New undergraduates often find that they have a lot more time to spend on independent work than they did at school or college. But how can you use that time well? When your lecturers say that they expect you to study your notes between lectures, what do they really mean? There is research on how mathematicians go about reading maths effectively. We'll look at a technique that has been shown to improve students' comprehension of proofs, and in this interactive workshop we'll practise together on some examples. Please bring a pen/pencil and paper!
This session is likely to be most relevant for first-year undergraduates, but all are welcome, especially those who would like to improve how they read and understand proofs.
In this interactive workshop, we'll discuss what mathematicians are looking for in written solutions. How can you set out your ideas clearly, and what are the standard mathematical conventions? Please bring a pen or pencil!
This session is likely to be most relevant for first-year undergraduates, but all are welcome.
What should you expect in intercollegiate classes? What can you do to get the most out of them? In this session, experienced class tutors will share their thoughts, and a current student will offer tips and advice based on their experience.
All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. (Students who attended the Part C/OMMS induction event will find significant overlap between the advice offered there and this session!)
A matrix M of real numbers is called totally positive if every minor of M is nonnegative. This somewhat bizarre concept from linear algebra has surprising connections with analysis - notably polynomials and entire functions with real zeros, and the classical moment problem and continued fractions - as well as combinatorics. I will explain briefly some of these connections, and then introduce a generalization: a matrix M of polynomials (in some set of indeterminates) will be called coefficientwise totally positive if every minor of M is a polynomial with nonnegative coefficients. Also, a sequence (an)n≥0 of real numbers (or polynomials) will be called (coefficientwise) Hankel-totally positive if the Hankel matrix H = (ai+j)i,j ≥= 0 associated to (an) is (coefficientwise) totally positive. It turns out that many sequences of polynomials arising in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive; in some cases this can be proven using continued fractions, while in other cases it remains a conjecture.
The moments of characteristic polynomials play a central role in Random Matrix Theory. They appear in many applications, ranging from quantum mechanics to number theory. The mixed moments of the characteristic polynomials of random unitary matrices, i.e. the joint moments of the polynomials and their derivatives, can be expressed recursively in terms of combinatorial sums involving partitions. However, these combinatorial sums are not easy to compute, and so this does not give an effective method for calculating the mixed moments in general. I shall describe an alternative evaluation of the mixed moments, in terms of solutions of the Painlevé V differential equation, that facilitates their computation and asymptotic analysis.
joint work with Moshe Kamensky and Silvain Rideau
Let $p$ be a fixed prime number and let $SCVF_p$ be the theory of separably closed non-trivially valued fields of
characteristic $p$. In the talk, we will see that, in many ways, the step from $ACVF_{p,p}$ to $SCVF_p$ is not more
complicated than the one from $ACF_p$ to $SCF_p$.
At a basic level, this is true for quantifier elimination (Delon), for which it suffices to add parametrized $p$-coordinate
functions to any of the usual languages for valued fields. It follows that all completions are NIP.
At a more sophisticated level, in finite degree of imperfection, when a $p$-basis is named or when one just works with
Hasse derivations, the imaginaries of $SCVF_p$ are not more complicated than the ones in $ACVF_{p,p}$, i.e., they are
classified by the geometric sorts of Haskell-Hrushovski-Macpherson. The latter is proved using prolongations. One may
also use these to characterize the stable part and the stably dominated types in $SCVF_p$, and to show metastability.
As we have learned over the last 10 years, many exact results for various observables in three-dimensional N=2 supersymmetric theories can be extracted from the computation of "supersymmetric partition functions" on curved three-manifold M_3, for instance on M_3= S^3 the three-sphere. Typically, such computations must be carried anew for each M_3 one might want to consider, and the technical difficulties mounts as the topology of M_3 gets more involved. In this talk, I will explain a different approach that allows us to compute the partition function on "almost" any half-BPS geometry. The basic idea is to relate different topologies by the insertion of certain half-BPS line defects, the "geometry-changing line operators." I will also explain how our formalism can be related to the Beem-Dimofte-Pasquetti holomorphic blocks. [Talk based on a paper to appear in a week, with Heeyeon Kim and Brian Willett.]
Poor economies not only produce less; they typically produce things that involve fewer inputs and fewer intermediate steps. Yet the supply chains of poor countries face more frequent disruptions - delivery failures, faulty parts, delays, power outages, theft, government failures - that systematically thwart the production process.
To understand how these disruptions affect economic development, we model an evolving input-output network in which disruptions spread contagiously among optimizing agents. The key finding is that a poverty trap can emerge: agents adapt to frequent disruptions by producing simpler, less valuable goods, yet disruptions persist. Growing out of poverty requires that agents invest in buffers to disruptions. These buffers rise and then fall as the economy produces more complex goods, a prediction consistent with global patterns of input inventories. Large jumps in economic complexity can backfire. This result suggests why "big push" policies can fail, and it underscores the importance of reliability and of gradual increases in technological complexity.