Fall 2022 seminar schedule
Fri Dec 2, 4:00pm · NCF 574 — A friendly introduction to cluster algebra, Part 2
In the previous talk, we discussed motivational examples of cluster algebra using frieze patterns of order 5 with relation x_{k-1}x_{k+1} = (x_k^{d1}+1) or (x_k^{d2}+1) for even/odd k. When (d1,d2)=(1,1),(1,2),(1,3) we get a periodic frieze pattern (finite type). We continue by defining the cluster algebra of geometric type and concrete examples, including counterexamples to Fermat numbers 2^{2^k}+1 being prime and to the Markov triple equation x1^2+x2^2+x3^2=3x1x2x3. The finite-type classification will also be stated.
Wed Nov 16, 11:00am · NCF 574 — A friendly introduction to cluster algebra, Part 1
Cluster algebra was invented by Fomin and Zelevinsky in the early 2000s to study total positivity and canonical bases in Lie theory. Since then, it has expanded into a vibrant research area with numerous applications to algebraic geometry, number theory, knot theory, (quiver) representation theory, and mathematical physics. The talk gives an inviting introduction with motivating examples and the formal definition of geometric-type cluster algebra.
Wed Nov 9, 11:00am · NCF 574 — Linear forms in Logarithms: Baker's theory
Gelfond–Schneider (1934) proved transcendence results like 2^{√2}, e^{π}, and i^i. Baker (1967) generalized to n logarithms and showed effective lower bounds for nonzero linear forms, with consequences such as transcendence of products e^{β0}∏α_i^{β_i}. Effective bounds of the form |∑b_i log α_i| ≥ (eB)^{−C} play a key role in Diophantine applications. (See full statement and context on the original page.)
Fri Nov 4, 4:00pm · NCF 574 — Compositions of Involutions in the Symmetric Group, Part 2
We consider the cycle structure of compositions of pairs of involutions in the symmetric group chosen uniformly at random. Every permutation can be factored into the product of two involutions, and the number of such factorizations depends on the cycle structure. The number of factorizations of a random permutation into two involutions is asymptotically lognormally distributed; other statistical properties and connections to swapping algorithms are discussed.
Fri Oct 21, 11:00am · NCF 574 — Compositions of Involutions in the Symmetric Group, Part 1
An involution is a bijection that is its own inverse. The composition of two involutions in the symmetric group can be modeled as modified graphs where each vertex has degree 2. Furthermore, every permutation can be written as the composition of two involutions, but not in a unique way. Using pairs of random involutions to generate random permutations introduces a sampling bias that favors permutations with many small cycles.