# LisMath Seminar

## Past sessions

### From racks to pointed Hopf algebras

A quandle is a group-like algebraic structure whose axioms reflect the Reidemeister moves from Knot Theory [2]. This means that quandles naturally give rise to knot invariants, which can be efficiently used in distinguishing knots. Therefore, we expect that even a partial classification of quandles will improve knot detection and distinction techniques. In [1], several important steps are made towards a complete classification of quandles. In this talk, after properly introducing Knot Theory, we present some of these results along with examples and applications.

Bibliography:

[1] N. Andruskiewitsch and M. Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003) 177-243

[2] D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg. 23 (1982), 37-65

lismath_seminar_antonio_lages.pdf

### Nonabelian Cohomology

For a given groupoid $G$ and $M$ a $G$-module, the $n$-th cohomology is defined as the set of homotopy classes $H^n(G,M)=[F_{\star}^{st} (G), K_n(M,G);\phi ]$, where $F_{\star}^{st} (G)$ is the free crossed resolution of $G$, and $\phi : F_1^{st} (G)\to G$ is the standard morphism.

In this talk we assign a free crossed complex to a cover $\mathcal{U}$ of the topological space $X$, so we get the notion of nonabelian cohomology.

We finish our talk by introducing a long exact sequence for nonabelian cohomology.

Bibliography:

[1] R. Brown , P. Higgins and R. Sivera. Nonabelian algebraic topology. European Mathematical Society, 2010

[2] T. Nikolaus and K. Waldorf. Lifting problems and transgression for non-abelian gerbes, Advances in Mathematics 242, pp. 50–79, 2013

[3] L. Breen. Bitorseurs et cohomologie non abélienne, The Grothendieck Festschrift, pp. 401–476, 2007.

Nonabelian cohomology.pdf

### Quiver Representations

A quiver is a directed graph where multiple arrows between two vertices and loops are allowed. A representation of a quiver $Q$, over a field $K$, is an assignment of a finite dimensional $K$-vector space $V_i$ to each vertex $i$ of $Q$ and a linear map $f_a:V_i\rightarrow V_j$ to each arrow $a:i\rightarrow j$. Given a quiver $Q$, the set of all representations of $Q$ forms a category, denoted by $\mathrm{Rep}(Q)$. A connected quiver is said to be of finite type if it has only finitely many isomorphism classes of indecomposable representations.

Quiver representations have remarkable connections to other algebraic topics, such as Lie algebras or quantum groups, and provide important examples of moduli spaces in algebraic geometry [3].

The main goal of this work would consist, first, of good comprehension of the category $\mathrm{Rep}(Q)$. Then, the student would cover the basics on quiver representations to be able to prove Gabriel's theorem [1], following a modern approach, as in [2]:

A connected quiver is of finite type if and only if its underlying graph is one of the ADE Dynkin diagrams $A_n$, $D_n$, for $n \in \mathbb N$, $E_6$, $E_7$ or $E_8$. Moreover, the indecomposable representations of a given quiver of finite type are in one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.

These basic concepts involve topics such as the Jacobson radical, Dynkin diagrams or homological algebra of quiver representations.

Bibliography:

[1] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Mathematica 6, pp. 71–103 (1972).

[2] H. Derksen and J. Weyzman, An Introduction to Quiver Representations, Graduate Studies in Mathematics 184, American Mathematical Society (2017).

[3] A. Soibelman, Lecture Notes on Quiver Representations and Moduli Problems in Algebraic Geometry, arXiv:1909.03509 (2019)

JavierOrts-QuiverRepresentations.pdf

### Homotopical Dynamics

In this presentation we will talk about some relations between algebraic topology and dynamical systems. We will explore the Flow type suspension and see that it agrees with the homotopical suspension for attractor-repeller homotopy data. Some other results will also be explored.

Bibliography:

[1] O. Cornea. Homotopical dynamics: Suspension and duality. Ergodic Theory and Dynamical Systems, 20(2), 379-391 (2000).

[2] J.F. Barraud, O. Cornea. Homotopical dynamics in symplectic homology. In: Biran P., Cornea O., Lalonde F. (eds) Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology. NATO Science Series II: Mathematics, Physics and Chemistry, vol 217 (2006), Springer.

Presentation_LISMATH_FRED_2019_2020.pdf

### An Introduction to Gerbes

Gerbes being generalisations of bundles over a manifold can be regarded as a geometric realisation of three dimensional cohomology classes of a manifold. Considering the example of circle bundles on a manifold $M$, we recall that such bundles can be described from different perspectives as either

• certain locally free sheaves on $M$
• cocycles $g_{\alpha \beta} : U_{\alpha} \cap U_{\beta} \rightarrow U(1)$ associated to an open cover $\{ U_{\alpha} \}$ of $M$
• principal $U(1)$ bundles over $M$

In a similar fashion also gerbes allow such characterizations, generalising the same ideas. This talk will focus mostly on the different definitions of gerbes and their applications in field theory.

Bibliography:

[1] M. K. Murray, An introduction to bundle gerbes, arXiv:0712.1651.

[2] G. Segal, Topological structures in string theory, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 359, number 1784, pp. 1389–1398, 2001, The Royal Society.

[3] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, 2007, Springer Science & Business Media.

[4] J. Fuchs, T. Nikolaus, C. Schweigert and K. Waldorf, Bundle gerbes and surface holonomy, arXiv:0901.2085.

An Introduction to Gerbes.pdf

### Polar Functions and Measure Transitions

In this talk we will discuss several properties of polar functions. These are functions that exhibit some kind of positive or negative definiteness. In particular, it is known that these functions are characterized as an integral transformation of a unique measure in maximal strips of the complex plane. The only restriction to the size of these strips are the singularities of the function. However, if the function is polar on either side of a pole then the respective measures are related. The goal of this talk is to explain this relation.

Bibliography:

[1] J. Buescu, A.C. Paixão.The Measure Transition Problem for Meromorphic Polar Functions. Submitted, 2019.

Polar functions talk.pdf

### On a new class of fractional partial differential equations

In this talk, I will discuss a class of fractional partial differential equations. Such fractional partial differential equations are obtained from extending the theory regarding the Riesz fractional gradients. I will first introduce the fractional differential operators $\nabla^s$ and $\div^s$. I will then explain a notion of fractional gradient, which has the potential to extend many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way. These ideas can then be used to establish analogous results for fractional partial differential equations.

Bibilography:

[1] Giovanni E. Comi and G. Stefani. A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I. 2019. arXiv: 1910.13419 [math.FA].

[2] Giovanni E. Comi and Giorgio Stefani. A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up. In: Journal of Functional Analysis 277.10 (2019), pp. 3373- 3435. issn: 0022-1236. doi: 10.1016/j.jfa.2019.03.011.

[3] José Francisco Rodrigues and Lisa Santos. On Nonlocal Variational and Quasi-Variational Inequalities with Fractional Gradient. In: Applied Mathematics & Optimization 80, no. 3 (2019), pp. 835?852. doi: 10.1007/s00245- 019-09610-0.

[4] Tien-Tsan Shieh and Daniel Spector. On a new class of fractional partial differential equations. In: Advances in Calculus of Variations 8 (2015), pp. 321-366. doi: 10.1515/acv-2014-0009.

[5] Tien-Tsan Shieh and Daniel Spector. On a new class of fractional partial differential equations II. In: Advances in Calculus of Variations 11 (2018), pp. 289-307. doi: 10.1515/acv-2016-0056.

[6] Miroslav Silhavy. Fractional vector analysis based on invariance require- ments (critique of coordinate approaches). In: Continuum Mechanics and Thermodynamics 32, Issue 1 (2020), pp. 207-288. doi: 10.1007/s00161-019- 00797-9.

LisMath_seminar_presentation.pdf

### Morse Homology and Floer Homology

Morse theory relates the topology of a smooth manifold with the critical points of Morse functions. Under Morse-Smale transversality one can define a chain complex generated by critical points which com- putes the singular homology of closed manifolds; in particular this implies the Morse inequalities on the number of critical points of Morse functions. Floer homology originated as a version of Morse homology for the symplectic action on the (infinite dimensional) free loop space on a symplectic manifold, which culminated in proving a conjecture by Arnol’d on the number of 1-periodic orbits of non-degenerate Hamiltonians on closed symplectic manifolds. Other examples of applications and generalizations are Viterbo’s theorem on the Floer homology of cotangent bundles, or $S^1$-equivariant Floer homology.

Bibiliography:

[1] Floer, A. (1988a). A relative Morse index for the symplectic action. Comm. Pure Appl. Math., 41(4):393-407.

[2] Floer, A. (1988b). The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math., 41(6):775-813.

[3] Audin, M. and Damian, M. (2014). Morse theory and Floer homology. Universitext. Springer, London; EDP Sciences, Les Ulis.

[4] Abbondandolo, A. and Schwarz, M. (2006). On the Floer homology of cotangent bundles. Comm. Pure Appl. Math., 59(2):254-316.

[5] Abbondandolo, A. and Schwarz, M. (2014). Corrigendum: On the Floer homology of cotangent bundles. Comm. Pure Appl. Math., 67(4):670-691.

[6] Abouzaid, M. (2015). Symplectic cohomology and Viterbo’s theorem. In Free loop spaces in geometry and topology, volume 24 of IRMA Lect. Math. Theor. Phys., pages 271-485. Eur. Math. Soc., Zurich.

[7] Bourgeois, F. and Oancea, A. (2017). $S^1$ -equivariant symplectic homology and linearized contact homology. Int. Math. Res. Not. IMRN, (13):3849-3937.

miguelmsantos_29042020.pdf

### Machine Learning Driven Optimal Stopping

Optimal stopping problems constitute a subset of stochastic control problems in which one is interest in finding the best time to take a given action. This framework has relevant contributions extending across different fields, namely finance, game theory and statistics. Recently the literature on machine learning has grown at a very large pace, specially in what concerns the usage of its techniques in other fields beyond computer science, in the hope that those might shed some light in long persisting problems such as, for instance, the well known curse of dimensionality. In light with this trend the literature on both stochastic control and optimal stopping has presented several contributions by either incorporating reinforcement learning techniques (Machine Learning Control) or by making use of neural networks to estimate the optimal stopping time of a given problem.

Bibliography:

[1] G. Peskir and A. Shiryaev, Optimal stopping and free-boundary problems, 2006, Springer.

[2] W. H. Fleming and H. M. Soner, Controlled Markov processes and viscosity solutions vol. 25, 2006, Springer Science & Business Media.

[3] S. Becker, P. Cheridito and A. Jentzen, Deep optimal stopping, Journal of Machine Learning Research, vol. 20, 2019.

[4] S. Becker, P. Cheridito, A. Jentzen and T. Welti, Solving high-dimensional optimal stopping problems using deep learning, arXiv preprint arXiv:1908.01602.

[5] H.J. Kappen, An introduction to stochastic control theory, path integrals and reinforcement learning, AIP conference proceedings, vol. 887, nr. 1, pp. 149–181, 2007.

[6] E. Theodorou, J. Buchli and S. Schaal, A generalized path integral control approach to reinforcement learning, Journal of Machine Learning Research, vol. 11, Nov, pp. 3137–3181, 2010.

LisMath_Seminar_2020.pdf

### Hydrodynamics for SSEP with non-reversible slow boundary dynamics: the critical regime and beyond

In this talk we present the Law of Large numbers for three quantities (local density, current and mass) for the Symmetric Simple Exclusion Process (SSEP) on the lattice $\{1, . . . , N − 1\}$ with “nonlinear” boundary dynamics. Informally, we let a particle jump only to its neighbor site if such site is empty. Then, we let the system be in contact with two reservoirs, which inject/remove particles from a window of size $K$ from the boundaries, at rates depending on the site of injection/removal. We let a particle enter to the first free site, and leave from the first occupied site. This boundary dynamics impose strong correlations between particles, which leads to the sudy of most physical quantities of the system being a challenge. Multiplying the boundary rates by $N^{-\theta}$, one observes macroscopically phase transitions on those quantities in the following way. Under a $N^2$ time-scale, macroscopically the local density behaves as a weak solution to the heat equation. For $\theta \in [0, 1)$ we have Dirichlet B.C., nonlinear Robin for $\theta = 1$, and Neumann for $\theta \gt 1$. For the current, we microscopically derive Fick’s Law, which depends on the B.C. for the density, while for the mass, we see that instead the time scale $N^{1+ \theta}$ is the most natural one, and obtain an ODE. We present only results for $\theta \geq 1$. We then show that starting from the stationary measure, we obtain steady state solutions of the aforementioned equations.

Bibliography:

[1] Gonçalves, P., Erignoux, C., Nahum, G.: Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part I, the critical regime and beyond, arXiv:1912.09841 (2019).

lismath-15-04-2020.pdf

### A Supercharacter Theory for approximately finite algebra groups

By an algebra group over a field $\mathbb{K}$ it is meant a group of the form $G = 1+A$, where $A$ is a nil algebra over $\mathbb{K}$ and product rule given as $(1+a)(1+b) = 1+a+b+ab$; the group $G = 1 + A$ is said to be an approximately finite algebra group if there is a family $\{G_n\}_{n \in \mathbb{N}}$ of finite algebra subgroups for which $G$ is the direct limit $\lim_{\rightarrow} G_n$.

Assuming mild conditions on a topological group, there is a well defined notion of characters that extend the usual Character Theory of finite, or more generally compact groups; in this setting indecomposable characters play the role of irreducible characters as they fully determine the Character Theory and serve as minimal group invariants. However, the set of indecomposable characters may be too large or even too complicated to characterize, for this reason it is of interest to consider a smaller family of characters that mimics the behaviour of indecomposable ones.

In this talk we generalize the definition of a Supercharacter Theory for finite groups into the topological group scenario, and using essentially ergodic theoretical tools we define and characterize a Supercharacter Theory for an arbitrary approximately finite algebra group.

### Classifying and counting $N=2$ black holes

We discuss BPS black holes in an $N=2$ supergravity model. We use the exact symmetries of the model to classify BPS orbits, and we propose a microstate counting formula, based on modular forms, that reproduces the entropy of dyonic black holes in this model.

### Finding a nice supercharacter theory for the Sylow $p$-subgroups of the symmetric group

After a brief introduction to the history of supercharacters, we start with cyclic groups and several procedures will be explored as direct, semidirect and wreath product in order to study the supercharacter theory arising from the orbit method on algebra groups, aiming to find a pleasant combinatorial model for that. Some examples will be discussed in full detail.

### Schur averages in random matrix ensembles

There is a well known relation between matrix integrals over the classical Lie groups and the determinants of Toeplitz and Hankel matrices. We generalize this connection to the averages of Schur polynomials over these groups, which correspond to the minors of the underlying Toeplitz and Hankel matrices. We will show how this approach can be exploited to obtain several results in the theory of symmetric functions and representation theory. Some applications to Chern-Simons theory will also be discussed.

### Surfaces of general type with canonical map of high degree

The study of the canonical map of surfaces of general type is a classical subject in the theory of algebraic surfaces. The canonical map was first studied by A. Beauville in 1979. He proved that if the image of the canonical map of a minimal surface of general type is a surface, this surface either has geometric genus zero (I) or is canonically embedded (II). Furthermore, he showed that the degree of the canonical map is less than or equal to 36. In the last decade, the problem of constructing examples of surfaces of general type with canonical map of high degree has been studied by many authors. Nonetheless, there still remain many questions left open. In this talk, we present some new examples of surfaces with non-birational canonical map in two classes (I), (II).

### On Gelfand-Graev type characters of unit groups arising from finite algebras

Our work is divided in two parts: the first part explores the induction of supercharacters from $U_n(q)$ to $GL_n(q)$ as a possible generalization of the Gelfand-Graev character, as well as some related problems which were posed during our research; the second part explores the possibility of adapting Gelfand and Graev’s construction to finite groups arising as the units of a finite dimensional unital algebra over some finite field.

### Decomposition of time series

A time series is a realization of a stochastic process. In general, time series are decomposed into their natural components (trend, seasonal, cyclical and irregular components) before further analysis. Neverthless, a decomposition into unusual components can also be useful in several areas of research. In this talk, different methods for time series decomposition into unusual components will be present. The Empirical Mode Decomposition (EMD) and the Independent Component Analysis (ICA) are two examples. Moreover, some of these methods will be tested with a simulated time series and the results will be discussed.

Bibliography:

[1] Cleveland, R.B., Cleveland, W.S., and Terpenning, I. (1990). STL: A seasonal-trend decomposition procedure based on loess. Journal of Official Statistics 6(1):3.

[2] Mijovic, B., De Vos, M., Gligorjevic, I., Taelman, J., and Van Huffel S. (2010). Source separation using single-channel recordings by combining empirical-mode decomposition and independent components analysis. IEEE Transactions on biomedical engineering, 57(9):2188-2196.

LisMath_MASilva.pdf

### Ordering protoalgebraic logics

The Leibniz hierarchy arose as an attempt to rank which types of logics are more amenable to be studied from an algebraic point of view and plays a central role in modern abstract algebraic logic. Within this hierarchy, the class of protoalgebraic logics resides at the very bottom, not being included in any other class. In this sense, protoalgebraicity in one of the weakest properties of logics that makes them amenable to most of the standard methods in algebra. In this seminar we will study the order properties of this lattice of logics with some interesting results.

Bibliography:

[1] Josep Maria Font, Ordering protoalgebraic logics, Journal of Logic and Computation 26 (2016)

[2] Josep Maria Font, Abstract Algebraic Logic - An Introductory Textbook, 2016

lismathseminar_pedro_filipe.pdf

### String theory and integrability

The goal of this seminar is to introduce the concept of integrability, showing how it can be applied to obtain solutions for dynamical systems. We will then apply the tools that we build in the first part of the talk to attack a problem that is relevant in topological string theory, the problems of the KP and KdV hierarchy. We will introduce those problems in a self-contained way, explore the use of the technique of the Lax pairs to solve them and finally we will see how those solutions are of use in matrix models, that appear in minimal string theories of interest.

Bibliography:

[1] E. Witten, Two-dimensional gravity and intersection theory on moduli space, Survey in Differential Geometry, 243-310 (1990)

[2] R. Dijkgraaf, Intersection theory, integrable hierarchies and topological field theory

[3] P. van Moerbecke, Integrable foundations of string theory, In: Lec- tures on Integrable Systems, ed by O. Babelon, P. Cartier, Y. Kosmann- Schwarzbach (World Sci. Publishing, River Edge, NJ, 1994) pp 163-267

[4] R. Dijkgraaf, E. Witten, Developments in topological gravity

[S_Baldino]_String_Theory_and_Integrability.pdf

### The twistor correspondence and the ADHM construction on $S^4$

We shall give a brief overview of Yang-Mills theory, discussing some of its applications to other areas of geometry. We then focus on the construction of holomorphic bundles corresponding, via the twistor transform, to instanton solutions of the Yang-Mills equation.

Bibliography:

[1] M. F. Atiyah, N. J. Hitchin, I. M. Singer - Self-duality in four-dimensional Riemannian geometry (1978)

[2] R. S. Ward, R. O. Wells Jr. - Twistor Geometry and Field Theory (1991)

[3] M. F. Atiyah - Geometry of Yang-Mills Fields (1979)

[4] N. J. Hitchin - Linear field equations on self-dual spaces (1980)