LisMath Seminar 

In this seminar, I would like to present the paper of Yael Karshon on Periodic Hamiltonian flows on four dimensional manifolds. We will explore the classification of periodic Hamiltonian flows on compact symplectic 4-manifolds through the use of labelled graphs and show that two such spaces are isomorphic if and only if they have the same graph. Moreover, if time allows we will also see that all these spaces are Kähler manifolds.

Bibliography:

[1] Y. Karshon, Periodic Hamiltonian flows on four dimensional manifolds,
https://arxiv.org/abs/dg-ga/9510004

[2] Y. Karshon, Hamiltonian actions of Lie groups, Ph.D. thesis, Harvard University, April 1993.

[3] A. Cannas da Silva, Lectures on Symplectic Geometry, Springer-Verlag Berlin Heidelberg, 2008.

[4] T. Broecker and K. Janich, Introduction to differential topology, Cambridge University Press, 1982.

[5] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, 2nd edition, Oxford Univ. Press, 1998.

A germ of a singular surface $S$ is QO if there is a finite projection $p:S\to \mathbb C^2$ such that its discriminant is normal crossings. A QO surface admits a very special parametrization, that is a natural generalization of the Puiseux parametrization of a plane curve. The purpose of this seminar is to find criteria to determine if a surface $S$ with a parametrization $(u,v) \mapsto (x(u,v),y(u,v),z(u,v))$ is QO. For instance, if the semigroup of the surface $S$ is the semigroup of a QO surface, is the surface $S$ QO?

Bibliography:

[1] Gonzalez Perez, The semigroup of a quasi-ordinary hypersurface, http://www.mat.ucm.es/~pdperezg/semi4

[2] Gonzalez Perez, Quasi-ordinary Singularities via toric Geometry, http://www.mat.ucm.es/~pdperezg/PhD-Es-gonzalez.pdf

[3] Patrick Popescu-Pampu, On the analytic invariance of the semigroup of a QO hyperface singularity, http://math.univ-lille1.fr/~popescu/04-Duke.pdf

Quasisymmetric functions are a generalization of symmetric functions, i.e., functions that are invariant under a certain action of the symmetric group. Like the latter, quasi symmetric functions form a graded algebra, over a commutative ring. The aim of this seminar is to introduce the basic notions on these functions, as well as some bases for this algebra. Within the basie, we highlight the quasisymmetric Schur functions, a recent and natural refinement of the classic Schur functions.

Bibliography:

[1] J. Haglund, K. Luoto, S. Mason, S. van Willigenburg, 'Quasisymmetric Schur functions', Journal of Combinatorial Theory, Series A, 118 (2) (2011), 463-490.

[2] K. Luoto, S. Mykytiuk, S. van Willigenburg, 'An Introduction to Quasisymmetric Schur Functions', Springer, 2013.

The aim of this seminar is to describe the behaviour of the canonical and pluricanonical maps of algebraic curves, and to explain the structure of the so-called canonical ring of curves.

Bibliography:

[1] B. Saint-Donat, On Petri's Analysis of the Linear System of Quadrics through a Canonical Curve, Mathematische Annalen 206 (1973): 157-176.

[2] E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of Algebraic Curves - Volume I, Springer-Verlag.

We present a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable, and we describe how Khovanov homology can emerge upon adding a fifth dimension.

Bibliography:

[1] Edward Witten, Two lectures on the Jones Polynomial and Khovanov Homology

The Cheeger constant of a manifold (or domain), first introduced about 50 years ago, is a quantitative measure of how easily the manifold can be partitioned into two pieces. It is also closely related to the first non-trivial eigenvalue of the Laplace-Beltrami operator on the manifold and thus links partitioning problems with spectral geometry. The proposed seminar talk explores some of these connections, as well as contemporary research results giving generalisations to graphs, higher-order partitions and eigenvalues.

Bibliography:

[1] P. Buser, Ann. Sci. École Norm. Sup. (4) 15 (1982), 213-230.

[2] J. Cheeger, Problems in Analysis, Princeton Univ. Press (1970), 195-199.

[3] B. Kawohl and V. Fridman, Comm. Math. Univ. Carol. 44 (2003), 659-667.

[4] C. Lange, S. Liu, N. Peyerimhoff and O. Post, Calc. Var. PDE 54 (2015), 4165-4196.

[5] J. Lee, S. Oveis Gharan and L. Trevisan, Proc. 2012 ACM Symposium on Theory of Computing, ACM, NY (2012), 1117-1130.

[6] L. Miclo, Invent. Math. 200 (2015), 311-343.

First we are going to give a brief introduction to syntax, semantics and axiomatization in modal logic and in temporal logic. Then, making use of the mosaic method, we are going to prove complexity and Hilbert completeness results in modal logic and in temporal logic over linear flows of time. 

Bibliography

1. Blackburn, P., de Rijke, M., Venema, Y. Modal Logic. Cambridge University Press, 2001. 
2. M. Marx, S. Mikulas, and M. Reynolds. The mosaic method for temporal logics. In TABLEAUX, 2000.
3. S. Mikulas. Taming first-order logic. Journal of the IGPL, 6(2):305-316, 1998. 

There is an interesting technique known as the Monte Carlo method that can be used to solve many types of problems [1]. Monte Carlo method is a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. It is mainly used in three distinct problem classes [2]: optimization, numerical integration, and generating draws from a probability distribution. We can use Monte Carlo algorithm for finding the solution of Dirichlet partial differential equations (DPDE) [3, 4] and to find the solution of Schro ̈dinger equation in quantum mechanics, Brownian motion [5, 6] as well as statistical mechanics. Since the main objective of statistical mechanics is to show how the properties of matter (macroscopic properties such as pressure, heat capacity, entropy, etc [7,8]) can be calculated from the properties of individual molecules (positions, molecular geometry and intermolecular forces, etc), we can use the statistical methods (such as Monte Carlo method, Green’s function [9,10]) to understand the relation between the behavior of atoms, energy and the treatment of huge numbers of distinguishable and indistinguishable particles, for example the particles in crystals net, the particles of gas (different in the particles position). Here, we present a brief overview of the Monte Carlo method and then illustrate its use in PDE and integration. 

Bibliography

1. Stanley, J. Farlow, Partial differential equations for Scientists and Engineers, Dover Publications, INC. NewYork (1982).
2. Kroese, D. P., Brereton, T., Taimre, T. and Botev, Z. I., Why the Monte Carlo method is so important today. WIREs Computational Statistics 386-392 (2014). DOI:10.1002/wics.1314.
3. Vajargah, B. F. and Vajargah, K. F., Monte Carlo Method for Finding the Solution of Dirichlet Partial Differential Equations, Applied Mathematical Sciences, 453-462 (2007).
4. Talay, D., Monte Carlo Methods for PDE’s. In Encyclopaedia of Mathematics, M. Hazewinkel (Ed.). Kluwer Academic Press (1997).
5. Jacod, J.; Lejay, A. and Talay, D., Estimation of the Brownian dimension of a continuous Ito process, Bernoulli, 469-498 (2008) DOI: 10.3150/07-BEJ6190.
6. Maire, S. and Talay, D., On a Monte Carlo method for neutron transport criticality computations, IMA Journal Numerical Analysis, 657-685 (2006).
7. Kraeft, W. D. and Bonitz, M., Thermodynamics of a Correlated Confined Plasma, Journal of Physics: Conference Series, 78 (2006).
8. Reed, T. M. and Keith, E.Gubbins, Applied statistical mechanics (1991).
9. Hussein, N. A., Eisa, D.A., Osman, A.-N. A. and Abbas, R. A., Quantum Binary and Triplet Distribution Functions of Plasma by using Green’s Function Contrib. Plasma Physics, 815 - 826 (2014) / DOI 10.1002/ctpp.201400016.
10. Hussein, N. A., Osman, A.-N. A., Eisa, D. A. and Abbas, R. A., The quantum thermodynamic functions of plasma in terms of the Green’s function, Natural Science, 71-80 (2014) http://dx.doi.org/10.4236/ns.2014.62011. 

In 1962, I.M. Gelfand and M.I. Graev constructed explicitly a character for $\operatorname{SL}(n,q$) and showed that it is multiplicity free [1]. In 1967, R. Steinberg generalised this construction for certain Chevalley-Dickson groups [2]. An even more general construction holds in the setting of finite groups of Lie type. In this talk, we define the Gelfand-Graev character for $\operatorname{GL}(n,q)$, and adapt the multiplicity free proof as given in [3]. For this, we make a quick introduction to some of the important tools needed from representation theory of associative algebras and finite groups. 

Bibliography

1. I. M. Gelfand, M. I. Graev, Construction of irreducible representations of simple algebraic groups over a finite field, Dokl. Akad. Nauk SSSR, 147 (1962).
2. R. Steinberg, Lectures on Chevalley Groups, Yale University, 1967.
3. R. W. Carter, Finite groups of Lie type: conjugacy classes and complex characters, Wiley Interscience, 1993. 

The first part of this talk will be an overview of some of the basic aspects of the theory of random matrices. This will include examples of the most common matrix ensembles, which are spaces of matrices whose entries are random variables. Focusing on an example of major importance, the Gaussian Unitary Ensemble, we will show how the probability distribution on these matrices is closely related to another probability distribution on their eigenvalues. We will also explore the relationship between random matrix theory and the theory of orthogonal polynomials. In the second part of the talk, to showcase the impact of random matrix theory on other fields, we will introduce Toeplitz determinants and discuss the Szeg Limit Theorem. If time allows it, we will comment some more general results in this direction.

Bibliography

1. P. Deift and D. Gioev, Random Matrix Theory: Invariant Ensembles and Universality, Courant Lecture Notes in Mathematics, 18 (2009). 
2. E. Basor, Toeplitz determinants, Fisher-Hartwig symbols, and random matrices, in Recent Perspectives in Random Matrix Theory and Number Theory, 309-336, Cambridge University Press (2005). 
3. P. Deift, A. Its and I. Krasovsky, Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results, Comm. Pure Appl. Math., 66, 13601438 (2013) [arXiv:1207.4990v3 [math.FA]]
4. D. Bump and P. Diaconis, Toeplitz Minors, J. Combin. Theory Ser. A, 97(2), 252-271 (2002).

Both finite (ramified) coverings of algebraic varieties  and quotients of algebraic varieties by finite group actions  are  important sources of examples in algebraic geometry. In this seminar these notions will be discussed with special emphasis on the case of Abelian coverings and Galois coverings. In addition some examples will be presented.

Bibliography

1. H. Cartan, Quotient d'un espace analytique par un groupe d'automorphismes, A symposium in honor of S. Lefschetz, Algebraic geometry and topology. pp. 90-102. Princeton University Press, Princeton, N. J. 1957.
2. F. Catanese, Singular bidouble covers and the construction of interesting algebraic surfaces. Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 97?120, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, 1999.
3. R. Hartshorne, Algebraic Geometry, G.T.M. 52, Springer-Verlag, New York, (1977).
4. M. Namba, On finite Galois coverings of projective manifolds,  J. Math. Soc. Japan, vol. 41, nº 3 1989.
5. R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math., 417 (1991), 191-213.

In the study of quantum field theory naturally arises the necessity to deal with infinite-dimensional oscillatory integrals. The purpose of the seminar is to show how the procedure of analytic continuation can make sense of such integrals in some particular cases. We will start with an example of an oscillatory integral in finite dimension given by the Airy function and try to adapt the same strategy to the infinite dimensional case of Chern-Simons action. If time allows it, we will also give an overview of the implications of this idea in physics and knot theory.

Bibliography

1. E. Witten, Analytic continuation of Chern-Simons theory, Chern-Simons gauge theory 20 (2011): 347-446, arXiv:1001.2933.
2. H. Murakami, An introduction to the volume conjecture and its generalizations, arXiv:0802.0039.
3. M. V. Berry and C. J. Howls, Hyperasymptotics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences,  Vol. 430. No. 1880. The Royal Society, 1990.
4. M. V. Berry and C. J. Howls, Hyperasymptotics for integrals with saddles, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 434. No. 1892. The Royal Society, 1991.
5. C. J. Howls, Hyperasymptotics for multidimensional integrals, exact remainder terms and the global connection problem, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences,  Vol. 453. No. 1966. The Royal Society, 1997.

In virtue of the ergodic method, the asymptotic representation theory allows for the study of characters of groups defined as inductive limits of finite groups, and stands on the knowledge of the irreducible characters of these finite groups. Our aim is to recover a similar method by replacing irreducible characters for supercharacters of the finite unitriangular groups, and this should be understood as a first step to extend the method to other infinite groups.

Bibliography

1. A. M. Vershik, S.V. Kerov, Asymptotic theory of characters of the symmetric group,  Functional Analysis and Its Applications, October 1981, Volume 15, Issue 4, pp 246-255.
2. P. Diaconis, I. M. Isaacs, Supercharacters and superclasses for algebra groups, Trans. AMS, 360 (5), 2359-2392 (2008).

There are nice ways of regarding the representation theory of the symmetric groups, such as the branching graph. This theory has some elements in common with the representation theory of the rook monoid, which have been studied more recently.

Bibliography

1. Kleshchev, Linear and Projective Representations of Symmetric Groups, Cambridge Tracts in Mathematics 163 (2005).
2. J A Green, Polynomial Representations of GLn, Springer Lecture Notes in Mathematics, 2000.
3. Okounkov, Vershik, A New Approach to Representation Theory of Symmetric Groups, Selecta Mathematica, New series, vol. 2 n. 4, 1996.
4. L. Solomon, Representations of the rook monoid, Journal of Algebra, 2002.
5. O. Ganyushkin, V. Mazorchuk, B. Steinberg, On the irreducible representations of a finite semigroup, Proc. Amer. Math. Soc., v 137, 2009.

In 1966, V.I. Arnold [1] showed that the Lagrangian flows corresponding to the Euler equations can be characterised as geodesics in the group of diffeomorphisms of the underlying manifold when considered with the $L^2$ metric. One can consider different metrics in this infinite-dimensional group (or in other groups) and derive many other conservative systems as the corresponding geodesics.

For the dissipative systems, replacing geodesics by irregular (stochastic) paths, it is possible to consider stochastic variational principles whose critical points are the Lagrangian trajectories of the systems. The velocity corresponds in these contexts to the drift of the stochastic paths. In [2, 3, 4], stochastic variational principle for Navier-Stokes equations, Camassa-Holm equations and Leray-alpha equations have been described.

In the first part of the seminar, we will give a stochastic variational principle for one-dimensional viscous Burgers equations considering a $L^q$ metric. After that, we will prove the existence of the critical diffusion. Finally, we will give an alternative probabilistic derivation of Burgers equations via a stochastic forward-backward differential systems. In [5,6], the Navier-Stokes equation was studied using this relation.

Bibliography

1. V. I. Arnold, Sur la géometrie differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits.  Ann. Inst. Fourier., 16, 316-361 (1966)
2. F. Cipriano, A. B. Cruzeiro, Navier-Stokes equation and diffusions on the group of homeomorphisms of the torus. Comm. Math. Phys., 275, no.1, 255-269 (2007).
3. M. Arnaudon, X. Chen, A. B. Cruzeiro, Stochastic Euler-Poincaré reduction. J. Math. Physics., 55, 081507 (2014).
4. A. B. Cruzeiro, G. P. Liu, A stochastic variational approach to the viscous Camassa-Holm and related equations. arxiv:1508.04064.
5. A. B. Cruzeiro, E. Shamarova, Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of the torus. Stoch. Proc. and their Applic., 119, 4034-4060(2009)
6. A. B. Cruzeiro, Z. M. Qian, Backward Stochastic Differential Equations Associated with the Vorticity Equations. J. Funct. Anal., 267, no.3, 660--677(2014)

Bibliography

1. M. Boyarchenko, Character sheaves and characters of unipotent groups over finite fields, American Journal of Mathematics (to appear)
2. M. Boyarchenko and V. Drinfeld, Character sheaves on unipotent groups in positive characteristic: foundations,  arXiv:0810.0794v2

Groupoids were introduced by Brandt in his 1926 paper [1] and since then they have been used in a wide variety of areas of mathematics, from ergodic theory and functional analysis to homotopy theory, algebraic geometry, differential geometry, differential topology and group theory. Specially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways: A groupoid can be seen as a group with a partial function replacing the binary operation or a category in which every morphism is invertible. In this talk, going through the notion of symmetry, I would like to justify the argument that the theory of groupoids does not differ widely in spirit and aims from the theory of groups and how groupoids describe symmetry. Of course, I will give the basic definitions and important examples in a wide range.

Bibliography:

[1] H. Brandt, Ueber eine Verallgemeinerung des Gruppenbegriffes, Math. Ann. 96, 360- 366 (1926).

[2] R. Brown, From Groups to Groupoids: A Brief Survey, Bull. London Math. Soc. 19, 113-134 (1987).

[3]A. Weinstein, Groupoids: Unifying Internal and External Symmetry, Notices Amer. Math. Soc. 43 (1996).

Bibliography

1. M. Aguiar et al. "Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras". Advances in Mathematics 229(4), (2012), 2310-2337
2. C. André, "Supercharacters of unitriangular groups and set partitions combinatorics" ECOS2013, 2da Escuela Puntana de Combinatoria, Universidad Nacional de San Luis, Argentina, July 22-August 2, 2013.