An Introduction to Supersymmetric String Theory

• Speaker: Gabriel Love

The Import and Classification of D-Branes in String Theory

• Speaker: Ben Ward

Abstract:  This Presentation will cover an introduction to D-Branes, looking at their place as boundary conditions for open strings. We will consider D-Branes as the object of the derived category of coherent sheaves in Calabi-Yau Manifolds. Through this presentation, we will construct a specific example using K-theory, and show its application in Type II sting theory more  broadly.

Arithmetic of Calabi-Yau Manifolds

• Speaker: Frey Boulton

Abstract: Calabi-Yau manifolds have properties of interest in number theory through the intersection with arithmetic geometry. I will explore the properties of C-Y manifolds over \Q, particularly those with interesting relations to number theory, and the applications to studying black holes through mirror symmetry.

Functorial definition of TQFTs

• Speaker: Yile Zhang

Abstract:  In this talk, we work through the following definition of topological quantum field theories (TQFTs):

"An $n$-dimensional topological quantum field theory is a symmetric monoidal functor from $A := (n\mathrm{Cob}, \coprod, \emptyset, T)$ to $B := (\mathrm{Vect}_{K},\otimes, K, \sigma)$."

Specifically, we do the following: First, we recall a few basic definitions from category theory, which allow us to work through the definition of symmetric monoidal categories and functors. Then, we demonstrate that $A$ and $B$ are indeed symmetric monoidal categories. Finally, we define TQFTs as objects in the category of symmetric monoidal functors between $A$ and $B$. If time permits, we will try to reconcile the above functorial definition of TQFTs with the definition proposed by Atiyah in 1988.

Mathematical Aspects and Development of Quantum Field Theory from Quantum mechanics

• Speaker: Peter Ilyk

Abstract: During its rise, the mathematical foundations of quantum mechanics were not structurally sound, as the tools developed were based on physical observations and empirical findings, with rigour only occurring as an afterthought. Throughout the century, physicists and mathematicians worked to ground quantum mechanics under a set of postulates based on functional analysis and operator theory to ensure a solid foundation.  This presentation will develop quantum mechanics as a (0+1)-dimensional QFT and explore the main postulates based on Hilbert space and operator theory. We will develop the path integral formalism of quantum mechanics and see how this gives rise to (1+1)-dimensional (bosonic and fermionic) QFT. We will introduce the Yang-Mills Lagrangian and examine how BRST quantisation is used to retain the physicality of the theory by removing redundant degrees of freedom.

Low dimentional Calabi-Yau Manifold

• Speaker: Kai Wu

Lunch Break

Yang-Mills equations and/or Morse theory

• Speaker: Shuxing Che

TBA

Lagrangian-Floer Homology

• Speaker: Noah Holicky

Abstract: An idea of central importance in mathematical and theoretical physics, in particular string theory, is the intersection of two Lagrangian submanifolds. In 1987 Andreas Floer developed a homology theory by constructing chain complexes generated by the intersection points of any two Lagrangian submanifolds. The main result in homological mirror symmetry (Kontsevich, 1994) states; the bounded derived categories of coherent sheaves on a Calabi-Yau manifold are isomorphic to the derived Fukaya category of a mirror Calabi-Yau manifold. This result is implicitly dependent on Lagrangian-Floer Homology for the derived Fukaya category is formalised through Lagrangian-Floer Homology.

QFT from a physicist viewpoint

• Speaker: Max Fleming

TBA

Mirror Maps

• Speaker: Liam Murray

Abstract: Supersymmetry suggests an involution on the moduli spaces of mirror pairs $(X,\check{X})$ such that we get the relations $H^q(X,\Lambda^p T_X)\cong H^q(\check{X},\Lambda^p \Omega_{\check{X}})$. More specifically, this gives rise to an isomorphism between the tangent spaces and hence local isomorphisms of the Kahler and Complex moduli spaces of the pairs. Such local isomorphisms are called "mirror maps", and were essential to Candelas et al's celebrated prediction of the number of rational curves on the quintic threefold.  The story of these mirror maps will let us explore ideas such as the Picard-Fuchs equation and Yukawa couplings. Ultimately we'll discover that the task of computing rational curves on one threefold is secretly a calculation about the variation of Hodge structure on a completely different threefold.

Construction of Clifford Algebra and Spin(n)

• Speaker: Zizheng Yang

Abstract: Construct Clifford algebras on vector spaces (primarily Rn\mathbb{R}^nRn). Define the Spin group in three different ways and compute some low-dimensional Spin(n) groups using various methods.

Celebratory tea, chocolate and cookies