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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.

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.

This seminar takes places across multiple time zones: 9:30 AM PT/ 10:30 AM MT / 11:30 AM CT Register via Zoom to receive the link (and reminders) for this event and the rest of the series.

A matrix is called totally positive if all of its minors are positive. These matrices are important in mathe- matics, but checking whether a large matrix is totally positive can be challenging. In this talk, we explore whether certain matrices associated with a totally positive matrix, such as its compounds or those obtained through Dodgson condensation, also remain totally positive. In general, we find that this is not the case. However, we provide some helpful sufficient conditions that make it easier to verify when a matrix is totally positive using these techniques. This is joint work with Shaun Fallat and Charles R. Johnson.

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This seminar takes places across multiple time zones: 9:30 AM PT/ 10:30 AM MT / 11:30 AM CT Register via Zoom to receive the link (and reminders) for this event and the rest of the series.

For more information see PIMS site.

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.

When particles interacts with attractive-repulsive dynamics, which can model birds flocking, space dust, or a variety of other phenomena, they tend to form a nice distribution. Understanding these distributions is an active area of applied analysis that is highly related to the classical concept of equilibrium measures that arises in approximation theory and random matrix theory. In this talk we discuss the numerical computation of such measures via new results on power law kernels applied to orthogonal polynomials that facilitate exploration of regimes where analysis is not yet available.

This seminar takes places across multiple time zones: 9:30 AM PT/ 10:30 AM MT / 11:30 AM CT Register via Zoom to receive the link (and reminders) for this event and the rest of the series.

In this talk, I will discuss recent progress on the directed Oberwolfach problem. A directed variant of the famous Oberwolfach problem, the directed Oberwolfach problem considers the fol- lowing scenario. Given m people and t round tables of size m1,m2 ...,mt such that mi ⩾ 2, does there exist a set of m − 1 seating arrangements such that each person is seated to the right of every other person precisely once? First, I will demonstrate how this problem can be formulated as a type of graph-theoretic problem known as a cycle decomposition problem. Then, I will discuss a partic- ular style of construction that was first introduced by R. Häggkvist in 1985 to solve several cases of the original Oberwolfach problem. Lastly, I will show how this approach can be adapted to the directed Oberwolfach problem, thereby allowing us to obtain solutions for previously open cases. Results discussed in this talk arose from collaborations with Andrea Burgess, Peter Danziger, and Daniel Horsley.