April 21, 2024

Undergraduate Network Meeting

Upcoming Workshop: Research in Algebra

Date and Time: January 22, 2011, 10am-4pm
Place: Hamilton Hall, McMaster University

Organizers: Richard Cerezo (Toronto), Gregory Cousins (McMaster), Michelle Hurst (McMaster)
Faculty Advisors: Matthias Neufang, Jim Colliander

The Fields Undergraduate Network (FUN) 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.
FUN hosts monthly meetings to explore different areas of mathematical research. The content and host university will vary from month to month. All interested undergraduates are encouraged to come. We would especially like to see students from student math societies or course unions participate in organizing these events.


Deirdre Haskell (McMaster University)
Joel Kamnitzer (University of Toronto)
Matt Valeriote (McMaster University)


10:00 a.m. Networking/Socializing
10:30 a.m. Warm-up Panel Discussion
11:00 a.m. Deirdre Haskell (McMaster Univeristy)
An Infinite Pigeonhole Principle (and Grothendieck Rings in Model Theory)
12:00 p.m. Lunch Break
1:30 p.m. Joel Kamntizer (University of Toronto)
Representation Theory of Semisimple Groups: Classical, Quantum, Geometric, Categorical
2:30 p.m. Panel Discussion
3:00 p.m. Matt Valeriote (McMaster University)
An Algebraic Approach to the Dichotomy Conjecture


Deirdre Haskell, McMaster University
An infinite pigeonhole principle (and Grothendieck rings in model theory).

One statement of the pigeonhole principle is that any injective function from a finite set to itself is surjective. This can also be taken to be the definition of a set being finite, and thus, by definition, infinite sets do not satisfy a pigeonhole principle. However, if we restrict the allowable functions to ones which are natural in some sense, then it is possible that some classes of infinite sets will have a pigeonhole principle. In this talk, I will make this idea precise. I will introduce the Grothendieck ring, whose non-triviality is equivalent to the infinite pigeonhole principle, and discuss various examples where the Grothendieck ring can be calculated. In particular, I will show how model theory provides us with very natural collections of sets and functions about which it makes sense to ask for an infinite pigeonhole principle.


Joel Kamnitzer, University of Toronto
Representation theory of semisimple groups: classical, quantum, geometric, categorical

The representation theory of semisimple Lie groups is a classical subject going back to Weyl. I will describe the basics of this theory.

Then I will describe recent geometric approaches to this subject and how they lead to categorical enhancement of representations.


Matt Valeriote, McMaster University
An Algebraic Approach to the Dichotomy Conjecture

Constraint satisfaction is a paradigm for specifying combinatorial problems in various areas of mathematics and computer science. One builds a constraint satisfaction instance by declaring a finite set of variables, a domain over which the variables are to be interpreted, and a finite collection of “constraints” which place restrictions on the values the variables may take in relation to the others. Given such a specification, one can ask whether a solution exists (i.e., an assignment of values to variables satisfying all the constraints). In general this sort of decision problem is NP-hard and hence presents a computational challenge to solve.

Many purely theoretical problems have arisen within the context of constraint satisfaction research, most famously the Dichotomy Conjecture which asserts that whenever the domain of a collection of instances are fixed to some finite set D and the allowable constraints are restricted to some finite set of relations on D, the “does a solution exist” question either remains NP-hard or becomes solvable in polynomial time. Spectacular advances towards a proof of the Dichotomy Conjecture have been achieved recently via the so-called “algebraic approach.” Under the algebraic approach one associates universal algebras, or collections of such algebras, to robust classes of restricted constraint satisfaction problems. It turns out that some well-studied properties of universal algebras perfectly capture the complexity of the corresponding problems. In my talk I will discuss the algebraic approach to the Dichotomy Conjecture, providing the necessary algebraic background along the way.


List of Confirmed Participants as of January 18, 2011:

Full Name University Name
Adkins, Chris McMaster University
Barreto, Jorge University of Toronto at Mississauga
Boyko, Mariya University of Toronto (Mississauga)
Brin, Marina University of Toronto
Carmichael, Keegan University of Western Ontario
Cerezo, Richard University of Toronto
Cleary, Erin McMaster University
Cousins, Gregory McMaster University
da Silva, Sergio University of Toronto
Dai, Chen University of Toronto at Scarborough
Gagliardi, Noam Brock University
Gill, Kanwar Anmol Singh University of Toronto at Scarborough
Ginsberg, Dan University of Toronto
Gracie, Mitchell University of Western Ontario
Grzadkowski, Michal University of Waterloo
Han, Changho University of Toronto
Hou, Tianhe University of Toronto at Scarborough
Kidwai, Omar University of Toronto
Lam, Chris McMaster University
Lin, Shichu University of Toronto at Scarborough
Lynch, Ray University of Toronto
Man, Elina University of Toronto at Scarborough
May, Brandon McMaster University
Menonkariyil, Mathew Brock University
Milcak, Juraj University of Toronto
Miners, Simon Brock University
Mirza, Qamar University of Toronto
Mohammadi, Mohammadreza University of Toronto
Mohsin, Faizan Khalid University of Toronto
Pascuzzi, Vince Brock University
Prawira, Daniel University of Toronto at Scarborough
Qian, Jiayun  
Terrana, Alexandra McMaster University
Tour, Dennis McMaster University
Vlasova, Jelena University of Toronto at Mississauga
Walton, Laura McMaster University
Yee, Yohan McMaster University


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