Mart Abel.
"On equivalences between some categories of sheaves, bundles and
presheaves"
The
talk will be about some results obtained in the papers by Mart Abel
on equivalences of categories. These equivalences were used to
generalize the Serre-Swan-Mallios Theorem but could be useful also
for people dealing with sheaves, bundles or presheaves in geometry
or in physics.
Erik
Backelin
Singular localization of quantum groups (relation with modular
represenation theory)
Rafael Diaz
"Categorifications of the Weyl algebras"
We
introduce several categorifications of the Weyl algebras paying
careful attention to the fact that the Weyl algebras, as universal
enveloping algebras of the Heisenberg Lie algebras, are actually
Hopf algebras. We study representations of the categorified Weyl
algebras and solve the categorified Schrödinger and Heisengber
equations. We also consider quantum inverses, quantum monoids and
quantum operads.
References:
[1]
http://arxiv.org/PS_cache/math/pdf/0509/0509674v3.pdf
[2]
http://www.ieja.net/papers/2009/V5/4-V5-2009.pdf
[3]
http://arxiv.org/PS_cache/math/pdf/0606/0606041v6.pdf
Cesar Galindo
"Module categories over graded tensor categories"
A
graded tensor category over a group G will be called a strongly
G-graded tensor category if every homogeneous component has at least
one multiplicatively invertible object. In this talk we shall
describe the simple module categories over a strongly G-graded
tensor category and faithfully G-graded fusion categories as induced
from module categories over tensor subcategories associated with the
subgroups of G.
Pinhas Grossman
"Quantum subgroups of the Haagerup fusion categories"
We
answer three related questions concerning the Haagerup subfactor and
its even parts, the Haagerup fusion categories. Namely we find all
simple module categories over each of the Haagerup fusion categories
(in other words, we find the `"quantum subgroups" in the sense of
Ocneanu), we find all subfactors whose principal even part is one of
the Haagerup fusion categories, and we compute the Brauer-Picard
groupoid of Morita equivalences of the Haagerup fusion categories.
In addition to the two even parts of the Haagerup subfactor, there
is exactly one more fusion category which is Morita equivalent to
each of them. This third fusion category has six simple objects and
the same fusion rules as one of the even parts of the Haagerup
subfactor, but has not previously appeared in the literature. We
also find the full lattice of intermediate subfactors for every
subfactor whose even part is one of these three fusion categories,
and we discuss how our results generalize to Izumi subfactors.
Reference:
http://arxiv.org/abs/1102.2631
Javier Lopez
Geometry over the field with one element
The
interpretation of certain limit phenomena in the theory of algebraic
groups as behaviour over a ``field with one element'' was proposed
initially in 1956 by Jacques Tits. In the nineties, Yuri Manin
proposed that the spectrum Spec F_1 of such a conjectural field
should take the role of the ``absolute point'', and the affine line
over F_1 the role of the absolute Tate motive appearing in the
theory of motives proposed by Deninger and Kurokawa as a natural
framework in which Weil's proof of the Riemann Hypothesis for
function fields might be translated to the complex case.
Christophe Soul\'e proposed in 2004 the first model of a geometry
over the field with one element producing (up to a constant factor)
the zeta functions that Manin predicted. In the last years a number
of different approaches have arisen, including our own approach
through torified varieties, that contains most of the interesting
examples and provides a more geometrical formalization of the ideas
developed by Alain Connes and Katia Consani in 2009.
As of today, we still lack a complete theory that satisfies all the
desirable properties. In this series of talks we will present the
history and motivation for an extension of algebraic geometry
happening "under the spectrum of Z", give a comparison of the main
approaches attempted up to date and present some results concerning
combinatorial properties of varieties over F_1.
References:
J. López Peña and O Lorscheid: "Mapping F_1-land: An overview of
geometries over the field with one element" arXiv:0909.0069
Joaquin Luna-Torres
"Basic properties of interior operators in categories"
For a
topological space it is well-known that the associated interior
operators provides equivalent descriptions of the topology; but it
is not generally true in other categories. The notion of interior
operator in a general categorical setting was introduced in [9]. In
this paper we study some formal properties of this notion: open and
codense subobjects, idempotent,cohereditary, productive interior
operators, and composition of interior operators.
References:
[1] Jiri
Adamek, Horst Herrlich, George Strecker, Abstract and Concrete
Categories, John Wiley & Sons, New York, 1990.
[2] G. Birkhoff, Lattice Theory, American Mathematical Society,
Providence, 1940.
[3] N. Bourbaki, General Topology, Addison-Wesley Publishing,
Massachusetts, 1966.
[4] G. Castellini, Categorical Closure Operators, Birkhauser,
Boston /Basel /Berlin, 2003.
[5] D. Dikrajan, W. Tholen, Categorical Structures of closure
operators, Kluwer Aca- demic Publisher, Dordrecht / Boston / London,
1995.
[6] sc J. Dugundji, Topology, Allyn and Bacon, Inc., Boston / London
/ Sydney / Toronto, 1966.
[7] U. Hohle, A. Sostak, Fixed-Basis Fuzzy Topologies In:
Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory,
Kluwer Academic Publisher, Boston, 1999.
[8] P. T. Johnstone, Stone spaces, Cambridge University Press,
Cambridge, 1982.
[9] J. Luna-Torres, C. O. Ochoa, Interior Operators and Topological
Categories, arXiv:1010.4460v1 [math.CT], 2010.
[10] S. MacLane, Categories for the Working Mathematician,
Springer-Verlag, New York / Heidelberg / Berlin,1971.
[11] S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic, A
rst introduction to Topos theory, Springer-Verlag, New York /
Heidelberg / Berlin,1992.
[12] J. R. Montan~ez, Funtores elevadores y coelevadores de
estructuras, Tesis de Doc- torado, Universidad Nacional de Colombia,
2007.
[13] K. Kuratowski, Topology, Vol. 1,Vol 2, Academic Press, New York
and London, 1968.
Martin Mombelli
"Classification
of representations of finite pointed tensor categories"
In
this talk I will present some recent advances in the classification
of representations of certain class of finite pointed tensor
categories.
A
pointed tensor category is a tensor category such that all simple
objects are invertible, and thus they form a group with
multiplication given by the tensor product. For any
finite-dimensional pointed Hopf algebra H the category Rep(H*) is
pointed.
I
will recall the technique presented in [Mo1] to classify module
categories over Rep(H) when H is a finite-dimensional pointed Hopf
algebra. This technique was used in [Mo2] to classify module
categories over any Hopf algebra constructed from a quantum linear
space. Finally, I will talk about a recent work with C. Galindo on
the classification of module categories over certain class of
pointed tensor categories that are not the representation
category of Hopf algebras.
References:
[AM]
N. Andruskiewitsch and M. Mombelli, On module categories over finite-
dimensional Hopf algebras, J. Algebra 314 (2007), 383{418.
[EO]
P. Etingof and V. Ostrik, Finite tensor categories, Mosc. Math. J. 4
(2004), no. 3, 627-654.
[GM]
C. Galindo and M. Mombelli, Module categories over finite pointed
tensor categories, preprint arxiv:1102.4882.
[Mo1]
M. Mombelli, Module categories over pointed Hopf algebras, Math. Z.
266 (2010) 319-344.
[Mo2]
M. Mombelli, Representations of tensor categories coming from
quantum linear spaces, J. Lond. Math. Soc. (2) 83 (2011)
19-35.
[O1]
V. Ostrik, Module categories, Weak Hopf Algebras and Modular
invariants, Transform. Groups, 2 8, 177-206 (2003).
Arnold Oostra
"Connective completeness in topos"
Topos
Theory arose in the last decades of the 20th Century as a
categorical generalization of Sheaf Theory, which in turn has deep
roots in Geometry. Soon categorists realized that a topos also is a
suitable setting for Logic: although many logical concepts may be
considered in almost every category, a topos has a full internal
logic. Various topics like independence proofs and recursion have
been studied with great success in this context, but the research
about functional completeness is rather scarce. It is a well-known
fact in Classical Logic that every Boolean function (that is, every
operation in the two element set) is equivalent to some combination
of the usual propositional connectives. This property of Set does
not hold in every topos, but on this issue there are only some
results in the topos of sheaves over a topological space.
Besides showing a rather broad picture of the Geometry and Logic of
topos, in this talk we want to point out a purely categorical
technique that may give new results about functional completeness in
this context.
References:
[1]
Xavier Caicedo, Lógica de los haces de estructuras. Revista de la
Academia Colombiana de Ciencias Exactas, Físicas y Naturales XIX, 74
(1995) 569–585.
[2]
Xavier Caicedo, Investigaciones acerca de los conectivos
intuicionistas. Revista de la Academia Colombiana de Ciencias
Exactas, Físicas y Naturales XIX, 75 (1995) 705–716.
[3]
Xavier Caicedo, Conectivos intuicionistas sobre espacios topológicos.
Revista de la
Academia Colombiana de Ciencias Exactas, Físicas y Naturales XXI, 81
(1997) 521–534.
[4]
Peter Freyd and André Scedrov, Categories, Allegories. Amsterdam:
North-Holland, 1990.
[5]
Robert Goldblatt, Topoi: The Categorial Analysis of Logic.
Amsterdam: North-Holland, 1979.
[6]
Peter T. Johnstone, Sketches of an Elephant: A Topos Theory
Compendium. Oxford: Clarendon Press, 2002.
[7]
Saunders MacLane and Ieke Moerdijk, Sheaves in Geometry and Logic.
New York: Springer Verlag, 1992.
[8]
Arnold Oostra, Conectivos en Topos. Master’s Thesis. Bogotá:
Universidad Nacional de Colombia, 1997.
[9]
Arnold Oostra, Conectivos en el topos de grafos dirigidos. Boletín
de Matemáticas 3 Nº 2 (1996) 55–62.
[10]
Arnold Oostra, Una mirada al problema de los conectivos nuevos.
Boletín de Matemáticas 12 Nº 2 (2005) 81–97.
Eddy Pariguan
Computing
deformation quantization
We
present a general description of the quantum product on the Poisson
orbifold R^n/G using the Kontsevich start product. We give explicit
formulas for the product rule in the symplectic orbifold hxh/W where
h is a Cartan subalgebra of a classical Lie algebra g and W is the
Weyl group associated to h. Finally we present some algorithmic
results for product rule in the symmetric power of the Weyl algebra.
Mainak Poddar
Topological
generalizations of toric varieties
I
will describe various generalizations of toric varieties in topology
like quasitoric manifolds, torus manifolds, etc. I will discuss some
of their invariants and the question of existence of geometric
structures on them. Time permitting, I will talk about orbifolds.
Lucy Zhang
"Kitaev's quantum double models as extended topologial quantum field
theories"
The
Kitaev's quantum double models are an important family of lattice
models supporting anyonic excitations, and useful for topological
quantum computation. Earlier mathematical description of topological
quantum computation often relied on the language of modular functors
(with the exceptions of recent work by Alexander Kirillov and recent
work by Kitaev and Kong on the Levin-Wen model). Here, I will
describe Kitaev's quantum double models explicitly in the language
of topological quantum field theories (TQFTs) after providing some
background. You may ask why I am interested in putting the Kitaev
model in such a mathematical framework (TQFT instead of modular
functor). One reason is that a more-sophisticated subclass of TQFTs
(called the extended TQFTs) to which these "Kitaev TQFTs" belong,
admit a layered structure which is key to relating domain walls and
twists in the Kitaev model (now promising for enhancing the power of
topological quantum computation) to higher categorical structures.
With luck, I maybe able to tell you something about the
extended/higher structures on these Kitaev TQFTs.
The
mathematical framework of TQFTs is also relevant to describing
topological phases in condensed matter physics.
* The
Visit to Tayrona National Natural Park (Neguanje - Playa Cristal)
costs around $120.000 COP for each guest ($ 22.000 COP in addition for
foreigners.) The visit includes: entrance to the park, tourist guide,
transport (by bus and boat) and lunch. If you wish to participate
you must register by Tuesday 2 at the latest.
