# Tubim shkencor: Gjuha e simetrive - Një këndvështrim nga teoria e përfaqësimit dhe gjeometria

AKADEMIA E SHKENCAVE DHE E ARTEVE E KOSOVËS
Seksioni i Shkencave të Natyrës

TUBIM SHKENCOR

GJUHA E SIMETRIVE - NJË KËNDVËSHTRIM NGA TEORIA E PËRFAQËSIMIT DHE GJEOMETRIA
(Anglisht: The Language of Symmetries - A Viewpoint from Representation Theory and Geometry)

11 korrik 2014, Prishtinë

Simetritë janë të pranishme në natyrë dhe ato shfaqen edhe në studimet shkencore në fizikë, kimi dhe matematikë, ndër shumë lëmenj të tjerë. Në këtë Takim shkencor do të diskutohen aspekte të ndryshme të gjuhës së simetrive nga perspektiva e Teorisë së përfaqësimit dhe nga Gjeometria.

REFERUESIT
Dr. Ethan Cotterill, Universiteti Federal Fluminese, Brazil
Dr. Harald Grobner, Universiteti i Vjenës, Austri
Dr. Masoud Kamgarpour, Universiteti i Kuinslendit, Australi
Robert Kremser, Universiteti i Mynihut, Gjermani
Dr. Marie-Amelie Lawn, Universiteti i Teksasit, Austin, SHBA
Dr. Justin Noel, Universiteti i Regensburgut. Gjermani
Dr. Travis Schedler, Universiteti i Teksasit, Austin, SHBA

KËSHILLI ORGANIZATIV
Dr. Qamil Haxhibeqiri (ASHAK)
Dr. Masoud Kamgarpour (Universiteti i Kuinslandit)
Dr. Qëndrim Gashi (Universiteti i Prishtinës)

PJESËMARRJA
Këshilli organizativ i fton të interesuarit për pjesëmarrje në këtë ngjarje të regjistrohen përmes dërgimit të një emaili tek qamil.haxhibeqiri@uni-pr.edu ose qendrim.gashi@uni-pr.edu. Afati i fundit për regjistrim është data 20 qershor 2014. Regjistrimi dhe pjesëmarrja janë falas.

Takimi shkencor mbahet më 11 korrik 2014 në lokalet e Akademisë së Shkencave dhe Arteve të Kosovës, në Prishtinë, rruga “Agim Ramadani”, p. n., duke filluar në orën 9:00.

PROGRAMI
Programi i detajuar do të publikohet më vonë.

Më poshtë mund t'i gjeni titujt dhe abstraktet e prezantimeve (në anglisht):

Dr. Ethan Cotterill, Universiteti Federal i Minas Gerais, Brazil
Dimension counts for singular rational curves
Abstract: Rational curves are essential tools for classifying algebraic varieties. Establishing dimension bounds for families of embedded rational curves that admit singularities of a particular type arises arises naturally as part of this classification. Singularities, in turn, are classified by their value semigroups. Unibranch singularities are associated to numerical semigroups, i.e. sub-semigroups of the natural numbers. These fit naturally into a tree, and each is associated with a particular weight, from which a bound on the dimension of the corresponding stratum in the Grassmannian may be derived. Understanding how weights grow as a function of (arithmetic) genus, i.e. within the tree, is thus fundamental. We establish that for genus  , the dimension of unibranch singularities is as one would naively expect, but that expectations fail as soon as  . Multibranch singularities are far more complicated; in this case, we give a general classification strategy and again show, using semigroups, that dimension grows as expected relative to   when  . This is joint work with Lia Fusaro Abrantes and Renato Vidal Martins.

Dr. Harald Grobner, Universiteti i Vjenës, Austri
Rationality of special values of L-functions
Abstract: We will show some rationality results for special values of automorphic L-functions. These L-functions are analytic invariants of cuspidal automorphic representations, which encode important arithmetic information. Using geometrically defined periods, we will single out the transcendental part of these L-values, which is very well in the spirit of conjectures of Deligne, Gross and Beilinson.

Dr. Masoud Kamgarpour, Universiteti i Kuinslendit, Australi
Symmetries in mathematics: a personal perspective
Abstract: Studying symmetries is one of the most powerful tools available to scientists for making sense of the world. In this talk, I will start by explaining how symmetries of geometric objects give rise to familiar matrix groups. Following suggestions of physicists, we will replace points of our spaces with strings or loops, thus arriving at loop spaces and loop groups. I will then explain some of the methods available for understanding the structure of loop groups and their representations. We will then switch gear and consider symmetries of certain differential equations. Here, too, we will see the loop group intervening. Finally, I will explain an intriguing conjectural relationship, suggested by the Langlands' philosophy, between differential equations and representations of loop groups.

Robert Kremser, Universiteti i Mynihut, Gjermani
Convex rank one subsets in symmetric spaces
Abstract: We will give some rigidity results concerning the boundary of convex subsets in symmetric spaces of higher rank. The subsets in question are closed, convex and with nontrivial discrete boundary with respect to the Tits topology - hence convex rank one subsets.
We address the question: Does a convex rank one subset share the boundary with a convex subset of a rank one symmetric subspace?
Nontrivial convex rank one subsets, which are additionally invariant under an ($\mathbb{R}$)-Zariski dense subgroup of isometries of its ambient irreducible symmetric space, can only exist in rank 1 (Kleiner, Leeb). This result motivates the above question in the case of no additional data.

Dr. Marie-Amelie Lawn, Universiteti i Teksasit, Austin, SHBA
Immersions into Robertson-Walker spacetimes
Abstract: Robertson-Walker spacetimes are special cases of pseudo-Riemannian warped products which arise as exact solutions of the Einstein field equations in general relativity. We will consider hypersurfaces of such spaces and give a necessary and sufficient condition for a Riemannian manifold to locally isometrically immerse into them. If we have more time, we will speak about generalized associated families of surface immersions into 3D Robertson-Walker and homogeneous spaces.

Dr. Justin Noel, Universiteti i Regensburgut, Gjermani
On a nilpotence conjecture of J. P. May
Abstract:  In 1986 Peter May made the following conjecture: Suppose that $R$ is a ring spectrum with power operations (e.g., a commutative $S$-algebra). Then the elements in the kernel of the integral Hurewicz homomorphism $\pi_* R \rightarrow H_* (R;Z)$ are nilpotent.  The proof is short, simple and only uses results that have been around since the late 90's. As corollaries we obtain nilpotence results in various cobordism rings including $\Omega^{Spin}_*$ and $\Omega^{String}_*$, the non-existence of commutative $S$-algebra structures on certain complex oriented ring spectra, and a partial analogue of Quillen's $\mathcal{F}$-isomorphism theorem for Lubin--Tate theories. This project is joint with Akhil Mathew and Niko Naumann.

Dr. Travis Schedler, Universiteti i Teksasit, Austin, SHBA
The Springer Correspondence and Poisson Homology
Abstract: The Springer Correspondence relates irreducible representations of the Weyl group of a semisimple complex Lie algebra to the geometry of the cone of nilpotent elements of the Lie algebra. The zeroth Poisson homology of a variety is the quotient of all functions by those spanned by Poisson brackets of functions.  I will explain a conjecture with Proudfoot that assigns a grading to the irreducible representations of the Weyl group via the Poisson homology of the nilpotent cone.  This conjecture is a kind of symplectic duality between this nilpotent cone and that of the Langlands dual. An analogous statement for hypertoric varieties is a theorem, which relates a hypertoric variety with its Gale dual, and assigns a second grading to its de Rham cohomology, which turns out to coincide with a different grading of Denham using the combinatorial Laplacian.