LisMath Seminar 

Interacting Particle Systems provide a framework, in a Markovian universe, for the study of phenomena arising from the collective behaviour of a very large number of agents interacting with each other. In this talk I am going to present two results of my PhD work in the context of "non-linear" dynamics.

The SSEP is a nearest neighbour exclusion process defined in a lattice, where particles exchange positions with its empty direct neighbours only, and at most one particle is allowed per site.
The PMM is a kinetically constrained model with exclusion constraints, but depending also on the particular configuration of all the windows, with equal length, that contain the pair of sites that are to exchange occupations. Precisely, fixed some length, the rate corresponds to the number of completely full windows. If this number is zero, the exchange is suppressed. For both the SSEP and the PMM one can prove that "density field $\approx$ differential equation" in probability. This result, in varied topology, is known as the Hydrodynamic Limit. The SSEP yields the heat equation, while the PMM the porous media equation. We define a Markov generator that interpolates continuously the one of the PMM and the SSEP, extending further to a fast diffusion model. The Hydrodynamic Limit is shown and as a result, under a scaling limit, the density field $\rho$ is associated with the diffusion coefficient $ \rho^m $ for any $m > -1$.

This is further generalized in the direction of $\rho^n(1-\rho)^k$ for any $ n , k $ positive integers, through combinatoric arguments. Specifically, it is presented a linear system that characterizes the rates for which the resulting models satisfy the gradient property. This is then extended in a natural way to a long range dynamics.

A maximal space is a finite $C_2$-CW complex $X$ whose cohomology and the cohomology of its fixed points are related by the following equality:

$$\sum_{q=0}^{\dim X^{C_2}} \dim_{F_2} H^q(X^{C_2};F_2) = \sum_{q=0}^{\dim X} \dim_{F_2} H^q(X;F_2).$$

It is a fact that the action of $C_2$ on $H^*(X;F_2)$ is trivial when $X$ is a maximal space.

This class of spaces was generalised by Krasnov in [5] by allowing the action on cohomology to be non-trivial and modifying the above equality conveniently to take this into account. The new spaces are called Galois-maximal spaces. The equation that characterises them is the following:$$\sum_{q=0}^{\dim X^{C_2}} \dim_{F_2} H^q(X^{C_2};F_2) = \sum_{q=0}^{\dim X} \dim_{F_2} H^1(C_2, H^q(X;F_2)).
$$

Maximal and Galois-maximal spaces have always been of great interest in Real algebraic geometry [2], where the CW complex is now the complex locus of a variety over $\mathbb{R}$ and the $C_2$ action is given by complex conjugation, but have became especially relevant in the recent years due to the following result by Biswas and D'Mello [3]:

Theorem 1:
If $X_g$ is a maximal curve, then the symmetric product $SP_n X_g$ is a maximal variety for $n=2,3$ and $n\geq 2g-1$.

This problem was fully solved by Franz [4] for the case of maximal varieties, and by Baird [1] in the case of Galois-maximal curves.
The main result of this talk will address the general case for Galois-maximal varieties:

Theorem 2:
If $X$ is a Galois-maximal variety, then $SP_n X$ is a Galois-maximal variety for ever $n$.

References:

[1] Baird, T. J., “Symmetric products of a real curve and the moduli space of Higgs bundles”, J. Geom. Phys. 126, 7–21 (2018).

[2] Bertrand, B. “Asymptotically maximal families of hypersurfaces in toric varieties”, Geom. Dedicata. 118, 49–70 (2006).

[3] Biswas, I., D’Mello, S.,“M-curves and symmetric products”, Proc. In- dian Acad. Sci. Math. Sci. , 127, No. 4, 615–624 (2017).

[4] Franz, M., “Symmetric products of equivariantly formal spaces”, Canad. Math Bull. 61, No. 2,272–281 (2018).

[5] Krasnov, V. A., “Harnack–Thom inequalities for mappings of real al- gebraic varieties”, Izv. Akad. Nauk SSSR Ser Mat., 47, No. 2, 268–297 (1983).

In this talk we elaborate on our last doings: the definition of a new family of quandles and the production of infinitely many examples of such quandles. These quandles are defined for having a profile made up of lengths pairwise distinct (which makes them connected quandles, thanks to our previous work) and also that each length divides the larger lengths. We prove that if we fix the number of lengths to be $c$, then the profile of these quandles is made up of $c$ lengths, the first two being $1$ and $l$ (where $l+1$ equals a prime power) and then $l_i = l(l+1)^{i-2}$. We call them Nested Quandles because they have subquandles with similar profiles. Time permits we elaborate on the examples of these quandles that we constructed.

The subject of principal bundles with connections (gauge theory) and their higher versions are of special interest in mathematics and physics. Using the known strategies of constructing non abelian bundle gerbes using the plus construction, we give meaning to the tricategory of nonabelian 2-crossed module 2-bundle gerbes using a higher notion of the plus construction. We will view nonabelian 2-crossed module 2-bundle gerbes also as bigroupoids with a 3-group action, and we call them principal 3-bundles. Also we will describe the cocycle conditions, and the classifying space of a 2-crossed module, which will allow us to see isomorphism classes of principal 3-bundles over a smooth manifold $X$ as homotopy classes of classifying maps from $X$ to the classifying space of the 2-crossed module. In the final part we will also discuss 3-connections, where we will compare two different definitions of holonomies for flat 3-connections.

Differential cohomology combining both homotopy-theoretic and geometric invariants is interpreted to be a refinement of ordinary cohomology. More precisely, a differential cocycle refines an ordinary cocycle just as a bundle with connection refines its underlying bundle. Deligne cohomology and Cheeger-Simons differential characters are the most common models of differential cohomology.

From a different perspective, parallel transport systems have been used successfully to model connections on higher bundles, relying on the concept of thin homotopy. Using the language of diffeological spaces and their homotopy theory, we will examine the relationship between Cheeger-Simons differential characters and the smooth cohomology of skeletal diffeological spaces. Here the notion of thinness is directly built into the skeletal diffeology. We will finish by showing how the skeletal diffeology naturally appears in the recently conjectured geometric cobordism hypothesis by Grady-Pavlov, suggesting a link to geometric field theories.

In this talk, we introduce some ideas from Dyadic Harmonic Analysis and we apply these methods to obtain weighted norm inequalities. We focus mainly on Gagliardo-Nirenberg inequalities with the Lebesgue measure modified by a weight. The classical Gagliardo-Nirenberg inequality generalizes the Sobolev inequality and is an absolutely central tool in the study of PDEs. The first results using weighted norms were given by Caffarelli, Kohn and Nirenberg, and they were subsequently improved by Chang-Shou Lin, who settled the inequality with homogeneous radial weights and integer derivatives. Here we will explain how to use the modern methods of weighted inequalities in Harmonic Analysis to extend Lin's inequality to the fractional derivatives setting. Moreover, these methods are flexible enough to give rise to similar inequalities with other weights, including non-homogeneous weights.

One can define the class of non tangential Hardy-Littlewood maximal operators as follows:

$$ M f(x) = \sup_{|x-y|< \alpha r} \frac{1}{2r}\int_{y-r}^{y+r} |f(t)|dt,
$$ where the supremum is taken in $r>0$ and $y$ for a given parameter $\alpha\in [0,1]$. This gives us a maximal averaging operator in $\mathbb R$. In this talk, we will explore how this operator behaves when applied to functions of finite total variation. One would expect that, by taking averages, the function becomes smoother in some sense, and thus, total variation should decrease.

We will present what is known for each $\alpha$, and give a sketch for an alternative proof of the result in [1]. We finish by showing how one could adapt this result for a discrete version of the same operator.

References:

[1] J. Ramos, Sharp total variation results for maximal functions, Ann. Acad. Sci. Fenn. Math. 44 (2019) 41-64

[2] E. Carneiro and D. Moreira, On the regularity of maximal operators, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4395–4404.

[3] J. Kinnunen, The Hardy-Littlewood maximal function of a Sobolev function, Israel J. Math. 100 (1997), 117–124.

[4] O. Kurka, On the variation of the Hardy-Littlewood maximal function, Ann. Acad. Sci. Fenn. Math. 40 (2015), 109–133.

[5] J. Bober, E. Carneiro, K. Hughes, and L. B. Pierce, On a discrete version of Tanaka’s theorem for maximal functions, Proc. Amer. Math. Soc. 140 (2012), 1669-1680.

The degeneracies of 1/4 BPS states with unit torsion in heterotic string theory compactified on a six-torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form $\Phi_{10}$ of weight 10. We use the symplectic symmetries of the latter to construct a fine-grained Rademacher type expansion which expresses these BPS degeneracies as a regularized sum over residues of the poles of $1/\Phi_{10}$. This construction shows how the polar data of a family of mock Jacobi forms are explicitly built from the Fourier coefficients of $1/\eta^{24}$ by means of a continued fraction structure, confirming previous results and providing a guide for the computation of the exact quantum entropy of $1/4$ BPS black holes in four-dimensional supergravity.

In this work we consider the study of the nonadditve thermodynamic formalism and multifractal analysis for continuous time dynamical systems. We introduce a version of the nonadditive topological pressure for flows and we describe some of its main properties. In particular, we discuss how the nonadditive topological pressure varies with the data and we establish a variational principle in terms of the Kolmogorov-Sinai entropy. In the more specific context of almost additive families of continuous functions, we establish an appropriate version of the classical variational principle for the topological pressure, and obtain the existence and uniqueness of equilibrium and Gibbs measures for families with bounded variation.

Building on the construction of equilibrium measures, we establish a conditional variational principle for the multifractal spectra of an almost additive family with respect to a continuous flow $\Phi$ such that the entropy map $\mu \mapsto h_{\mu} (\Phi) $ is upper-semicontinuous. We also show that the spectrum is continuous and that in the case of hyperbolic flows the corresponding irregular sets have full topological entropy.

In the last part, we introduce a version of the nonlinear thermodynamic formalism for flows. In this context, building on the multifractal analysis developed earlier, we discuss the existence, uniqueness, and characterization of equilibrium measures for an almost additive family with tempered variation. Moreover, we consider with some care the special case of an additive family for which it is possible to strengthen some of the results.

The portfolio problem for a stochastic volatility model goes way back to Merton in his seminal paper "Lifetime Portfolio Selection under Uncertainty: the Continuous-Time Case" [1].

In this seminar we will look at the dynamic programming method to solve a variant of the portfolio problem in which consumption is allowed. We assume that the agent makes his investment and consumption decisions based on a power utility function.

Considering a simple portfolio, composed by a bond and a single stock on a market modelled by the $\alpha$-Hypergeometric Stochastic volatility model, we will derive the correspondent Hamilton-Jacobi-Bellman equation and discuss the existence of a classical solution and the techniques used to find such solution.

References:

[1] Merton R.C. “Lifetime Portfolio Selection under Uncertainty: the Continuous-Time Case,” The Review of Economics and Statistics, 51, No. 3, 247–257 (1969).

We discuss the hydrodynamic behavior of the long jumps symmetric exclusion process with a slow barrier. When jumps occur between a negative site and a non-negative site, the rates are slowed down by a factor of $\alpha n^{-\beta}$, where $\alpha >0$ and $\beta \geq 0$. The jump rates are given by a symmetric transition probability $p(\cdot)$. In [4], we study the case where $p(\cdot)$ has finite variance and obtain the heat equation with various boundary conditions, analogously to [6]. On other hand, in [5] we study the case where $p(\cdot)$ has infinite variance and obtain diverse partial differential equations given in terms of the fractional Laplacian and the regional fractional Laplacian, with different boundary conditions, similarly to [1] and [2]. Finally, we present how we can derive a fractional porous media equation from the symmetric exclusion process, as it is described in [3].

References:

[1] Bernardin C., Cardoso P., Gonçalves P., Scotta S., “Hydrodynamic limit for a boundary driven super-diffusive symmetric exclusion,” arXiv preprint arXiv:2007.01621, (2021).

[2] Bernardin C., Gonçalves P., Jiménez-Oviedo B., “A microscopic model for a one parameter class of fractional Laplacians with Dirichlet boundary conditions,” Archive for Rational Mechanics and Analysis, 239, No. 1, 1–48 (2021).

[3] Cardoso P., de Paula R., Gonçalves P., “Derivation of the fractional porous media equation from a microscopic dynamics,” In preparation, (2022).

[4] Cardoso P., Gonçalves P., Jiménez-Oviedo B., “Hydrodynamic be- havior of long-range symmetric exclusion with a slow barrier: diffusive regime,” arXiv preprint arXiv:2111.02868, (2021).

[5] Cardoso P., Gonçalves P., Jiménez-Oviedo B., “Hydrodynamic be- havior of long-range symmetric exclusion with a slow barrier: superdif- fusive regime,” arXiv preprint arXiv:2201.10540, (2022).

[6] Franco T., Gonçalves P., Neumann A., “Phase transition of a heat equation with Robin’s boundary conditions and exclusion process,” Transactions of the American Mathematical Society, 367, No. 9, 6131– 6158 (2015).

In this talk I will present some results concerning the evolution of a polarization, in the geometric quantization context, under the imaginary-time Hamiltonian flow of a specific family of Hamiltonians. I will first begin with the rudiments of toric geometry and geometric quantization and the work in [1] on imaginary-time flows, aiming for an accessible presentation.

References:

[1] José M. Mourão, João P. Nunes. On Complexified Analytic Hamiltonian Flows and Geodesics on the Space of Kaehler Metrics, International Mathematics Research Notices, Volume 2015, Issue 20, 2015, Pages 10624 - 10656.

Logical matrices are arguably the most common algebraic semantical structures used for propositional logics. After Lukasiewicz, a logical matrix consists of an underlying algebra, functionally interpreting logical connectives over a set of truth-values, together with a subset of designated truth-values. In 2005, Avron and Lev [1] introduced non-deterministic matrices (Nmatrices), a generalization of logical matrices where the connectives are interpreted, not as functions, but as multi-functions.

This notion was further generalized by Baaz et al. [2] by also allowing for partiality (PNmatrices). Since their introduction, these generalized structures have been gaining traction amongst logicians, as the added expressiveness allows for finite characterizations of a much wider class of logics and general recipes for various problems in logic, such as procedures to constructively update semantics when imposing new axioms [4,5], or effectively combining semantics for two logics, capturing the effect of joining their axiomatizations [3,6,7].

However, several problems whose computational and complexity status is well studied for traditional matrices, have not yet been studied in the wider context of PNmatrices. In this talk, we review the basic notions of algebraic semantics, introduce partial non-deterministic matrices and present our recent results regarding the computational and complexity status of some of these problems.

References:

[1] A. Avron and I. Lev. Non-deterministic multiple-valued structures. Journal of Logic and Computation, 15(3):241–261, 2005.

[2] M. Baaz, O. Lahav, and A. Zamansky. Finite-valued semantics for canonical labelled calculi. Journal of Automated Reasoning, 51(4):401– 430, 2013.

[3] C. Caleiro and S. Marcelino. Modular semantics for combined many- valued logics, 2021. Submitted.

[4] C. Caleiro and S. Marcelino. On axioms and rexpansions. In O. Arieli and A. Zamansky, editors, Arnon Avron on Semantics and Proof The- ory of Non-Classical Logics, volume 21 of Outstanding Contributions to Logic. Springer, Cham, 2021. doi:https://doi.org/10.1007/ 978-3-030-71258-7_3.

[5] A. Ciabattoni, O. Lahav, L. Spendier, and A. Zamansky. Taming para- consistent (and other) logics: An algorithmic approach. ACM Transactions on Computational Logic, 16(1):5:1–5:16, 2014.

[6] S. Marcelino. An unexpected boolean connective. Logica Universalis, 2021. doi:10.1007/s11787-021-00280-7.

[7] S. Marcelino and C. Caleiro. Disjoint fibring of non-deterministic ma- trices. In J. Kennedy and R. de Queiroz, editors, Logic, Language, In- formation, and Computation (WoLLIC 2017), volume 10388 of LNCS, pages 242–255. Springer-Verlag, 2017.

In this talk we will explore the resurgent properties of the inhomogeneous and "q-deformed'' Painlevé II equations. Both of these equations appear in the double-scaling limit of some matrix models and, in particular, the "q-deformed'' PII describes 2d super-gravity in some special background. After a brief explanation on how resurgent transseries allow us to construct non-perturbative solutions to these equations, we will give some explicit examples of Stokes Data and how they allow us to construct solutions on the whole complex plane. In particular, we will show the computation for the well-known Hastings-McLeod solution and the first one of the special function solutions. We will finish by mentioning the connection between these two equations via the Miura map.

In the talk I will present the objectives and results of the two projects for my PhD. The first project constitutes the semi-classical decoding of Minimal String Theory using a generalization of Hermitian Matrix Models to Super Matrix Models. The second project revolves around understanding the Minimal String Theory - Gauge Theory Correspondence in a new way: As different expansions of one underlying Partition Function. I will start out by explaining the semi-classical decoding of (2,3) Minimal String Theory. In this context it is only known how to produce half of the non-perturbative contributions predicted by resurgence from the Minimal String Theory - or its associated Hermitian Matrix Model. Here I will quickly review Hermitian Matrix Models and then explain the super generalization. I will outline how the Super Matrix Model is capable of predicting the whole non perturbative content of Minimal String Theories. Having established how Minimal String Theories interplay with resurgent transseries expansions I will move on to understanding the Minimal String Theory - Gauge Theory Correspondence. While this correspondence is usually established via integrability methods one of my objective is to understand such a duality in a new way: By identifying the above theories as different expansions of the $\tau$ function associated to Painlevé I. We denote those expansions by rectangular Framing and Diagonal Framing. While Rectangular Framing corresponds exactly to the semi-classical decoding of Minimal String Theory discussed above, Diagonal Framing corresponds to gauge theoretic considerations. To underline the above discussion I will then present the explicit map from Rectangular to Diagonal Framing which makes the Gauge Theory Partition Function appear. To finish the talk I will present future endeavors and open problems.

Starting from a generalization of the good-bad-ugly model, we showed that the solutions to the Einstein field equations in generalized harmonic gauge (GHG) admit polyhomogeneous expansions near null infinity. This allows us to find out, under a very general class of initial data, whether the peeling property holds for each gauge choice. In general, we find that it does not. However we find that the interplay between gauge and constraint addition can be exploited in order to recover peeling. The same method can be used to build a regularization of the Einstein field equations in GHG.

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.

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.

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

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 $\overline{\mathfrak{M}}_{K^2,\chi}$.