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| Fall 2022 seminar |
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| Spring 2023 seminar |
| Date | Place | Speaker | Title |
|---|---|---|---|
| Fri Dec 2, 4:00pm | NCF 574 | Bach Nguyen | A friendly introduction to cluster algebra, Part 2 |
| Summary In the previous talk, we discussed motivational examples of cluster algebra using frieze patterns of order 5 with relation \[x_{k-1}x_{k+1}=\begin{cases} x_k^{d_1} +1, & k \mbox{ even}\\x_k^{d_2}+1, & k \mbox{ odd} \end{cases} \] for some choice \((d_1,d_2)\in {\mathbb Z}_{\gt 0}^2.\) When \((d_1,d_2)=(1,1), (1,2), (1,3)\), we get a periodic frieze pattern (and this corresponds to the cluster algebra of finite type). In this talk, we will continue by defining the cluster algebra of geometric type and going over some concrete examples. In particular, we will see examples of cluster algebras which provide counterexamples to the Fermat numbers \(2^{2^k}+1\) being prime, and to the existence of solution for the Markov triple equation \(x_1^2+x_2^2+x_3^2=3x_1x_2x_3\). The classification of cluster algebras of finite type will also be stated. | |||
| Date | Place | Speaker | Title |
|---|---|---|---|
| Wed Nov 16, 11:00am | NCF 574 | Bach Nguyen | A friendly introduction to cluster algebra, Part 1 |
| Summary Cluster algebra was invented by Fomin and Zelevinsky in the early 2000s to study total positivity and canonical bases in Lie theory. Since then, they have been expanded into a vibrant research area with numerous application to other subjects in mathematics such as algebraic geometry, number theory, knot theory, (quiver) representation theory, and mathematical physics to name a few. This talk is designed to give an inviting introduction to the theory of cluster algebras. We will discuss many motivating examples and observe some interesting properties of cluster algebra. We will also be going over the formal definition of cluster algebra of geometric type. | |||
| Date | Place | Speaker | Title |
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| Wed Nov 9, 11:00am | NCF 574 | Pranabesh Das | Linear forms in Logarithms: Baker's theory |
| Summary In 1934, Gelfond and Schneider proved that Theorem 1. (Gelfond-Schneider) If \(\alpha\) and \(\beta\) are complex algebraic numbers with \(\alpha \neq 0, 1,\) and \(\beta\) not rational, then any value of \(\alpha^{\beta}\) is a transcendental number. This immediately proves that the numbers like \(2^{\sqrt{2}}, e^{\pi}, i^i\) are transcendental. The Theorem 1. is equivalent as saying if \(\log \alpha\) and \(\log \beta\) linearly independent over \(\mathbb Q\) for non-zero algebraic numbers \(\alpha\) and \(\beta\), then \(\log \alpha\) and \(\log \beta\) are linearly independent over the algebraic numbers. Gelfond-Schneider's method was specific to only two algebraic numbers. In 1967 Alan Baker established the following generalization of the Gelfond-Schneider theorem. Theorem 2. (Baker1) Let \(n\) be a positive integer. Let \(\alpha_1,\cdots,\alpha_n\) be non-zero algebraic numbers and \(\log \alpha_1,\cdots,\log \alpha_n\) any determinations of their logarithms. If \(\log \alpha_1,\cdots,\log \alpha_n\) are linearly independent over \(\mathbb Q\), then \(1,\log \alpha_1,\cdots,\log \alpha_n\) are linearly independent over the algebraic numbers. As an immediate consequence, we have Corollary 3. \(e^{\beta_0}{\alpha_1}^{\beta_1}{\alpha_2}^{\beta_2}\cdots{\alpha_n}^{\beta_n}\) is transcendental for all non-zero algebraic numbers \(\alpha_1,\cdots,\alpha_n,\beta_0,\beta_1\cdots,\beta_n.\) For applications to Diophantine problems, it is important that not only the above linear form is non-zero, but also that we have a strong enough lower bound for the absolute value of this linear form. We give a special case, where \(\beta_0 = 0\) and \(\beta_1,\cdots,\beta_n\) are rational integers. Theorem 4. (Baker2) Let \(\alpha_1,\cdots,\alpha_n\) be algebraic numbers different from \(0\) and \(1\). Further, let \(b_1,\cdots,b_m\) be rational integers such that \[b_1\log \alpha_1+\cdots +b_n\log \alpha_n \neq 0.\] Then \[\left|b_1\log \alpha_1+\cdots +b_n\log \alpha_n\right| \geq (eB)^{-C},\] where \(B:= \max(|b_1|,\cdots,|b_n|)\) and \(C\) is an effectively computable constant depending only on \(n\) and \(\alpha_1,\cdots,\alpha_n\). Baker was awarded a Fields medal for his work on linear forms in logarithms in 1970. In a series of two-three talks, we plan to discuss some of the results of this topic and its beautiful applications in Diophantine problems. | |||
| Click here to access a pdf file with the slides from the talk | |||
| Date | Place | Speaker | Title |
|---|---|---|---|
| Fri Nov 4, 4:00pm | NCF 574 | Charles Burnette | Compositions of Involutions in the Symmetric Group, Part 2 |
| Summary In the second part of this talk, we consider the cycle structure of compositions of pairs of involutions in the symmetric group chosen uniformly at random. We saw previously that every permutation can be factored into the product of two involutions, and that the number of such factorizations depends on the cycle structure of the permutation. From this, we will see that number of factorizations of a random permutation into two involutions is asymptotically lognormally distributed. Other statistical properties will be fleshed out as well. Connections to the analysis of swapping algorithms will also be considered through the use of a graph-theoretic model for pairs of involutions. | |||
| Date | Place | Speaker | Title |
|---|---|---|---|
| Fri Oct 21, 11:00am | NCF 574 | Charles Burnette | Compositions of Involutions in the Symmetric Group, Part 1 |
| Summary 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. Hence, using pairs of random involutions to generate random permutations introduces a significant sampling bias in the sense that the composition favors permutations with a lot of small cycles. | |||