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A learning seminar on topics at the intermediate graduate level. Possible topics for this year include Seiberg-Witten theory and twisted K-theory.
Time: Tuesdays, 4:00 pm - 5:00 pm (Semester 1) Thursdays, 4:00 pm - 5:00 pm (Semester 2) Location: Engineering Mathematics Building, Room EMG06 (Semester 1), Room EM212 (Semester 2) This seminar is organised by David Baraglia, kindly contact me in order to be included in the mailing-list. 2018 ScheduleSpeaker: Hokuto Konno Title: Seiberg-Witten 101 (part 1) (Notes attached) Abstract: This is an introductory lecture series on Seiberg-Witten theory. Speaker: Hokuto Konno Title: Seiberg-Witten 101 (part 2) Speaker: Hokuto Konno Title: Seiberg-Witten 101 (part 3) Speaker: Hokuto Konno Title: Seiberg-Witten 101 (part 4) Speaker: Guanheng Chen Title: Compactness property of the Moduli space of Seiberg Witten equations Abstract: I will introduce the analytic aspects of the Seiberg Witten equations and show that the moduli space is compact. Speaker: Guanheng Chen Title: Transversality of the moduli space of Seiberg Witten equations Abstract: In the seminar, I will show that the moduli space of the Seiberg Witten equations is a manifold for generic perturbation. Speaker: Guanheng Chen Title: Transversality of the moduli space of Seiberg Witten equations 2 Abstract: In the seminar, I will show that the moduli space of the Seiberg Witten equations is a manifold for generic perturbation. Speaker: Guanheng Chen Title: Adjunction inequality Abstract: I will introduce the adjunction inequality for three and four manifolds. Speaker: Guanheng Chen Title: Thom conjecture Abstract: Thom conjecture states that the genus of the algebraic curve is a lower bound for the genus of any smooth embedded surface representing the same homology class. I will introduce how to use Seiberg Witten theory to prove this conjecture. Speaker: Guanheng Chen Title: Thom conjecture II Abstract: I will present the proof of a wall crossing formula of Seiberg Witten invariants. Speaker: Danny Stevenson Title: An introduction to bundle gerbes. Abstract: I'll give a brief introduction to bundle gerbes, emphasising their applications in twisted K-theory. Speaker: David Roberts Title: Bundle gerbes, geometry and constructions. Abstract: This talk will work through several constructions of important families of bundle gerbes, and introduce the connective structures that bundle gerbes admit. Speaker: David Brook Title: An introduction to higher twisted K-theory Abstract: Topological $K$-theory is a branch of algebraic topology initially introduced by Atiyah and Hirzebruch in the late 1950s, in which two abelian groups $K^0(X)$ and $K^1(X)$ are assigned to a locally compact Hausdorff space $X$. The notion of $K$-theory is not limited to topological spaces alone; $C^*$-algebras have their own rich version of $K$-theory which can be viewed as a generalisation of topological $K$-theory to a noncommutative setting. These progressions then allowed twisted $K$-theory to be developed by Donovan and Karoubi in the 1970s, in which the algebra of interest is "twisted" by a bundle of compact operators, and much more recently this led to the development of higher twisted $K$-theory by Pennig and Dadarlat in 2016, in which the bundle of compact operators is replaced by a strongly self-absorbing $C^*$-algebra. In this talk, I will present key definitions and results about $C^*$-algebras and their $K$-theory, and introduce the basic notions of twisted $K$-theory and higher twisted $K$-theory. Speaker: Ahnaf Tajwar Tahabub Title: A glimpse into the world of Abelian Seiberg-Witten (SW) Theory and their Application to Super-conductors Abstract: To sidestep the computational difficulty of the Yang-Mills invariants for 4-manifolds, Seiberg and Witten developed two abelian non-linear partial differential equations that essentially provide the same information but under mild assumptions. However, the abelian nature of the $U(1)$-connection used for these SW equations make them far easier to study. Remarkably Witten showed that the dimension reduction (to two dimensions) gives precisely the Landau-Ginzberg equations for super-conductors. This relationship combined with that determined by Taubes with Pseudo-holormorphic curves provides better insight into the types of super-conductors. In this talk I shall precisely construct the SW equations for an orientable compact 4-manifold and show how the reduced equations gives rise to the famous Meissner Effect. This will motivate the introduction of the focus of my project, the non-abelian SW equations using an SU(2)-connection, and hence allow for discussion of their possible use for higher temperature superconductors and Bardeen-Cooper-Schrieffer theory. Speaker: Johnny Lim Title: Pontryagin duality in higher Aharonov-Bohm effect. Abstract: Pontryagin duality in cohomology depicts the close relation between the ordinary integral homology and its character group the circle-valued cohomology theory. In particular, there is a non-degenerate pairing map between them that provides a way for 'phase' computation. One of the famous examples being the Aharonov-Bohm effect in quantum mechanics, which is the manifestation of Pontryagin duality in cohomology in degree 1. The extension of this notion to a generalised cohomology theory, such as K-theory, then plays a role in higher Aharonov-Bohm effect in Type II String theory. Moreover, an analytic Pontryagin duality pairing in K-theory can be formulated explicitly and in fact it encodes much more data than that of in cohomology. In this talk, I will explain this notion in more detail and discuss the Pontryagin duality pairing between even K-homology K_0 and circle-valued K^0 theory. Speaker: Matthias Ludewig Title: The cobordism proof of the Atiyah-Singer index theorem Abstract: We present a concise proof of the Atiyah-Singer index theorem, by following the original arguments of Atiyah and Singer (which were later elaborated on in the book "Seminar on the Atiyah-Singer index theorem" by Palais, Seeley and others). Although nowadays the proofs using K-theory or the heat kernel technique are the more usual choice, the first proof using cobordism has its own beauty and features several intermediate results, which are of independent interest. Speaker: Matthias Ludewig Title: The cobordism proof of the Atiyah-Singer index theorem Abstract: We present a concise proof of the Atiyah-Singer index theorem, by following the original arguments of Atiyah and Singer (which were later elaborated on in the book "Seminar on the Atiyah-Singer index theorem" by Palais, Seeley and others). Although nowadays the proofs using K-theory or the heat kernel technique are the more usual choice, the first proof using cobordism has its own beauty and features several intermediate results, which are of independent interest. Speaker: Miles Simon (University of Magdeburg) Title: An introduction to Ricci flow Abstract: In these three talks we give an introduction to Ricci flow and present some applications thereof. After introducing the Ricci flow we present some theorems and arguments from the theory of linear and non-linear parabolic equations. We explain why this theory guarantees that there is always a solution to the Ricci flow for a short time for any given smooth initial metric on a compact manifold without boundary. We calculate evolution equations for certain geometric quantities, and present some examples of maximum principle type arguments. In the last lecture we present some geometric results which are derived with the help of the Ricci flow. Speaker: Miles Simon (University of Magdeburg) Title: An introduction to Ricci flow Abstract: In these three talks we give an introduction to Ricci flow and present some applications thereof. After introducing the Ricci flow we present some theorems and arguments from the theory of linear and non-linear parabolic equations. We explain why this theory guarantees that there is always a solution to the Ricci flow for a short time for any given smooth initial metric on a compact manifold without boundary. We calculate evolution equations for certain geometric quantities, and present some examples of maximum principle type arguments. In the last lecture we present some geometric results which are derived with the help of the Ricci flow. Previous Years: |