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Long DNA and RNA molecules encode the genetic code of viruses and living organisms. We study the changes in DNA topology mediated by essential processes such as DNA packing and transcription of DNA into RNA. These processes are highly regulated, and even small structural changes can lead to catastrophic effects. We use techniques from knot theory and topology, aided by discrete and computational methods, to model these biological processes. Our emphasis is on the geometry and topology of DNA and RNA. In this lecture, I first discuss discrete methods in the study of biopolymers. The rest of the presentation will be devoted to the R-loop grammar, a formal grammar model that allows us to predict the formation and entanglement of DNA:RNA hybrids that arise during transcription. The presentation is accessible to students and suitable for a diverse interdisciplinary audience.

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The array

3     -3    4     -4    5     -5    0     0

4     -4    -3    3     0     0     5     -5

-5    5     0     0     -3    3     -4    4

has an interesting feature: its columns are each vectors satisfying the linear relation 3x + 4y + 5z = 0, and its rows are permutations of each other. Are there arrays of integers like this for other simple linear relations? This question was raised by Chris Smyth in 1986 in connection with an audacious conjecture in algebraic number theory about linear relations between Galois conjugates. We explain how to prove Smyth's conjecture, and what it has to do with eigenvalues of sums of permutation matrices, weightings on hypergraphs, and (depending on time and audience indulgence) a local-to-global principle for equations in random variables.

Joint work with Will Hardt.

Starting with one red ball and one black ball in an urn, repeat the following - choose a ball from the urn at random, observe the colour, put it back in the urn together with an additional ball of the same colour. This simple model is called Polya’s urn and is an example of a random process with 'reinforcement' (e.g. if red is selected first then after this first iteration we have 2 red balls and 1 black ball in the urn, so red is more likely to be selected again in the second iteration).

Beginning with Polya’s urn, this talk will take a tour through some of the weird and wonderful behaviour that has been observed or conjectured for various random processes with reinforcement.

Registration:
Participants register once on Zoom and may attend any of the colloquium talks. Please remember to download the calendar information to save the dates on your calendar. PIMS will resend the confirmation from Zoom prior to each event date.

I will discuss a result with Bonatti and Crovisier from 2009 showing that the C^1 generic diffeomorphism f of a closed manifold has trivial centralizer; i.e. fg = gf implies that g is a power of f. I’ll discuss features of the C^1 topology that enable our proof (the analogous statement is open in general in the C^r topology, for r>1). I’ll also discuss some features of the proof and some recent work, joint with Danijela Damjanovic and Disheng Xu that attempts to tackle the non-generic case.