of Waterloo
Dosseva, Annamaria University of Waterloo
Drabek, Rafal  
Dranovski, Anne University of Toronto
Du, Chen Fei University of Waterloo
Duong, Adrian University of Waterloo
Kabir, Ifaz University of Waterloo
Lacharité, Marie-Sarah University of Waterloo
Li, Bing University of Toronto
Li, Yifan University of Waterloo
Liang, Jiayu University of Toronto
Ma, David University of Waterloo
Mauger, Philippe University of Waterloo
McLaughlin, David University of Waterloo
Mohammadi, Mohammadreza University of Toronto
Ng, Keith University of Toronto
Pashley, Bryanne University of Waterloo
Pistone, Jamie University of Toronto
Poon, Alexander McMaster University
Rhee, Donguk University of Waterloo
Sagatov, Sergei University of Toronto
Schaeffer, Luke University of Waterloo
Tham, Emin  
Wesolowski, Michael University of Waterloo
Yeung, Tiffany University of Toronto
Zhu, Ren University of Waterloo

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