Summer Seminar Series

Analytic torsion on manifolds is, by the Cheeger-Muller theorem, in fact a topological invariant. This invariant captures a subtle feature of the manifold which homology misses. The topological torsion, which is called the Reidemeister torsion, can be used to distinguish lens spaces which are homotopic and hence have the same homology. In the talk I will go over preliminaries of Riemannian geometry, spectrum of the Laplacian and regularized determinants which are required to define Analytic torsion.

References:

1. Analytic torsion and the Cheeger-Muller theorem - Elizabeth Jagersma

2. Talk notes, Video recording

Morse theory gives a nice way of decomposing a closed manifold into a CW complex giving a simple method to compute the homology of the manifold. I will go over the necessary definitions and results needed to get to this CW complex. Proofs will not be given as it involves some analysis, but I will draw pictures for simple examples to demonstrate the results. Time permitting I will also talk about Witten Laplacians and localization.

References:

1. Morse Theory - Shintaro Fushida-Hardy

2. Talk notes, Video recording

The construction and classification theory of compact orientable surfaces is a well-known and essential tool in elementary topology. We will discuss analogue theorems in the non-compact setting, namely:

1. Ian Richards representation theorem: There exists a well-organized class C of non-compact orientable surfaces such that given any non-compact orientable surface S, class C contains a (homeomorphic) copy of S.

2. Kerékjártó's classification theorem: Every non-compact orientable surface has three invariants that are necessary and sufficient to determine it uniquely (up to homeomorphism)

References:

1. On Classification of Noncompact Surfaces - Ian Richards

2. Notes, Video

I will start by introducing the concept of random matrices and some of their applications. Then, we will discuss some of the most studied random matrix ensembles and their properties. Finally, I will state the famous Wigner's Semicircle Law. Time permitting, I'll briefly talk about the proof.

Reference: Talk Video

Kronecker’s Jugendtraum (Jugendtraum is German for “youthful dream”) describes a central problem in class field theory, to explicitly describe the abelian extensions of an arbitrary number field K in values of transcendental functions.

Class field theory gives a solution to this problem in the case where K=Q, the field of rational numbers. Specifically, the Kronecker-Weber theorem gives that any number field sits inside one of the cyclotomic fields Q(e^{2 pi i/n) for some n. Refining this only slightly gives that we can explicitly generate all abelian extensions of Q by adjoining values of the transcendental function e^{2 pi i z} for certain points z in Q/Z.

Not much is known if we replace base field by some other number field. I will talk about the case when the base field is an imaginary quadratic field, in which case Kronecker’s Jugendtraum has been solved by the theory of “complex multiplication”. The specific transcendental functions which generate all these abelian extensions are the modular j-function (as in elliptic curves) and Weber’s w-function.

In the first talk, I'll talk about Elliptic curves, Elliptic functions, and Complex Multiplication. In the second talk, I'll talk about some necessary Algebraic number theory, and then give a proof of the First fundamental theorem of CM which gives an explicit description of the Maximal Abelian unramified extension in this case (Also known as Hilbert Class Fields).

And if time permits, I'll talk about explicit construction of Hilbert class fields of Imaginary Quadratic fields.

References:

1. Cox David A., Primes of the Form x^2 + ny^2 , second edition, Pure and Applied Mathematics, John Wiley & Sons, Ltd, 2013.

2. Janusz, Gerald J. Algebraic number fields. Second edition. Graduate Studies in Mathematics, 7. American Mathematical Society, Providence, RI, 1996. x+276 pp.

3. Zagier D.,Traces of singular moduli, 2002

4. Notes, Video

Recently, Hong, Mertens, Ono, and Zhang proved a conjecture of C˘ald˘araru, He, and Huang that expresses the Taylor series of the modular j-function around the elliptic points i and ρ = e^{πi/3} as rational functions arising from the signature 2 and 3 cases of Ramanujan’s theory of elliptic functions to alternative bases. We extend these results and give inversion formulas for the j-function around i and ρ arising from Gauss’ hypergeometric functions and Ramanujan’s theory in signatures 4 and 6.

References:

1. Inversion Formulas for the j-function Around Elliptic Points - Alejandro De Las Penas Castano, Badri Vishal Pandey

2. Slides, Video

The talk is going to discuss the Operator System, Unital Completely Positive (UCP) maps on a Operator System, Dilation of an operator, Dilation of a UCP map and some important results which gives a very nice description of the UCP maps.

In the second part of the talk I shall discuss about convexity as a tool to study many interesting things about Operator Algebras and briefly talk about classical Croquet Theory.

Towards the last part I shall briefly motivate and introduce the "Non Commutative Convexity".

Reference: Video

Video

von Neumann algebras are C* algebras with an extra added structure. In this introductory talk, we will motivate the idea of von Neumann algebras, and cover the basic definitions theory and examples. In the end, we take a look at the standard form of a tracial von Neumann algebra.

Reference: Video, Slides

The Atiyah-Singer index theorem is one of the great achievements in modern mathematics. It gives an analytic formula for topological quantities making the analytic and the topological index equal. In this series of talks I will try to motivate and/or define all the terms mentioned. The first talk will be about vector bundles over manifolds: definitions and examples, connections and curvature. I will briefly talk about characteristic classes.

References

1. Vector Bundles & K-Theory - Allen Hatcher

2. Differential Geometry - Loring Tu

3. Video, Notes

References:

1. Video

References

1. Vector Bundles & K-Theory - Allen Hatcher

2. Video