# LisMath Seminar

## Past sessions

### Hydrodynamic behavior of a degenerate microscopic dynamics with slow reservoirs

In recent years, there has been intensive research activity around the derivation of partial differential equations with boundary conditions from interacting particle systems. This derivation is known as hydrodynamic limit. In this seminar we will discuss how to derive the porous medium equation with Dirichlet, Neumann, and Robin boundary conditions, from a random microscopic dynamics with slow reservoirs.

### Classification of Hamiltonian circle actions on compact symplectic orbifolds of dimension 4

In this talk I want to present my thesis work towards the classification of Hamiltonian circle actions on compact orbifolds of dimension 4, with isolated cyclic singularties.

I will start by explaining what orbifolds are and how circle actions are defined on them. Then I will explain how to associate a graph to any such space, and show that two 4 dimensional orbifolds with Hamiltonian circle actions are isomorphic, if and only if their graphs are isomorphic. I will then give a list of minimal models from which many orbifolds with a Hamiltonian circle action can be constructed.This work is inspired and based on the work of Yael Karshon, who classified Hamiltonian circle actions on compact manifolds of dimension four.

### Spectral theory, clustering problems and differential equations on metric graphs

We present our thesis work dealing with several topics in PDE theory on metric graphs. Firstly, we present our framework and present existence results for nonlinear Schroedinger (NLS) type energy functionals as generalizations and unification of various results obtained by several authors, most notably from [1] and [2], among others. Secondly, we consider spectral minimal partitions of compact metric graphs recently introduced in [3]. We show sharp lower and upper estimates for various spectral minimal energies, estimates between these energies and eigenvalues of the Laplacian and discuss their asymptotical behaviour. Thirdly, we present Pleijel's theorem on the asymptotics of the number of nodal domains $\nu_n$ of the $n$-th eigenfunction(s) of a broad class of operators of Schroedinger type on compact metric graphs. Among other things, these results characterize the accumulation points of the sequence $(\frac{\nu_n}{n})_{n\in\mathbb N}$, which are shown always to form a finite subset of $(0,1]$. Finally, we introduce a numerical method for calculating the eigenvalues for a special operator in the beforementioned class, the standard Laplacian, based on a discrete graph approximation.

References

[1] Adami, Serra and Tilli, Journal of Functional Analysis 271 (2016), 201-223

[2] Cacciapuoti, Finco and Noja, Nonlinearity 30 (2017), 3271-3303

[3] Kennedy et al, Calculus of Variations and Partial Differential Equations 60 (2021), 61

### Group actions on surfaces of general type and moduli spaces

The main numerical invariants of a complex projective algebraic surface $X$ are the self-intersection of its canonical class $K^2_X$ and its holomorphic Euler characteristic $\chi(\mathcal{O}_X)$. If we assume $X$ to be minimal and of general type then $K^2_X\geq 2\chi(\mathcal{O}_X)-6$ by Noether's inequality.

Minimal algebraic surfaces of general type $X$ such that $K^2_X=2\chi(\mathcal{O}_X)-6$ or $K^2_X=2\chi(\mathcal{O}_X)-5$ are called Horikawa surfaces and they admit a canonical $\mathbb{Z}_2$-action. In this talk we will discuss other possible group actions on Horikawa surfaces. In particular, $\mathbb{Z}_2^2$-actions and $\mathbb{Z}_3$-actions on Horikawa surfaces will be studied.

In the case of $\mathbb{Z}_2^2$-actions we will not settle for Horikawa surfaces and results regarding the geography of minimal surfaces of general type admitting a $\mathbb{Z}_2^2$-action will be discussed. They will yield some consequences on the moduli spaces of stable surfaces $\overline{\mathfrak{M}}_{K^2,\chi}$.

### A cactus group action on shifted tableau crystals and a shifted Berenstein-Kirillov group

Gillespie, Levinson and Purbhoo recently introduced a crystal-like structure for shifted tableaux, called the shifted tableau crystal. Following a similar approach as Halacheva, for crystals of finite Cartan type, we exhibit a natural internal action of the cactus group on this structure, realized by the restrictions of the shifted Schützenberger involution to all primed intervals of the underlying crystal alphabet. This includes the shifted crystal reflection operators, which agree with the restrictions of the shifted Schützenberger involution to single-coloured connected components, but unlike the case for type A crystals, these do not need to satisfy the braid relations of the symmetric group.

In addition, we define a shifted version of the Berenstein-Kirillov group, by considering shifted Bender-Knuth involutions. Paralleling the works of Halacheva and Chmutov, Glick and Pylyavskyy for type A semistandard tableaux of straight shape, we exhibit another occurrence of the cactus group action on shifted tableau crystals of straight shape via the action of the shifted Berenstein-Kirillov group. We also conclude that the shifted Berenstein-Kirillov group is isomorphic to a quotient of the cactus group. Not all known relations that hold in the classic Berenstein-Kirillov group need to be satisfied by the shifted Bender-Knuth involutions, namely the one equivalent to the braid relations of the type A crystal reflection operators, but the ones implying the relations of the cactus group are verified, thus we have another presentation for the cactus group in terms of shifted Bender-Knuth involutions.

### Integrability and holography in dimensionally reduced theories of gravity in 2 dimensions

This talk is divided into two parts. In the first part, we consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics. We prove that the canonical Wiener-Hopf factorization of a matrix obtained from a general monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the gravitational field equations. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields distinct solutions to the field equations.

In the second part, we approach the problem of constructing the holographic dictionary for the AdS$_2$/CFT$_1$ correspondence for higher derivative gravitational actions in AdS$_2$ spacetimes obtained by an $S^2$ reduction of 4-dimensional $\mathcal{N}=2$ Wilsonian effective actions with Weyl squared interactions. We focus on BPS black hole solutions, for which we show how the Wald entropy of these black holes is holographically encoded in the dual CFT. Additionally, using a 2D/3D lift we show that the dual CFT$_1$ is naturally embedded in the chiral half of the CFT$_2$ dual to the AdS$_3$ spacetime.

### Geometry and topology of generalized polygon spaces

We consider the space of polygons with edges in a complex projective space, generalizing the classical space of polygons with edges in $\mathbb{R}^3$.

Using the Gelfand-McPherson correspondence and wall crossing techniques we will prove a recursive formula for the Poincaré polynomial of these spaces and compute their symplectic volume.

### Nonuniform Hyperbolicity in Difference Equations: Admissibility and Infinite Delay

We consider a nonautonomous dynamical system given by a sequence of bounded linear operators acting on a Banach space. We introduce the notion of an exponential dichotomy which is central in the stability theory of dynamical systems. Our results give a characterization of the existence of an exponential dichotomy in terms of the invertibility of a certain linear operator between so-called admissible spaces. Using this characterization, we show that the notion of an exponential dichotomy is robust for sufficiently small linear perturbations.

We also introduce the notion of an exponential dichotomy for difference equations with infinite delay. This requires considering an appropriate class of phase spaces that are Banach spaces of sequences satisfying a certain axiom motivated by the work of Hale and Kato for continuous time. We present a result that establishes the existence of stable manifolds for any sufficiently small perturbation of a difference equation having an exponential dichotomy.

Finally, we briefly describe the formulation of the previous results for the more general case of a tempered exponential dichotomy. This is a nonuniform version of an exponential dichotomy that is ubiquitous in the context of ergodic theory.

References

[1] L. Barreira, J. Rijo, C. Valls. Characterization of tempered exponential dichotomies. J. Korean Math. Soc., 57(1):171–194, 2020.

[2] L. Barreira, J. Rijo, C. Valls. Stable manifolds for difference equations with infinite delay. J. Difference Equ. Appl., 26(9-10):1266–1287, 2020.

### 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.

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