Presentations by the MAMME students 2020

As it is costumary, the participants of the course on Differentiable Manifolds at the MAMME are going to present their short projects on various topics related to the contents of the course. The presentations will take place on a Google Meet conference (link to be announced). Here is a list of the presentations that will take place:

May 18th: Differential Topology, Lie groups and applications

Click here to watch this day's talks by streaming.

Exotic Spheres

Speaker: Oriol Almirall Lladó

Time: 15:00 - 15:20

Abstract: Given a topological manifold, it is discussed the existence of distinct differential structures on it. In other words, the existence of homemorphic manifolds that are not diffeomorphic between them. The intention is to explain how J. Milnor proved the existence of distinct differential structures over the 7-sphere, and the geometric construction of this 'exotic spheres'.

Bibliography:

Milnor, John. On manifolds homeomorphic to the 7-sphere. Annals of Mathematics, (2), 64, (1956), 399-405.
Milnor, John. Differentiable structures on spheres. American Journal of Mathematics, 81, (1959), 962-972.

Lefschetz fixed-point theorem

Speaker: Roger Bergadà Batlles

Time: 15:25 - 15:45

Abstract: The Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. With this essay, we will acquire the knowledge necessary to prove the theorem.

Bibliography:

Victor Guillemin and Alan Pollack. Differential topology. PrenticeHall, Inc., Englewood Cliffs, N.J., 1974, pp. xvi+222.

Lie groupoids and algebroids

Speaker: Sergi Galve Regolf

Time: 15:50 - 16:10

Abstract: Category theory is a branch of mathematics that, among other things, should be understood as a language or tool that allows mathematicians to see things with a higher perspective, which sometimes makes problems easier to deal with.

In this project, we follow the categorical approach to differential geometry. More concretely, we want to understand why it is useful to generalize the theory of Lie groups and Lie algebras using a categorical framework. We introduce Lie groupoids and Lie algebroids and we provide some examples to understand them better. Moreover, we also present,
in a short way, some of the theory of integrability of Lie algebroids, which generalizes the same theory made on Lie algebras.

Bibliography:

M. Crainic, R. Loja: Lectures on Integrability of Lie Brackets. ArXiv:math/0611259, v1, 9 Nov 2006. (Main reference)

I. Moerdijk, J. Mrcun: Introduction to Foliations and Lie Groupoids. Cambridge University Press. New York, 2003.

Topology and Hydrodynamics

Speakers: Jordi Frías and Josep Gallegos

Time: 16:10 - 17:00

Abstract: We give an overview of how topology turns out to be a powerful toolbox in the field of ideal fluid dynamics. First, we review how the Euler equations are formulated over Riemannian 3-manifolds. We prove Arnold's theorem on C^ω Euler fields. We also survey some results on contact topology (the study of totally non-integrable plane distributions) which allow us to prove the existence of nonvanishing steady solutions to the Euler equations for closed and oriented 3-manifolds. For the sake of completeness, we briefly introduce some properties of the solutions of the Euler equations in R² and R³ as well as some concepts from Riemannian geometry.

Bibliography:

On the contact topology and geometry of ideal fluids, R. Ghrist

Selected topics on the topology of ideal fluid flows, Daniel Peralta-Salas

May 20th: Facets of Cohomology and Morse Theory

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Chevalley-Eilenberg Cohomology

Speaker: Lucia Constantini

Time: 15:00 - 15:20

Abstract: Due to its properties, for any Lie group we can compute a Lie algebra, whose underlying vector space coincides with the tangent space of the Lie group at its identityelement. The Lie algebra can thus be seen as a linearised version of the global object (its corresponding Lie group) and entirely captures the the latter's local structure.

Building on this key realisation, we notice that the examination of questions regarding the topological nature of a Lie group and the associated manifolds can be reduced to algebraic questions in Lie algebras.

The way one can do this is by first using *de Rham's theorems* to transform matters concerning homology groups into ones concerning arbitrary differential forms. Then these can be turned intoinvariant differential forms using invariant integration. Finally, the problem could be reduced further by looking at the multilinear forms on the Lie algebra.

In simplifying and analysing these topological questions, we inevitably encounter some purely algebraic objects that can be constructed from any Lie algebra over a field ofcharacteristic zero. Such objects can be sorted in equivalence classes using cohomology groups.

Bibliography:

Claude Chevalley and Samuel Eilenberg. Cohomology theory of lie groups
and lie algebras.Transactionsof the American Mathematical Society,
63(1):85–85, Jan 1948.

G. Hochschild and G. D. Mostow. Cohomology of lie groups.Illinois
Journal of Mathematics,6(3):367–401, 1962

De Rham Cohomology and Poincaré duality

Speakers: Guillermo Gamboa and Sarah Zampa

Time: 15:25 - 16:05

Abstract: It is often difficult to compare the topological structure of two smooth manifolds directly using only topological notions. The de Rham cohomology is a mathematical method that allows for algebraic computations over smooth manifolds in order to extract topological information. It introduces a group structure on the set of differential forms of a smooth manifold through means of an equivalence relation. In practice, the computational tools used to create this quotient group rely on the Mayer-Vietoris sequences and the homotopy invariance of the de Rham groups. We formally introduce these sequences and provide examples on how to compute the de Rham cohomology groups of different manifolds. We also study the notion of good covers on a manifold which gives way to stating and discussing, through means of examples, the finiteness properties of the de Rham cohomology groups for compact manifolds. At last, we introduce the Poincaré Duality and prove its main properties and results.

Bibliography:

R. Bott and L.W. Tu. Differential Forms in Algebraic Topology. Graduate Texts in Mathematics. Springer New York, 1995.

V. Guillemin and P. Haine. Differential Forms. World Scientic Publishing Company Pte. Limited, 2019.

L.W. Tu. An Introduction to Manifolds. Universitext. Springer New York, 2010.

Morse theory and persistence

Speakers: Gerard Gnutti and Anna Sopena

Time: 16:15 - 16:55

Abstract: Persistent homology is a method that allows to study topological properties of data as shapes and functions. This tool is inspired by Morse theoretic reasoning. In this essay we introduce basic concepts of Morse homology and
we relate them with the concept of persistence.

Bibliography:

J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963

Edelsbrunner, Herbert, and John Harer. "Persistent homology - a survey". Contemporary mathematics 453 (2008): 257-282.

https://math.uchicago.edu/~may/REU2018/REUPapers/Lingareddy.pdf

May 22nd: Complex, Symplectic Geometry and Lie theory in interaction with physics

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Tailoring with complex spheres

Speaker: David Adame Carrillo

Time: 15:00 - 15:20

Abstract: Modern theoretical physics is a fascinating and intriguing world that has indeed inspired several advances in mathematics. A remarkable topic therein is Conformal Field Theory (CFT): A technique that has proven to be useful in a number of scenarios ranging from ferromagnetism to string theory. When selecting a framework in which to rigorously construct CFT, complex geometry is a solid candidate, for it encodes the conformal symmetries of the theory in a natural manner. In this text, we draw a path within the landscape of complex geometry in two dimensions. We present both
basic concepts and more advanced tools so as to prove that any two compact genus-0 Riemann surfaces are, in some sense, equivalent. On our way to rigorous CFT, we also introduce and characterise a basic technique in the
field: the sewing operation.

Bibliography:

H.M. Frakas, I. Kra. - Riemann surfaces

A.I. Bobenko - Introduction to compact Riemann surfaces (In: A.I. Bobenko, et al. - Computational approach to Riemann surfaces)

Y.-Z. Huang - Two-dimensional conformal geometry and vertex operator algebras

An introduction to symplectic billiards

Speakers: Aida Chaikh and Júlia Perona

Time: 15:25 - 16:05

Abstract: We study symplectic billiards that differs from Birkhoff billiards by the generating function, the symplectic area and the length, respectively. We introduce and explore some properties of symplectic billiards and its similarities and differences with Birkhoff billiards.

Bibliography:

P. Albers and S. Tabachnikov. Introducing symplectic billiards. August 25, 2017.

A Representation theory for the Standard Model

Speaker: Jordi Manyer

Time: 16:15 - 16:35

Abstract: The fundamental laws governing our physical world should be simple. Most of the physics community nowadays agrees on that regard, even though there is no real explanation on why it should be like this. Beauty (in the mathematical sense) and simplicity are therefore things any worthy theory should strive to have.

In that regard, the standard model of particle physics is often criticised as being a loose recompilation of particles lacking in beauty compared with General Relativity, which is constructed on a breathtakingly simple relation between geometry and matter. This criticism might have finally come to an end with the recent thesis by C. Furey, which may have lay
the ground for the construction of a complete representation theory for the standard model.

Representation theory is a branch on mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. The link between representation theory and the standard model is not a new thing: It was first found by Wigner in the 1930s, who showed that the different quantum states of an elementary particle give rise to an irreducible representation of the Poincaré group. The problem was expanding these results to the whole theory.

By solving the non-associativity problem of the octonions, C. Furey may have opened the door to find a complete representation of the standard model using the so-called Dixon algebra, T = RxCxHxO, which would be a true milestone for our understanding of symmetries in the universe. In this text, we will explore old and new results alike concerning representation theory for the standard model, trying to understand the symmetries lying in the Dixon algebra.

Bibliography:

C.Furey, Standard Model Physics from an Algebra?, PhD Thesis 2014

Symplectic Methods for Ray Tracing and aberration Computation in Geometrical Optics

Speaker: Kevin Martínez Añón

Time: 16:40 - 17:00

Abstract: In this essay, a deep insight in the symplectic nature of Geometrical Optics is presented together with some examples of the new perspectives it offers into the classical problems of ray tracing and theory of aberrations. First, we prove the equivalence between ray tracing in Paraxial Optics and the study of symplectic linear transformations of
R^{2n}, and we explore some of the implications of Hamilton's formalism in this linear framework. Finally, the full Hamiltonian method is developed in accordance to Fermat's Principle in the more general non-linear context of Geometrical Optics and it is employed in understanding all Seidel's five optical aberrations: spherical aberration, coma, astigmatism, curvature of field and distortion.

Bibliography:

V. Guillemin and S. Sternberg (1984): Symplectic Techniques in Physics

Introduction to Lagrangian cobordisms

Speaker: Laura González

Time: 17:05 - 17:25

Abstract: This essay studies the Lagrangian cobordism. It gives a required background on symplectic geometry and Lagrangian manifolds. Finally, it illustrates the notion of flexibility on immersed Lagrangians, with an example of Lagrangian Tori in the 2-dimensional complex space.

Bibliography:

Charette, François. "What is a monotone Lagrangian cobordism?." Séminaire de théorie spectrale et géométrie* 31 (2012-2014): 43-53. <https://eudml.org/doc/275818>.

Biran, Paul; Cornea, Octav. Lagrangian cobordism. I. J. Amer. Math. Soc. 26 (2013), no. 2, 295-340.