| DATE |
SPEAKER |
TITLE (click to show abstract) |
| January 23 |
Pavel Etingof, MIT |
Periodic pencils of flat connections and their \(p\)-curvature
Abstract: A periodic pencil of flat connections on a smooth algebraic variety \(X\) is a linear family of flat connections
\(\nabla(s_1,...,s_n)=d-\sum_{i=1}^r\sum_{j=1}^ns_jB_{ij}dx_i\), where \(\lbrace x_i\rbrace\) are local coordinates on
\(X\) and \(B_{ij}: X\to {\rm Mat}_N\) are matrix-valued regular functions. A pencil is periodic if it is generically
invariant under the shifts \(s_j\mapsto s_j+1\) up to isomorphism. I will explain that periodic pencils have many
remarkable properties, and there are many interesting examples of them, e.g. Knizhnik-Zamolodchikov, Dunkl,
Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus
fixed points. I will also explain that in characteristic \(p\), the \(p\)-curvature operators
\(\lbrace C_i,1\le i\le r\rbrace\) of a periodic pencil \(\nabla\) are isospectral to the commuting endomorphisms
\(C_i^*:=\sum_{j=1}^n (s_j-s_j^p)B_{ij}^{(1)}\), where \(B_{ij}^{(1)}\) is the Frobenius twist of \(B_{ij}\).
This allows us to compute the eigenvalues of the \(p\)-curvature for the above examples, and also to show that a periodic
pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.
|
| January 30 |
Alex Sherman, University of Sydney |
Queer Kac-Moody algebras and d=2 Ramond Superconformal algebras
Abstract: Kac-Moody (super)algebras, especially those of finite growth, have a very rich theory over the complex numbers. Determined by only a square matrix (the Cartan matrix), and including the class of simple Lie algebras (sl(n), so(n), etc.) they have deep connections to representation theory, tensor categories, physics, combinatorics, number theory, and more.
The queer superalgebra, a super-generalization of gl(n), does not fit into the Kac-Moody framework due to a maximal torus
which is not purely even. We remedy this by defining a new Kac-Moody construction with the most general type of torus
possible in the super setting. We will then explain the classification results that we obtain, including one on
finite-growth queer Kac-Moody algebras.
Special appearances will be made by two d=2 superconformal algebras in the Ramond sector.
|
| February 6 |
No seminar |
See this SLMath workshop instead
|
| February 13 |
Alexander Shapiro, University of Edinburgh |
Toda and Ruijsenaars integrable systems
Abstract: The phase space of the Ruijsenaars integrable system can be
identified with (a Hamiltonian reduction of) the moduli space of
\(GL_n\) local systems on a punctured torus. The latter admits a
structure of a cluster variety. On the algebraic level, this leads to
an injective homomorphism from a spherical subalgebra of the double
affine Hecke algebra into the quantized algebra of global functions on
the named cluster variety. From an analytic point of view, it allows
for a unitary equivalence between Toda and Ruijsenaars quantum
integrable systems. In particular, we will show that the
eigenfunctions of Macdonald operators can be presented as a matrix
coefficient of an order 4 element in the modular group evaluated on a
pair of Toda eigenfunctions. Finally, we will discuss how to work on
polynomial rather than analytic level, and obtain Macdonald
polynomials from the Whittaker ones. During this talk we will focus on
the \(n=2\) case when no Hamiltonian reduction is required. This talk is
based on a joint work with P. DiFrancesco, R. Kedem, S. Khoroshkin,
and G. Schrader.
|
| February 20 |
Subho Chatterjee, UC Davis |
Probabilities and Supergeometry: Measurement theory for
dynamical discrete systems
Abstract: Discrete probabilistic systems, like bits on a computer or
faces of a coin, abound in nature. We propose a geometric model
describing dynamics and measurement theory for such systems. Our
approach is covariant with respect to choices of clocks and
laboratories. The configuration space is a super phasespacetime
modelled by an odd dimensional symplectic supermanifold, observables
are superfunctions and states are suitable (star) squares of
superfunctions. The data of an odd dimensional symplectic
supermanifold canonically incorporates dynamics. We also obtain
dynamical probabilities using convex polyhedral cones and find that
they obey Markov-like evolution.
arxiv no.: 2311.05711
|
| February 27 |
Milen Yakimov, Northeastern University |
Reflective centres of module categories and quantum K-matrices
Abstract:
Braided monoidal categories have applications in various situations,
in particular their universal R-matrices give solutions of the quantum Yang-Baxter equation and representations
of braid groups of type A. There are powerful methods for constructing them: Drinfeld doubles of Hopf algebras
and Drinfeld centres of monoidal categories. On the other hand, universal K-matrices, leading to solutions of the
reflection equation and representations of braid groups of type B are much less well understood. We will describe a
construction of reflective centers of module categories. It gives rise to braided module categories and a quantum double
construction for universal K-matrices.
This is a joint work with Robert Laugwitz and Chelsea Walton.
|
| March 5 |
Sarah Witherspoon, Texas A&M University |
Varieties for tensor categories
Abstract:
We will give a brief survey of three types of topological spaces
associated to objects in certain tensor categories, some in more
limited settings than others, namely the Balmer support, the
cohomological support, and the rank varieties. These are important
tools in representation theory, and are best understood in the
classical setting of finite group representations. Much is known more
generally, while there are still many questions. We will include some
recent results, focusing particularly on the question of how the
spaces associated to a tensor product of objects are related to those
of the objects.
|
| March 12 |
Dmitry Kaledin, HSE University |
Tensor categories and the Grothendieck construction
Abstract: Tensor categories of various types are usual described via
explicit associativity and/or commutativity isomorphisms that have to
satisfy higher constraints. This is rather heavy technically and not
very canonical. I am going to advertise an alternative packaging for the
same data based on a purely categorical gadget called the Grothendieck
construction. If time permits, I will also try to show how this leads to
a very transparent understanding of deformations.
|
| March 19 |
Ben Davison, University of Edinburgh |
R matrices and shuffle algebras
Abstract: For any quiver Q there is an associated shuffle algebra
structure on the ring of symmetric functions, which can be realised as
the cohomological Hall algebra of the category of representations of
the quiver. Mimicking Green's coproduct for finitary Hall algebras,
one obtains also a kind of localised coproduct, along with an easy
construction of an R-matrix for the resulting (localised) Hopf
algebra. This provides a very flexible framework for producing new
quantum groups, using the formalism of Faddeev, Reshetikhin and
Takhtadzhyan, after a choice of favoured modules for the shuffle
algebra. I will explain what is known: that using a specific variant
of this construction we obtain the new Yangians defined by Maulik and
Okounkov (via recent joint work with Botta). I will also present some
conjectures regarding the algebra we obtain this way for certain other
choices of cohomological Hall algebras and modules over them.
The University of Edinburgh is a charitable body, registered in
Scotland, with registration number SC005336. Is e buidheann
carthannais a th’ ann an Oilthigh Dhùn Èideann, clàraichte an Alba,
àireamh clàraidh SC005336.
|
| March 26 |
No seminar |
Spring break
|
| April 2 |
Julia Pevtsova, University of Washington |
Are categories of modular representations of finite groups (locally) regular?
Abstract: For a symmetric tensor triangular category T we suggest a notion of “local regularity” which
agrees with the classical concept for a derived category D(A) for a commutative ring A. We study this property
for stable categories of a finite group scheme (in positive characteristic) and discover that it is related
to many other, such as dualizability and various finiteness conditions. I’ll review some fundamental principles
of tensor triangular geometry and the stratification of the stable category of a finite group G over a field of
positive characteristic. Then we specialize to the fibers of the stable category at homogeneous prime ideals p in
the cohomology ring and show that they are regular.
Joint work with D. Benson, S. Iyengar, H. Krause.
|
| April 9 |
Iva Halacheva, Northeastern University |
Bethe algebras, cacti, and crystals
Abstract: The Bethe subalgebras of the Yangian Y(gl(n)) form a family of maximal
commutative subalgebras indexed by points of the Deligne-Mumford
compactification of the moduli space M(0,n+2). When considering a
point C in the real locus of this parameter space, the corresponding
Bethe subalgebra B(C) acts with simple spectrum on a given tame
representation of Y(gl(n)). This results in an unramified covering,
whose fiber over C is the set of eigenlines for the action of B(C). I
will discuss the identification of each fiber with a collection of
Gelfand-Tsetlin keystone patterns, which carry a gl(n)-crystal
structure, as well as the monodromy action realized by a type of
cactus group. This is joint work with Anfisa Gurenkova and Leonid
Rybnikov.
|
| April 16 |
Yuri Berest, Cornell University |
Representation homology of spaces and the strong Macdonald conjectures
Abstract: In 1982, I. G. Macdonald published a series of beautiful
combinatorial conjectures related to classical root systems (or
equivalently, compact Lie groups).
These conjectures were in the focus of research in representation
theory and geometry for over 30 years. In the early 1990s, B. Feigin
and P. Hanlon proposed
a homological refinement of the Macdonald conjectures (nowadays known
as the strong Macdonald conjectures) that were studied by many
mathematicians
and eventually settled by S. Fishel, I. Grojnowski and C. Teleman in
2008. In this talk, I will give a topological interpretation of the
strong
Macdonald conjectures and present a series of new conjectures
suggested by topology that remain wide open.
|
| April 23 |
Jonathan Kujawa, Oregon State University |
The Lie superalgebra generated by transpositions
Abstract: The symmetric group has been the object of study since forever. Nevertheless, there are still new things to say.
Using the commutator, you can view the group algebra of the symmetric group as a Lie algebra. In 2003,
Marin described this Lie algebra and the subalgebra generated by the transpositions. Since the symmetric group naturally
splits into even and odd permutations, you can also ask about the graded version of the commutator.
This makes the group algebra into a Lie superalgebra.
In 2023, Chris Drupieski and I obtained the super analogue of Marin’s results.
|
| April 29 (Monday) 12-1PM, 891 Evans. |
Alberto Elduque, University of Zaragoza |
From the Albert Algebra to Kac's Jordan superalgebra via tensor categories
Abstract: Kevin McCrimmon proved the surprising result that Kac's ten-dimensional simple
Jordan superalgebra satisfies the super version of the Cayley-Hamilton
equation of degree 3 in characteristic 5. This was fundamental in the
discovery of a new simple Lie superalgebra in characteristic 5.
In this talk it will be shown that over fields of
characteristic 5, Kac's superalgebra may be obtained from the Albert
algebra, i.e. the Jordan algebra of 3 by 3 hermitian matrices over the
octonions, by looking at it as an algebra in the tensor category of
representations of the cyclic group of order 5 and passing to the
associated semisimple category: the Verlinde category.
|
| April 30 |
Alistair Savage, University of Ottawa |
Diagrammatics for real supergroups
Abstract: We introduce diagrammatic monoidal supercategories
controlling the representation theory of real forms of the general
linear, orthosymplectic, periplectic, and isomeric supergroups. As a
consequence, we obtain first fundamental theorems for these real
supergroups and equivalences between monoidal supercategories of
tensor supermodules over the real forms of a complex supergroup. This
is joint work with Saima Samchuck-Schnarch.
|