Representation Theory Session
Alexis Bouthier (Sorbonne University Paris)
Title:
Singular support for ind-schemes
Abstract:
We give a construction of a singular support for ind-schemes that generalizes the one of
Beilinson and Saito.
Then we explain how it can be used to compute the singular support of some affine Springer
sheaves and obtain an affine version of results of Mirkovic-Vilonen.
Alexei Latyntsev (University of Southern Denmark)
Title:
Quantum vertex algebras and cohomological Hall algebras
Abstract:
There is an extremely rich history of interaction between string theory and the mathematics
of moduli spaces, for instance cohomological Hall algebras/algebras of BPS states, or
vertex/chiral algebras. In this talk, I will explain a link between two of these: Joyce's
vertex algebras attached to the moduli stack of objects in an abelian category, and one
dimensional CoHAs. This is based on my recent paper 2110.14356, whose main result says
that the cohomologies of such stacks are ``quantum vertex algebras": the
factorisation/vertex analogues of quasitriangular bialgebras.
The main technical tool is a ``bivariant" Euler class which makes torus localisation work
in this context.
I will discuss applications of these techniques to CoHAs of coherent sheaves on a curve
and future directions.
Wille Liu (Academia Sinica)
Title:
Translation functors for trigonometric double affine Hecke algebras
Abstract:
Double affine Hecke algebras were introduced by Cherednik around 1995 as a tool to study
the Macdonald polynomials.
The trigonometric double affine Hecke algebras (TDAHA), degenerate version of the former,
have also been found related to several other areas, notably representation theory
of \(p\)-adic groups.
In this talk, I will be focusing on specific aspects of the representation theory of the
TDAHA.
Given a root system and two families of complex parameters \(c\) and \(c'\) such that
\(c - c'\) takes values in \(\mathbf{Z}\), there is an equivalence of the derived categories
of modules of the resulting TDAHA:
\(\mathrm{D}^{\mathrm{b}}(H_c\mathrm{-mod}) \cong
\mathrm{D}^{\mathrm{b}}(H_{c'}\mathrm{-mod})\),
called translation functor.
After a brief introduction to the TDAHA, I will talk about a construction of translation
functors.
Yuchen Fu (RIMS, Kyoto University)
Title:
Factorization Modules and Quantum Category \(O\)
Abstract:
Given a braided Hopf algebra \(A\) in some braided monoidal category \(C\), we explain how to
establish an equivalence between the category of left modules over its Majid double
bosonization DBos(\(A\)) and the category of factorization modules over a factorization
algebra Fact(\(A\)).
For \(C\) an abelian category, this was a celebrated result by Bezrukavnikov, Finkelberg and
Schechtman; our construction, which uses different methods, generalizes it to the derived
setting.
We will also explain how this allows us to express objects in the quantum BGG category \(O\)
in terms of factorization modules.
Ana Kontrec (RIMS, Kyoto University)
Title:
Representation theory and duality properties of some minimal affine \(\mathcal{W}\)-algebras
Abstract:
One of the most important families of vertex algebras are affine vertex algebras and their
associated \(\mathcal{W}\)-algebras, which are connected to various aspects of geometry and
physics.
Among the simplest examples of \(\mathcal{W}\)-algebras is the Bershadsky-Polyakov vertex
algebra \(\mathcal{W}^k(\mathfrak{g}, f_{min})\), associated to \(\mathfrak{g} = sl(3)\)
and the minimal nilpotent element \(f_{min}\).
In this talk we are particularly interested in the Bershadsky-Polyakov algebra
\(\mathcal W_k\) at positive integer levels, for which we obtain a complete classification
of irreducible modules.
In the case \(k=1\), we show that this vertex algebra has a Kazama-Suzuki-type
dual isomorphic to the simple affine vertex superalgebra \(L_{k'} (osp(1 \vert 2))\) for
\(k'=-5/4\). This is joint work with D. Adamovic.
Kota Murakami (Kyoto University)
Title:
Categorifications of deformed symmetrizable generalized Cartan matrices
Abstract:
Motivated from studies of the representation theory of quantum loop algebras,
Geiss-Leclerc-Schröer introduced the notion of the generalized preprojective algebra
associated with a symmetrizable generalized Cartan matrix and its symmetrizer.
We study a several parameter deformation of a symmetrizable generalized Cartan matrix as a
numerical aspect of the multi-graded module category of the generalized preprojective
algebra.
In particular, we will interpret some numerical formula about this matrix in terms of
braid group symmetries of our graded module category.
This is a joint work with Ryo Fujita (RIMS).
Topology Session
Francesco Fournier-Facio (Eidgenössische Technische Hochschule Zürich)
Title:
Aut-invariant quasimorphisms on groups
Abstract:
For every group \(G\), there is a natural action of \({\rm Aut}(G)\) on the space of
homogeneous quasimorphisms of \(G\). This action is very poorly understood, in particular
it is hard to produce fixpoints, i.e. Aut-invariant quasimorphisms, which can be used to
estimate Aut-invariant norms on groups.
I will report on joint work with Ric Wade (Oxford) where we construct Aut-invariant
quasimorphisms on all Gromov-hyperbolic groups, and more.
Kohei Kikuta (Osaka University) Title:
Autoequivalence groups of K3 surfaces and Mapping class groups
Abstract:
Autoequivalence groups of derived categories of K3 (complex) surfaces are interesting objects
in group theory.
Via homological mirror symmetry, one can see an analogy between autoequivalence groups and
mapping class groups of (real) surfaces.
In this talk, we explain this analogy, especially focus on spherical twists, spaces of
stability conditions and complexes of spherical objects, which are analogue of Dehn twists,
Teichmuller spaces and curve complexes, respectively.
This talk is partly based on a joint work with Federico Barbacovi.
Harry Petyt (University of Oxford)
Title:
Large-scale geometry of mapping class groups
Abstract:
Mapping class groups are classical objects in topology and group theory.
In recent years there has been a lot of interest in understanding their geometry,
and in particular in the question of which features of nonpositive curvature they display.
In this talk I'll discuss some recent results in this direction.
Partly based on joint work with Thomas Haettel and Nima Hoda.
Sheng Bai (Kyoto University)
Title:
Equivalence of state surfaces
Abstract:
State surfaces of a link are special spanning surfaces for the link corresponding
to Kauffman states of link diagrams.
We follow D. Bar-Nartan, J. Fulman and L. H. Kauffman's method to show that any two
connected state surfaces of the same link are related up to isotopy by addition of small
half-twisted bands.
We further show that every state surface is isotopic to a checkerboard surface.
We find that the self-linking number of state surface is the first grading in Khovanov
homology.
Using our main result, we prove that if two spanning surfaces for the same link have the
same boundary slope on each component of the link, then they are tube-equivalent.
Finally we recover some classical results from this theorem.
This is a joint work with Louis H. Kauffman.
Yichen Tong (Kyoto University)
Title:
Homotopy Commutativity in Hermitian symmetric spaces
Abstract:
A fundamental problem on \(H\)-spaces is to find whether or not a given \(H\)-space
is homotopy commutative.
It is proved that the loop spaces of some homogeneous spaces are homotopy nilpotent,
but we do not even know they are homotopy commutative or not.
In this talk we investigate the homotopy commtativity of loop spaces of irreducible
Hermitian spaces case-by-case. The method also applies to compute the homotopy nilpotency
of flag manifolds.
This is a joint work with Daisuke Kishimoto and Masahiro Takeda.
Operator Algebra Session
Srivatsav Kunnawalkam Elayavalli (University of California, Los Angeles)
Title:
The small at infinity boundary for von Neumann algebras
Abstract:
In this talk I will describe a generalized notion of a small at infinity compactification
a la Ozawa of a finite von Neumann algebra introudced by the speaker, Ding and Peterson.
The key technical novelty here was to bypass an old well known obstruction, involving the
absence of interesting derivations from \(B(L^2M)\) into the compact operators, by
considering the closure of the compacts in a topology of Magajna from the theory of strong
operator bimodules.
By effectively working in this topological framework we clarify old
problems of Anantharaman-Delaroche involving various viewpoints of the Haagerup property
for \(II_1\) factors, and also the notion of mixingness for bimodules.
This work additionally allows for several kinds of rigidity results via a generalized notion
of proper proximality (a la Boutonnet-Ioana-Peterson) for von Neumann algebras.
Some of the applications I will talk about from this work include:
absense of weakly compact cartan subalgebars for a wide class of II_1 factors;
a structure result for subfactors of \(L(G)\) where \(G\) is non amenable bi-exact,
settling a problem of Popa for the case of \(F_2\) and solid ergodicity for various
Gaussian actions, extending works of Boutonnet and Chifan-Ioana.
By developing an abstract upgrading result for the notion of relative proper proximality,
the speaker and Ding also recently obtained a new application of this framework,
involving the structure of free products.
I will discuss these results and touch on some key new ideas.
Koichi Oyakawa (Vanderbilt University)
Title:
Bi-exactness of relatively hyperbolic groups
Abstract:
Bi-exactness is an analytic property of groups defined by Ozawa and of fundamental
importance to the study of operator algebras.
In this talk, I will show that finitely generated relatively hyperbolic groups are bi-exact
if and only if all peripheral subgroups are bi-exact.
This is a generalization of Ozawa's result which claims that finitely generated relatively
hyperbolic groups are bi-exact if all peripheral subgroups are amenable.
Pieter Spaas (University of Copenhagen)
Title:
Obstructions to stability and lifting properties for groups with property (T)
Abstract:
We will start with discussing the notion of Hilbert-Schmidt stability for countable
discrete groups.
We will motivate its definition, discuss some examples, and establish a cohomological
obstruction to it for certain groups with property (T).
This will allow us to provide examples of groups that are not Hilbert-Schmidt stable.
We will then further discuss some related lifting properties for groups and their operator
algebras.
This is based on a joint work with Adrian Ioana and Matthew Wiersma.
Ryoya Arimoto (RIMS, Kyoto University)
Title:
On the type of the von Neumann algebra of an open subgroup of the Neretin group
Abstract:
The Neretin group is the totally disconnected locally compact group consisting of almost
automorphisms on the rooted tree.
In 2021, P.-E. Caprace, A. Le Boudec, and N. Matte Bon proved that the Neretin group is
not of type I and conjectured that a distinguished open subgroup of this group is not of
type I either.
In this talk, I will show that the group von Neumann algebra of this open subgroup is of
type II and answer their question.
Junichiro Matsuda (Kyoto University)
Title:
Algebraic connectedness and bipartiteness of quantum graphs
Abstract:
The notion of quantum graphs was introduced as a non-commutative analogue of classical
graphs in quantum information theory, and it has been developed in the interactions between
theories of operator algebras, quantum information, quantum groups, tensor categories,
non-commutative geometry, etc.
Similarly to the classical case, the degree of a regular quantum graph is shown to be the
spectral radius of the adjacency matrix. Thus it makes sense to consider the behavior of
the spectrum in \([-d,d]\) for \(d\)-regular undirected quantum graphs.
We generalize the well-known fact that the spectrum of the adjacency matrix can
characterize the connectedness and bipartiteness of a regular graph.
We also obtained the equivalence between bipartiteness, quantum and classical
two-colorability, and symmetry of the spectrum of connected regular quantum graphs.
Akihiro Miyagawa (Kyoto University)
Title:
The conjugate system for the q-Gaussians
Abstract:
The \(q\)-Canonical Commutation Relation (\(q\)-CCR) is an interpolation between the CCR
and the CAR with a parameter q. In the 1990s, M. Bożejko and R. Speicher found that the
\(q\)-CCR is represented on the \(q\)-Fock space.
The \(q\)-Gaussians are realized as the field operators with the vacuum state, which forms
a non-commutative distribution.
The von Neumann algebra generated by \(q\)-Gaussians has been studied for many years, and
it is known that this algebra shares several properties with the free group factor.
In terms of applications, the \(q\)-Gaussians are also related to a random matrix model of
quantum holography, the so-called SYK model.
On the other hand, a conjugate system is a notion of free probability introduced by
D. Voiculescu.
This carries important information about a non-commutative distribution of given operators
and has many implications for the generated von Neumann algebra.
In this talk, I will start to give an overview of the \(q\)-Gaussians and related topics.
In the sequel, I will explain the existence of a conjugate system for the q-Gaussians.
This talk is based on the joint work with R. Speicher.
Applied Mathematics Session
Daniel Ginsberg (Princeton University)
Title:
The stability of model shocks and the Landau law of decay
Abstract:
It is well-known that in three space dimensions, smooth solutions to the equations
describing a compressible gas can break down in finite time.
One type of singularity which can arise is known as a shock, which is a hypersurface
of discontinuity across which the integral forms of conservation of mass and momentum
hold and through which there is nonzero mass flux.
One can find approximate solutions to the equations of motion which describe expanding
spherical shocks.
We use these model solutions to construct global-in-time solutions to the irrotational
compressible Euler equations with shocks. This is joint work with Igor Rodnianski.
Naoki Sato (The University of Tokyo)
Title:
Nested invariant tori foliating a vector field and its curl: toward steady Euler
flows in toroidal domains without Euclidean isometries
Abstract:
This work studies the problem of finding a three-dimensional solenoidal vector field
such that both the vector field and its curl are tangential to a given family of toroidal
surfaces.
We show that this question can be translated into the problem of determining a periodic
solution with periodic derivatives of a two-dimensional linear elliptic second-order
partial differential equation on each toroidal surface, and prove the existence of smooth
solutions.
An example of smooth solution foliated by toroidal surfaces that are not invariant under
Euclidean isometries is also constructed explicitly, and it is identified as an equilibrium
of anisotropic magnetohydrodynamics.
The problem examined here represents a weaker version of a fundamental mathematical problem
that arises in the context of fluid mechanics and magnetohydrodynamics concerning the
existence of regular steady Euler flows and equilibrium magnetic fields in bounded domains
without Euclidean isometries. The existence of such configurations represents a key
theoretical issue for the design of the confining magnetic field in nuclear fusion reactors
known as stellarators.
The connection with the study of vortex flows over curved surfaces is also discussed.
Taichi Uemura (Stockholm University)
Title:
Normalization and coherence for \(\infty\)-type theories
Abstract:
\(\infty\)-type theories are a higher dimensional generalization of type theories
introduced by Nguyen and the speaker to tackle coherence problems in the
\(\infty, 1)\)-categorical semantics of type theories.
There, the coherence problem for an \(\infty\)-type theory is whether the initial
model of the \(\infty\)-type theory is \(0\)-truncated.
Normalization for a type theory is the property that every type or term in the type
theory has a unique normal form. An application is calculation of equality of
types and terms.
In this talk, I present a technique for normalization for \(\infty\)-type theories to
solve the coherence problems for \(\infty\)-type theories at some level of generality.
Normalization allows us to calculate path spaces of types and terms and to determine the
truncation level of the initial model.
Yuki Amano (RIMS, Kyoto University)
Title:
Locally Defined Independence Systems on Graphs
Abstract:
The maximization for the independence systems defined on graphs is a generalization of
combinatorial optimization problems such as the maximum \(b\)-matching, unweighted MAX-SAT,
matchoid problem, and maximum timed matching.
In this paper, we consider the problem under the local oracle model to investigate the
global approximability of the problem by using the local approximability.
We first analyze two simple algorithms FixedOrder and Greedy for the maximization under
the model, which shows that they have no constant approximation ratio.
Here algorithms FixedOrder and Greedy apply local oracles with fixed and greedy orders of
vertices, respectively.
These results can be generalized to the hypergraph setting. We then propose two
approximation algorithms for the \(k\)-degenerate graphs, whose approximation ratios are
\(\alpha +2k -2\) and \(\alpha k\), where \(\alpha\) is the approximation ratio of local
oracles.
We also propose an \((\alpha + k)\)-approximation algorithm for bipartite graphs, in which
the local independence systems in the one-side of vertices are \(k\)-systems with
independence oracles.
Jean-Simon Pacaud Lemay (RIMS, Kyoto University)
Title:
Tangent Categories: A Bridge between Differential Geometry and Algebraic Geometry
Abstract:
Differential geometry and algebraic geometry share many similarities,
with one common aspect being that differentiation is a fundamental notion for both.
The theory of tangent categories uses category theory to provide the foundations of
differential calculus over smooth manifolds.
As such, tangent categories have been able to formalize numerous aspects of differential
geometry such as: tangent bundles, vector fields, vector bundles, differential equations,
Lie brackets, connections, etc.
Due to their generality, tangent categories have also found applications in synthetic
differential geometry and computer science.
Recently, tangent categories have also found surprising connections and applications to
algebraic geometry.
In this talk, I will give an introduction to tangent categories and discuss its relation
to algebraic geometry.
This will highlight how tangent categories provide a connecting bridge between
differential geometry and algebraic geometry.
Kento Yasuda (RIMS, Kyoto University)
Title:
Most probable path of an active Brownian particle
Abstract:
In this talk, I show the latest research on the transition path of a
free active Brownian particle (ABP) on a two-dimensional plane between
two given states.
The extremum conditions for the most probable path
connecting the two states are derived using the Onsager–Machlup
integral and its variational principle. We provide explicit solutions
to these extremum conditions and demonstrate their nonuniqueness
through an analogy with the pendulum equation indicating possible
multiple paths.
The pendulum analogy is also employed to characterize
the shape of the globally most probable path obtained by explicitly
calculating the path probability for multiple solutions.
We comprehensively examine a translation process of an ABP to the front
as a prototypical example. Interestingly, the numerical and
theoretical analyses reveal that the shape of the most probable path
changes from an \(I\) to a \(U\) shape and to the \(\ell\) shape with an increase in
the transition process time.
The Langevin simulation also confirms this shape transition.
If I have enough presentation time, I will
explain an additional topic, an enzyme molecule characterized by odd elasticity.
