# Warwick Imperial Autumn Meeting

#### 30 November 2019

Organisers: James Taylor (Warwick), Kobe Marshall-Stevens (Warwick), Ali Barkhordarian (Imperial)

WIMP is a free one day conference primarily aimed at late undergraduate, masters, and early PhD students from Warwick and Imperial, though of course others are welcome. It is on Saturday 30th November in the Zeeman building, University of Warwick. The aim of the conference is to introduce different areas of student's research, as well as featuring an academic's talk. We will start with a plenary talk and then divide into two streams of five talks, so there should be something of interest to everyone. All talks will be approximately 45 minutes in length with 15 minutes afterward for questions, discussion, and refreshment.

To register for the conference, please fill out the following form: WIMP2019-Registration-Form. This is helpful for booking and food organisation.

If you have any other questions please email j.taylor.26@warwick.ac.uk

**Getting to campus:** For general advice on how to get to the campus see here https://www2.warwick.ac.uk/study/undergraduate/visits/gettinghere/

**Directions to WIMP:** For those new to the campus, the day starts inside the Warwick Mathematics Institute. When you enter the building there is a staircase in front, slightly to the right. Go up the first flight of stairs and turn left. There is a common room where you can register.

**Schedule for the day:**

**09:30:** Registration (Upstairs common room: Zeeman Building)

**10:00-11:00**:Plenary

[MS.04] Dr Mario Micallef (Warwick) - Morse theory, a meeting point of analysis, topology and geometry.

**11:00-12:00: **

[MS.04] Daniele Mastrostefano (Warwick) - On the Duffin and Schaeffer Conjecture

[MS.05] Austin Hubbard (Imperial) - Riemann surfaces & Monodromy Representation

**12:00-13:00:**

[MS.04] Philippe Michaud-Rodgers (Warwick) - Congruent Numbers and Elliptic Curves

[MS.05] Arjun Sobnack (Warwick) - It's All Downhill From Here

**13:00-14:00**:

Lunch

**14:00-15:00**:

[MS.04] Xenia Dimitrakopoulou (Imperial) - On the Weil Conjectures

[MS.05] George Kontogeorgiou (Warwick) - Bidimensionality: What’s It Good For?

**15:00-16:00**:

[MS.04] Tudor Ciurca (Imperial) - Periods and the Galois theory of Integrations

[MS.05] Luke Peachey (Warwick) - On the Gauss-Bonnet-Chern Theorem

**16:00-16:30**:

Afternoon tea and coffee

**16:30-17:30**:

[MS.04] Enrico Ancilotto (Imperial) - Forcing or: how I learned to stop worrying and start loving Cohen extensions

[MS.05] ~~Lilybelle Cowland Kellock (Imperial) - Counting Markoff triples in p-adic non-integers ~~ The 27 Lines on a Cubic Surface - M.T. Set (Imperial)

**17:30**: (Common room)

-----

**Abstracts:**

**Dr Mario Micallef (Warwick) - Morse theory, a meeting point of analysis, topology and geometry.
**

A smooth manifold $X$ is a space which is locally Euclidean and on which we can do calculus. (Think of a sphere, or torus, or the orthogonal group.) Given a real valued function $f$ on $X$, one tries to construct $X$ from the level sets of $f$. The structure of these level sets changes only at critical points, i.e., points where the gradient of $f$ vanishes. I will describe informally how the change in the topology of the level sets near a critical point $p$ is related to the Hessian of $f$ at $p$. In a similar vein, one can study critical points of functionals on spaces of functions and maps. For instance, critical points of the length functional on the space of closed curves in $X$ correspond to closed geodesics. In this setting, Morse theory for the length functional relates homotopy groups and the curvature of $X$. (I will informally describe homotopy groups and curvature.)

**Daniele Mastrostefano (Warwick) - On the Duffin and Schaeffer Conjecture**

One of the questions Metric Number Theory is concerned with is understanding how well irrational numbers can be approximated by rational ones. Probably the most well known result in this context is the Dirichlet’s Approximation Theorem, which states that any irrational number is close to infinitely many rational fractions $\frac{p}{q}$ by at most $\frac{1}{q^2}$. Duffin and Schaeffer showed that we can improve the precision of the approximation while still finding a subset of the real numbers with full measure where this happens. Moreover, they formulated a conjecture on the sharpest version of their result.In the talk we will review some of the most important contributions in Diophantine Approximation Theory and give a sketch of the recently Maynard–Koukoulopoulos solution of the aforementioned conjecture.

**Philippe Michaud-Rodgers (Warwick) - Congruent Numbers and Elliptic Curves**

The study of congruent numbers dates back to Pythagoras. We say that a natural number $n$ is a congruent number if it is the area of a right angled triangle with rational side lengths. The number 5 is a congruent number, but 3 is not. Why is this? Can we classify congruent numbers? Is there a test for congruent numbers? In this talk I will discuss this problem in depth and examine its relation with a special class of elliptic curves. The talk will serve as a gentle general introduction to the theory or elliptic curves and we’ll discuss famous results such as the Mordell-Weil, Lutz-Nagell, and Reduction Theorems and see how they are used to solve the congruent number problem. The talk should be accessible to all (the only prerequisites being some modular arithmetic and some basic algebra, such as the classification of finitely generated abelian groups).

**George Kontogeorgiou (Warwick) - Bidimensionality: What’s It Good For?**

Bidimensionality is a relatively new theory which concerns the use of certain well-behaved graph parameters (dubbed “bidimensional”) for the design of efficient fixed parameter algorithms for NP-hard problems in graph theory. In this talk, we will initially introduce treewidth (the archetypical bidimensional parameter) and comment on its ubiquity in graph-theoretic theorems and algorithms, from dynamic programming to Courcelle’s theorem to the graph structure theorem. Then we will move on to the more general setting of minor-and contraction-bidimensional parameters, which we will use to prove fixed parameter tractability for classic problems like VERTEX COVER and DOMINATING SET. The goal of this talk is to explain the importance of bidimensionality as the true force behind many of treewidth’s useful properties, name some bidimensional parameters, and introduce a new and useful approach to the design of graph algorithms.

**Arjun Sobnack (Warwick) - It's All Downhill From Here**

Gradient flows form an interesting class of (ordinary and partial) differential equations, for which the Łojasiewicz-Simon Inequalities (1965 & 1983) are one of many useful tools. I will softly motivate and introduce gradient flows and present some applications of the Łojasiewicz-Simon Inequalities.

**Austin Hubbard (Imperial) - Riemann surfaces & Monodromy Representation**

The theory of Riemann surfaces provides a beautiful and intuitive link between complex analysis, algebraic geometry and topology. Riemann surfaces are 2 (real) dimensional manifolds with a holomorphic structure, they are necessarily orientable thus are easy to visualise. We will classify compact surfaces and look at monodromy representations as an easy way to study maps between Riemann surfaces.

**Tudor Ciurca (Imperial) - Periods and the Galois theory of Integration**

Periods can be thought of as complex numbers that we get by integrating "algebraic differential forms" over "algebraic subsets". In this talk we make the notion of a period number precise, by describing the period isomorphism. This is a version of De Rham's theorem but for smooth projective varieties, and it concerns algebraic De Rham cohomology. We describe how the periods we get by evaluating algebraic De Rham classes over singular chains coincides with integrals of rational functions over semi-algebraic sets, which is a more down-to-earth description of a period. We will see many examples, and the tools required to compute periods associated to a smooth projective variety. This is a good opportunity to see the machinery of algebraic geometry at play.

Time-permitting, we will state the conjectures of Kontsevich-Zagier and Grothendieck regarding periods, and how they may be used as a tool to develop a Galois theory of integration which allows us to determine the transcendentality of some complex number.

**Enrico Ancilotto (Imperial) - Forcing or: how I learned to stop worrying and start loving Cohen extensions**

Gödel’s famous incompleteness theorem proved that any theory that can conduct arithmetical operations, as defined by the Peano Axioms, will either be inconsistent will be incomplete. While the proof of the theorem only constructs an extremely pathological theorem which can’t be proved or disproved from certain axioms, it soon became clear that many more interesting theorems were also independent of the ZF axioms used to model modern mathematics., the most famous example is the Axiom of Choice. For this reason, a general technique to verify independence results is crucial to the development of our understanding of set theory. This technique is called forcing: it allows the addition of new sets to our universe in a way that lets us verify whether the new theory so developed has certain properties. The talk will present forcing and show how it can be applied to show the independence of the Continuum Hypothesis from the ZFC axioms. While not required, some familiarity with set theory and formal logic is recommended.

**Lilybelle Cowland Kellock (Imperial) - Counting Markoff triples in p-adic non-integers**

In this talk we will work out how to generate all solutions to the Markoff equation in the natural numbers. We then take a look at the field of p-adics and study solutions to a similar equation (differing by a coefficient 3) in the p-adic rationals (non-integers). We find a result analogous to D. Zagier’s “On the Number of Markoff Numbers Below a Given Bound” but for counting the number of triples in the orbits of a given solution in the p-adic rationals with p-adic absolute value below a given bound. We derive this result for certain orbits of solutions and we show that it is infinite otherwise. Our methods involve parameterising solutions, in order to categorise the orbits of solutions, and using the Euclid tree (the tree of the Euclidean algorithm) for counting, as D. Zagier does in his paper proving the analogous result.

**Xenia Dimitrakopoulou (Imperial) - On the Weil Conjectures**

The Weil Conjectures were a series of remarkable conjectures that took multiple decades to prove and influenced the development of modern algebraic geometry and number theory. In this talk, I will state and prove these conjectures in some specific cases.

**Luke Peachey (Warwick) - On the Gauss-Bonnet-Chern Theorem**

The Gauss-Bonnet Theorem is a simple relationship between the topological and geometric information of a Riemannian surface. We will explore this relationship further in higher dimensions and for general vector bundles. Only the basic definitions from geometry and topology will be assumed.

-----

Special thanks to the department for their generous funding and Ali Barkhordarian for organising the Imperial side of WIMP.