Academic Year 2022-23

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Organised by:

- Tathagata Ghosh
- Shuang Zheng

*Abstract*: You may, at various points in your life, be forced
to interact with a model theorist. While this is no cause for alarm on
its own, the notion that you may then be expected to sit in a room while
a model theorist gives a talk can be troubling to some. Have no fear,
for I will explain the basic machinery of model theory by exhibiting its
intuitive roots. The ideas of languages, models, and theories build on
very reasonable generalisations of structures found all over
mathematics, and by looking at models in that generality we are led to
powerful tools that have wide-reaching applications.

*Abstract*: In this talk, I’ll explain what mathematical gauge
theory is. I’ll (somewhat informally) discuss the mathematical
prerequisites, namely, the notions of smooth manifolds, differential
forms, vector and principal bundles, connections, curvature etc.
Finally, I’ll discuss the heart of gauge theory: Yang-Mills equations,
Instantons and monopoles using all the mathematical tools discussed
during the talk.

The objectives of this talk are three-folds. First, to explain the main ideas of gauge theory to non-specialists. Second, to discuss some particular ideas so that this talk will act as a prelude to many upcoming talks. Third, to connect these notions to various other brunches of mathematics to convince the non-geometers that these mathematical objects are important in their fields as well!

*Abstract*: An ordered field is real closed if any positive
element has a square root and all odd degree polynomials have a
solution. These properties can be translated quite easily into a model
theoretic structure so it makes sense so see what properties these
structures have.

It turns out that the theory of real closed fields has quantifier elimination, is model complete and o-minimal. These properties can be used to give slick proofs of certain classical results concerning semialgebraic sets such the Tarski–Seidenberg theorem (all semialgebraic sets are closed under projection).

The property of model completeness stated earlier means that if something is true in one real closed field, it’s true in any real closed field extension. We can leverage this to give short proofs of Hilbert’s 17th Problem (all semi-definite rational functions can be written as the sum of squares of rational functions) and a real version of the Nullstellensatz. All the material from this talk will be based on section 3.3 in the book “Model theory: An Introduction” by David Marker.

*Abstract*: The pigeonhole principle is the assertion that if
two pigeons were in one hole, then there was one hole that contained two
pigeons. This observation can be generalised. I’ll discuss some
generalisations.

*Abstract*: We will follow the elegant proof of Fermat’s
Christmas theorem by Zagier using the method of counting the fixed
points of some special involutions. Then we will discuss a “geometric”
proof of Fermat’s little theorem using counting techniques involving
group action of a cyclic group on its group algebra.

*Abstract*: In this talk, I’ll explain how to find the
cardinal sequence of a topological space. We are especially interested
in cardinal sequences of locally compact, scattered, Hausdorff
topological spaces (LCS spaces). Given a sequence
$\theta$
of cardinals, we’d like to find out whether there exists an LCS space
which has cardinal sequence
$\theta$
(or, failing that, whether the existence of such a space is consistent
with the axioms of set theory). I will discuss some existing results
about this problem, and some open questions.

*Abstract*: In this talk, I will outline a proof of (a weak
version of) the Odd Goldbach Conjecture, which states that every
sufficiently large odd number can be written as the sum of three primes.
I will explain the Hardy-Littlewood circle method, an important tool in
analytic number theory that forms the backbone of the proof. Along the
way, we will see why a different proof is needed to obtain an effective
bound for ‘sufficiently large’, and why the proof cannot be readily
adapted to prove the (Even) Goldbach Conjecture, which posits that every
even number can be written as the sum of two primes.

*Abstract*: Vortices are topological solitons on Riemann
surfaces initially studied as models of supercurrents in a strong
magnetic field. More recently, generalisations of vortices have been
used to define novel invariants on symplectic manifolds and have deep
ties to the theory of vector bundles over complex algebraic curves. In
this talk we will look at various examples of vortices in the Abelian
Higgs model, building up from global vortices on the complex plane to
gauged vortices on a compact Riemann surface. Time permitting we will
consider slight generalisations such as non-Abelian and symplectic
vortices.

*Abstract*: This talk will begin with an introduction to
almost cocommutative Hopf algebras, as a fertile soil on which to study
Yang-Baxter (YB) type equations that possess inherent symmetry. We then
introduce and apply the so called “RLL-method” with the aim of obtaining
solutions of the YBE acting in the tensor product of two “generic”
lowest weight representations of the symmetry algebras
$A = \mathrm{U}(\mathfrak{sl}_n), \mathrm{U}_q(\mathfrak{sl}_n)$.

The “generic” class of representations we are aiming for are
differential

($q$-difference)
representations of
$A$
on a function space in
$\frac{n(n-1)}{2}$
variables. The defining relations for a YB R-matrix can be interpreted
as a permutation condition allowing for its factorisation into
elementary transposition operators.

*Abstract*: The double ramification (DR) hierarchy provides a
new way to approach problems in the geometry and topology of moduli
spaces, as well as connections with mathematical physics, such as the
Eynard-Oratin topological recursion and integrable systems. The DR
hierarchy is related to the study of the DR cycle, which is a cohomology
class associated to a pair of integers called the ramification profile.
The hierarchy can be thought of as an infinite sequence of equations
which relate certain classes on the moduli space of algebraic curves,
and it encodes information about the geometry of the moduli space. The
double ramification hierarchy has also been used to study Gromov-Witten
invariants and to understand mirror symmetry.

In this talk, we will introduce moduli spaces and cohomological field theories (CohFT) to motivate the key concepts behind the double ramification cycle (DR cycle) and the DR hierarchy. We will review some basic examples of moduli spaces and cohomological field theories. To help us understand the DR classes and their properties, we will compute the first classes for the trivial CohFT, which leads to the KdV hierarchy. This will give us a glimpse of the intricate connections between the DR hierarchy, CohFTs, and integrable systems.

*Abstract*: The idea of a space, map or other construction
being ‘smooth’ rears its head throughout a variety of subjects in
mathematics. Smooth manifolds are a fundamental construction throughout
mathematical physics, differential geometry and geometric topology.
Unfortunately, the category of smooth manifolds is somewhat poorly
behaved; limits and colimits are rare at best, while internal hom’s are
entirely unheard of.

A solution to this conundrum is the theory of diffeological spaces, a structure whose category has all (co)limits and is Cartesian closed while containing smooth manifolds with smooth maps as a full subcategory. On closer inspection, this category turns out to host a veritable zoo of smooth structures, including among its members smooth manifolds with boundary and corners, orbifolds, infinite-dimensional manifolds, Hilbert spaces, diffeomorphism groups and much more. The constructions available on these objects are similarly diverse, ranging from differential-geometric notions like differential forms and de Rham cohomology to algebra-topoolgical ones like smooth homotopy groups. The power to apply all these constructions to all the above examples in the same category is rather startling.

In this talk, I will introduce the basic ideas behind diffeological spaces, giving a taste of the different examples and constructions possible in this powerful category.

*Abstract*: It’ll be something about provability and truth of
statements, but this abstract is false.

*Abstract*: It will be a basic talk about some computational
aspects of elimination theory, namely Gröbner bases and Buchberger’s
algorithm.

The main goal of the talk will be to provide a very gentle (but hopefully precise) introduction to the two aforementioned objects, although -time permitting- I may hint at applications of this theory to effectively computing some homological invariants of finitely generated modules over a ring of multivariate polynomials (over a field).

*Abstract*: Most mathematicians first formally encounter the
Axiom of Choice in the form of Zorn’s Lemma, even though they have been
using some form of choice without second thought. We will briefly look
at the controversial history of this axiom and how the matter has been
“settled” by logicians in a way that makes its acceptance a
philosophical matter. Then we will look at common implicit uses of
choice, as well as some counterintuitive consequences of it and the
chaos that is unleashed in its absence.