** Abstract: **Compact quantum metric spaces are noncommutative generalizations of the algebras of Lipschitz functions over compact metric spaces. Intriguing new topologies have been constructed over classes of quantum metric spaces and related structures, in the spirit of the Gromov-Hausdorff distance topology. In this talk, we propose to explore the construction of some of these topologies and see some recent applications to modules and morphisms over quantum metric spaces.

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** Abstract: ** The main theme of this talk is of ``emergent geometry''. We think of geometry
as a tool to study analysis. We argue that a linear elliptic second-order
partial differential operator on a vector bundle over a smooth manifold
determines the local geometry of the manifold. The essential properties
of the partial differential operators are determined by their leading symbols.
The operators with scalar leading symbols (so called Laplace type operators)
determine the Riemannian geometry. However, Riemannian geometry is inadequate
to study the operators with non-scalar leading symbols (so called non-Laplace
type operators). Such operators naturally
define a collection of Finsler geometries on the manifold, which can be
thought of as a non-commutative deformation of Riemannian geometry when
instead of a Riemannian metric there is a matrix valued self-adjoint
symmetric two-tensor that plays the role of a ``non-commutative''
metric. We generalize the basic concepts of Riemannian geometry to the
non-commutative setting. We define the natural partial differential operators
associated with it and study their spectral asymptotics.
We compute the first two non-zero heat trace coefficients for manifolds
without boundary. We propose a non-commutative deformation of the
Einstein-Hilbert action functional as a linear combination of the first two
spectral invariants. The critical points of this functional naturally
define the ``non-commutative'' generalization of Einstein equations.

** Abstract: ** Quantum mechanics is a theory of wave functions in Hilbert space. Many features that we generally take for granted when we use quantum mechanics -- classical spacetime, locality, the system/environment split, collapse/branching, preferred observables, the Born rule for probabilities -- should in principle be derivable from the basic ingredients of the quantum state and the Hamiltonian. I will discuss recent progress on these problems, including consequences for cosmology and quantum gravity.

** Abstract: **(Extended) Harper model describes 2D Bloch electrons in
perpendicular magnetic fields and is popular in various areas of math
and physics. We will overview recent results and open questions on both
models and corresponding one-dimensional families, focusing on sharp
arithmetic transitions.

** Abstract: ** Smale introduced the notion of an Axiom A system in the 1960's as part of an ambitious program to study the dynamics of smooth maps of manifolds. The interesting part of the dynamics occurs on invariant subsets, called basic sets. Typically, these are not submanifolds, but have some kind of fractal nature. Ruelle then gave a definition of a Smale space to axiomatize these systems. Among the most important of these are shifts of finite type. Bowen (following work of Sinai and others) showed that every basic set was the quotient of a shift of finite type. Manning used this to count the periodic points of a Smale space and proved that the Artin-Mazur zeta function was rational. This led Bowen to conjecture the existence of a homology theory for Smale spaces having a Lefschetz formula. Such a theory was given by Krieger for shifts of finite type by using ideas from C*-algebras and K-theory. In this talk, I will show how to combine Krieger's and Bowen's results to provide a homology theory, as proposed by Bowen.

** Abstract: ** I will define the Gromov-Hausdorff topology on rooted graphs, explain a result (joint with Joshua Frisch) on its restriction to the transitive graphs, and discuss an Economics question about social networks (joint with Elchanan Mossel and Allan Sly) where this topology makes a surprise appearance.

** Abstract: ** I will discuss the bulk algebra and topological D-brane category arising from the differential model of the open-closed B-type topological Landau-Ginzburg theory defined by a pair (X; W), where X is a non-compact Calabi-Yau manifold and W a holomorphic function with compact critical set. When X is a Stein manifold the D-brane category can be described by projective matrix factorizations defined over the ring of holomorphic functions of X. Further simplification can be made when X is holomorphically parallelizable.

** Abstract: ** Very much in the spirit of F_1-theory, there has been done a lot of deep research on common combinatorial characterizations of certain classes of varieties (say, defined over finite fields). In the second lecture, I want to describe a number of such characterizations, with a particular focus on the F_1-aspects of the story.

** Abstract: ** In a lecture course given at the end of the 1990s in Brussels, Tits explained his axiomatic approach to "Cremona planes": he defined an incidence structure consisting of "points" and "figures," and endowed this structure with a large set of synthetic axioms based on the geometry of (smooth, projective) models of C(X,Y). One of the main features of his approach was that the automorphism group of this incidence geometry precisely is the group of automorphisms of C(X,Y) fixing C elementwise, that is, the Cremona group. In the paper which was based on these lectures, and published in his Evres, Tits posed the question as to whether a "thin" Cremona plane exists. In September 2014, Deligne, after reading Tits's paper, answered the question in a letter to Tits, by providing a description of a thin Cremona plane (defined over F_1). In this lecture, we mention some aspects of this theory.

** Abstract: **In this talk, I will introduce the 2-dimensional Ginzburg-Landau vortices from both the mathematical and the physical point of view. I will then establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. Finally, I will compute the corresponding Berry phase in the large volume limit.

** Abstract: ** The interstellar medium (ISM) represents a chaotic, highly nonlinear system in which super-sonic turbulence, gravity, and "feedback" from both massive stars and accretion onto super-massive black holes play a critical role. Yet despite this complexity, there is remarkable regularity observed in a range of properties, and many of the most fundamental unsolved questions in star and galaxy formation revolve around understanding how the interplay of these physical processes produces such regular correlations. I'll discuss how many observed properties of the ISM can be understood as a fundamental consequence of basic properties of super-sonic turbulence, in a rapidly cooling, self-gravitating medium. In doing so, I'll show how the excursion-set formalism, commonly used to describe the formation of large-scale dark matter structure, can be applied to understand the origins of structure in the ISM and star formation, including the mass function and structural properties of giant molecular clouds, the distribution of masses of stars, and the clustering of star formation. This can also be used to study time-dependent evolution of structure even in highly non-linear systems, allowing us to understand many emergent properties of simulations with supersonic turbulence and gravity.

** Abstract: ** In this talk, I will begin with a review of p-adic numbers and their relation to the Bruhat-Tits tree, an infinite regular graph. A bulk/boundary correspondence can be naturally set up, with the boundary theory defined on a p-adic number field (instead of real numbers in standard Archimedean AdS/CFT constructions) and the bulk described by the Bruhat-Tits tree whose boundary is precisely the p-adic numbers. I will give an introduction to p-adic AdS/CFT, and discuss holographic correlation functions, emphasising the surprising similarities between the p-adic and Archimedean results, when correlation functions are expressed in terms of local zeta functions. I will end with a brief discussion of edge-length dynamics and a discrete analog of Einstein equations, based on a notion of Ricci curvature on graphs.

** Abstract: ** A general question behind the talk is to explore a good notion for
intrinsic curvature in the framework of noncommutative geometry
started by Alain Connes in the 80's. It has only recently begun (2014)
to be comprehended via the intensive study of modular geometry on the
noncommutative two tori. In this talk, we will focus on a class of
noncommutative manifolds obtained by deforming certain Riemannian
manifolds along a torus action. I will explain how to formulate some
basic notions in Riemannian geometry that are often described in local
charts (such as the metric tensor, scalar curvature) using the
language of functional analysis so that they will survive in the
noncommutative setting. The highlight is that under a noncommutative
conformal change of metric, we found not only the conformal change of
the scalar curvature in Riemannian geometry but also some exciting new
features: the quantum part of the curvature which is hidden in the
commutative setting. What is more striking is that the quantum part of
the curvature is defined by certain entire functions which play a
prominent role in many other areas in mathematics (e.g. in the theory
of characteristic classes).

** Abstract: ** In this talk we will discuss, for $p \in [1,\infty)$ and an
irrational number $\theta$, those Banach subalgebras of B(L^p)
which are generated by two unitaries (invertible isometries) $u$
and $v$ subject to the usual commutation relation $uv =
e^{2\pi i \theta}vu$. When $p$ is different from 2, it turns out
that such an algebra is not unique, and the description of the
possible norms is challenging. Our description allows us to
show that all the L^p irrational rotation algebras are simple,
have a unique trace, and have identical K-theory. Time
permitting, we will discuss further topics such as embeddability
into AF-algebras.

** Abstract: ** We introduce the notion of open quantum subgroups of locally compact quantum groups, and then show the two proposed constructions of induced representations, by Kustermans and by Vaes, are both equivalent in the setting of open quantum subgroups to Rieffel's (rather simple) construction of induced representations in the context of C*-algebras.
This talk is based on joint work with Pawel Kasprzak, Adam Skalski, and Piotr Soltan.

** Abstract: ** We compare different constructions of cyclic cocycles for the algebra of complete symbols of pseudodifferential operators and show that our comparison result leads to interesting index-theoretic consequences and a construction of invariants of the algebraic $K$-theory of the algebra of pseudodifferential symbols. This is a joint work with H. Moscovici.

** Abstract: **
Using the coherent state transform I willI establish the asymptotical behaviour of the Riesz mean for functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. Furthermore, I will prove the Weyl law for the eigenvalue counting function of these operators, therefore implying that their inverses are trace class. This is joint work with A. Laptev and L. A. Takhtajan.

** Abstract: ** We will show, how one may use groups acting on buildings to construct new C*-algebras and compute explicitly their K-theory.

** Abstract: ** The well-known analytic class number formula, linking the
special value at s=0 of the Dedekind zeta function of a number field to
its class number and regulator, has been the foundation and prototype for
the highly conjectural theory of special values of L-functions. We will discuss generalizations of the class number formula to the context
of equivariant Artin L-functions which capture refinements of the
Brumer-Stark and the Coates-Sinnott conjectures, as well as the Iwasawa
main conjecture. These generalizations relate
various algebraic-geometric invariants associated to a global field, e.g.
its Quillen K-groups and etale cohomology groups, to various special
values of its Galois-equivariant L-functions. They illustrate the subtle
interactions of number theory with complex and p-adic analysis, algebraic
geometry, topology and homological algebra.

** Abstract: **
I will introduce in this talk a new distance between Hilbert modules equipped with some metric data which generalize, for my purpose, the idea of a connection. My new distance extends the quantum Gromov-Hausdorff propinquity, a noncommutative analogue of the Gromov-Hausdorff distance which I introduced as a well-behaved distance with respect to the C*-algebraic structure of quantum metric spaces --- notions which I will briefly review in this talk as well. I will then discuss how my modular distance can be applied to prove the continuity of families of Heisenberg modules over quantum 2-tori when the modules are equipped with their natural connections. The modular propinquity represents an exciting step in my program of extending metric geometry to noncommutative geometry, opening the possibility to approximate not only (quantum) spaces, but their vector bundles as well, and is new even in the classical setting.

** Abstract: **
In this talk, I will discuss how the spectral distortions of the cosmic microwave background (CMB) can be used to probe the early universe. After a brief introduction on the relevance and characteristics of the large scale modes of the CMB, I will explain how we can measure these modes at other locations in the universe by using the polarization signal that they induce in the direction of galaxy clusters. The primary importance of these measurements is that they will allow us to inspect the large scale anomalies observed in the CMB and determine if they are coincidental or fundamental. This, in turn, will constrain certain inflation models or exotic cosmologies that attempt to explain these anomalies. I will also discuss the implications of the generalized aberration kernel formalism and illustrate how the detailed study of spectral distortions enables the next generation of microwave surveys to separate the currently indistinguishable kinematic and intrinsic CMB dipoles.

** Abstract: ** We begin with a brief review of important results related to the
strong subadditivity of entropy. We show several routes to
the proof of the convexity of relative entropy and the monotonicity
of Renyi's entropy. One route uses the triple matrix inequality,
which will play a role in the sequel, namely the quantum version of
the uncertainty principle, which extends the classical Maassen-Uffink
inequality. The next act in the play concerns the sandwiched Renyi
entropies and their monotonicity under CPTP maps. We then go on to the
\alpha-z entropies, their monotonicity and their open problems. Finally,
there are some new concavity/convexity theorems extending work of Hiai.
The talk is based on joint works with Eric Carlen and with Elliott Lieb.

** Abstract:** Classical Riemann-Hilbert correspondence gives an equivalence of the category of bundles with regular singular connections on a projective curve and the category of locally constant sheaves on the curve, with singularities of the connections removed. I am going to recall classical story as well as its generalization to the category of holonomic (possibly irregular) D-modules. It admits various interpretations relating the subject to the deformation quantization of Poisson surfaces, Fukaya categories, Legendrian links, etc. Quantum spectral curve (the term is due to physicists) is a holonomic module over the quantum torus. Therefore, we now consider difference equations instead of differential ones. Aim of the talk is to explain how the Riemann-Hilbert correspondence looks in this case. If time permits, I will explain higher-dimensional generalizations of all that as well its relation to holomorphic version of Floer theory, wall-crossing formulas and analytic properties of Feynman integrals.

** Abstract:** In the high-energy quantum-physics literature one finds statements such as ``matrix algebras converge to the sphere''. Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as compact quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. I will indicate briefly how this works.
But physicists want, even more, to treat structures on spheres such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. The main part of my talk will consist of indicating how to do this for vector bundles. One would like to be able to say that for two compact quantum metric spaces that are close together, to a given vector bundle on one of them there corresponds a unique vector bundle on the other. Even for ordinary compact metric spaces and ordinary Gromov-Hausdorff distance it is not so obvious how to do this.

** Abstract:** In this talk I will describe the new geometric picture for the scattering amplitudes in N=4 super Yang-Mills (SYM) theory. The building blocks are on-shell diagrams which are associated with the cells in the positive Grassmannian. Furthermore, all cells can be glued together giving rise to the Amplituhedron which is a certain map from the top cell of positive Grassmannian (and its generalizations). The form with logarithmic singularities on the boundaries of this space gives the scattering amplitudes in N=4 SYM theory.

** Abstract:**
Recent remarkable work by several researchers in the classification theory of simple separable nuclear C*-algebras has boiled down the problem of classifying the crossed products of free minimal actions of amenable groups to the single question of whether the nuclear dimension is finite. I will discuss different strategies for verifying finite nuclear dimension dynamically, with a focus on actions on the Cantor set and the connection to recent tiling results for amenable groups.

** Abstract:** Nevanlinna-Pick spaces are Hilbert function spaces which mirror some of the fine structure of the classical Hardy space on the unit disc. Their multiplier algebras are an important class of non self-adjoint operator algebras of functions.
I will talk about recent work with Raphael Clouatre, in which we investigate representations of these multiplier algebras.
In particular, we determine the boundary representations in the sense of Arveson of a special class of multiplier algebras.
As a consequence, we find that these algebras, despite being commutative, can only be embedded into non-commutative C*-algebras.

** Abstract:**The talk will deal with algebras which are commutants modulo normed ideals of n-tuples of operators and quotients of these by their ideal of compact operators. This will include duality properties, K-theory aspects and the property of countable degree-1 saturation.

** Abstract:** I will discuss asymptotic symmetries for the gravity theory in Minkowski space, Ward identities for S-matrix and their relation to soft theorems.

** Abstract:** The connection between quantum field theory and Grothendieck's theory of motives aims at understanding the number theoretic aspects of Feynman integrals via their period interpretations. The arrangements of the smooth quadric hypersurfaces (a.k.a. the graph hypersurfaces) and the certain hyperplane arrangements (i.e., the configuration spaces) have been examined in order to understand the Feynman integrals respectively in momentum and position space formulations. I will discuss a new one in this talk; arrangements of singular quadrics. I will discuss the motives of such arrangements associated to the Feynman integrals in phi^3 theory.

**Abstract:**
In two recent works, my student Merida and myself have designed a map F which associates to each graph $\Gamma$ a Deitmar scheme (which is a scheme defined over F_1, and hence through base extension a scheme over any field k. Denote these schemes by $\chi_k := F(\Gamma) \otimes_{\mathbb{F}_1} k$ (where k = F_1 is allowed, and simply gives F(\Gamma)). One of the goals is to study the class of schemes $\chi_k$ on the level of the graphs, and to ``translate'' data from the graphs to the schemes. I will indicate how an interesting zeta function can be defined on graphs by using F, and how automorphisms of the schemes $\chi_k$ can be understood if $\Gamma$ is a tree.

** Abstract:** It is hard not to see the theory of the field with one element F_1 intimately connected to its combinatorial side. That is the side I want to highlight in the first lecture.
In particular, besides some basics, I want to mention certain incidence geometries in this context, pass via hyperstructures to a series of problems for group actions on projective geometries, and end with ``fields with F_1-structure.''

** Abstract: **
I will speak on the Feynman periods, the values of Feynman integrals in (massless, scalar) phi^4 theory, from the number-theoretical perspective. Then I define a closely related geometrical object, the graph hypersurface. One can try to study the geometry of these hypersurfaces (cohomology, Grothendieck ring, number of rational points over finite fields) and to relate it to the periods. The most interesting results come out from the study of the c_2 invariant (on the arithmetical side).

**Abstract: **
Eta invariant and analytic torsion are the two prominent spectral invariants arising from index theory. The eta invariant is the boundary contribution to the index formula and measures the spectral asymmetry. The analytic torsion, on the other hand, is a certain combination of determinants of Laplacians, and gives analytic interpretation of the Reidemeister torsion, a topological invariant which is not homotopy invariant (and hence is very useful in finer classifications). Each has higher dimensional generalizations and noncommutative geometric extensions. In this talk, I will review these developments and discuss our work with Weiping Zhang on the relation between these two invariants.

** Abstract: **
We study the first law of thermodynamics in the setting of a finite dimensional quantum system coupled to an infinitely extended thermal reservoir. Using a two-time measurement protocol, in which measurements of the system and reservoir energy are made at time 0 and at some later time t, the first law states that in the limit of large t and small coupling energy, the change of energy in the reservoir should be equal to the negative change of energy of the system. It is well-known that this holds in the sense of expected values of the energy measurements. We define measures which encode the full statistics of the energy changes, and show that the measures themselves converge weakly, which is significantly stronger. To do this, we write the measures in terms of a relative modular operator and exploit the machinery of modular theory.

** Abstract: **
Kauffman brackets on a surface are quantum deformations of homomorphisms from the fundamental group of the surface to the Lie group SL_2(C). A fundamental example arises from Witten's topological quantum field theory interpretation of the Jones polynomial of knots in 3-dimensional manifolds. I will discuss properties of finite-dimensional Kauffman brackets when the quantum parameter q is a root of unity. This will include the construction of invariants, existence properties, and uniqueness theorems and conjectures.

**Abstract: **
In this talk, we will report on recent work connecting aspects
of geometric analysis on fractals and noncommutative fractal
geometry. We construct spectral triples and Dirac operators
on a class of fractals built on curves, including the Sierpinski
gasket, the harmonic gasket (which is ideally suited for
developing analysis on fractals and is a good model for the
elusive notion of a 'fractal manifold'), as well as suitable
quantum graphs, Cayley graphs and other infinite graphs.
We recover from the spectral triple the geodesic metric
intrinsic to the fractal, as that metric is shown to coincide
with the noncommutative metric naturally associated with the
spectral triple. This main result is especially interesting in the
case of the harmonic gasket, which will be the key example
used to illustrate our theory. This work is joint with Jonathan
Sarhad ("Journal of Noncommutative Geometry", vol. 8,
2014, pp. 947-985). It builds on and significantly extends earlier work of the author, joint with Eric Christensen and
Cristina Ivan (published in "Advances in Math.", vol. 217,
2008, pp. 1497-1507) in which we constructed geometric
Dirac operators allowing us to recover the natural geodesic
metric and the natural Hausdorff measure of the Euclidean
Sierpinski gasket (as well of other fractals built on curves). It
also builds on earlier work of the author (carried out in the
1990s) in which, in particular, a broad research program was
proposed for developing "noncommutative fractal geometry".
The new advance highlighted here is that we can now deal
with a significantly broader class of fractals, including the
harmonic Sierpinski gasket (which can be viewed as a kind
of "measurable Riemannian manifold", according to the
recent work of Jun Kigami), allowing us to get one step
closer to developing aspects of geometric analysis truly
connected with the study of fractal manifolds and their
intrinsic families of geodesic curves.

** Abstract: **
In the past few years, my work on special functions has led me to realize that there are close connections with noncommutative algebraic geometry. Although this initially began as a construction of special functions using noncommutative geometry, the ideas have led to a number of new results in the latter. In particular, I'll describe a new construction of a large family of noncommutative rational surfaces, with particular attention to the case of noncommutative elliptic surfaces, and derived equivalences thereof. I'll also explain how one can use derived equivalences to compute certain natural moduli spaces, and the consequences for special functions.

** Abstract: **
A II_1 factor is an infinite dimensional von Neumann algebra which has infinite center and admits a trace. As I will recall, examples of II_1 factors arise naturally
from countable groups (with infinite conjugacy classes), and their measure preserving actions on probability spaces. I will present a recent result showing the existence of uncountably many separable II_1 factors whose ultrapowers, with respect to arbitrary ultrafilters, are pairwise non-isomorphic. More precisely, the families of non-isomorphic II_1 factors originally introduced by McDuff (1969) are such examples. This is joint work with Remi Boutonnet and Ionut Chifan.

** Abstract: In this seminar I will present a gluing construction for Kahler constant scalar curvature metrics starting from a compact or ALE orbifolds with isolated singularities. Besides giving new families of examples, the connection with the Tian-Yau-Donaldson Conjecture and the K-stability of the resolved manifold will be discussed. Joint work with A. Della Vedova, R. Lena and L. Mazzieri.
**

** Abstract: In joint work with A. Guionnet (and further joint work in progress with A. Guionnet and Y. Dabrowski) we construct an analog of Brenier's monotone transport in the non-commutative case. This is done by analyzing a non-commutative Monge-Ampere PDE. Among applications of our work are isomorphisms between C* and W*-algebras associated to certain classes of free Gibbs states. In particular, so-called q-deformed free group factors of Bozejko and Speicher are isomorphic to free groups factors if |q| is sufficiently small.
**

** **

** Abstract:**
In this talk I will discuss two branches of scattering amplitudes which have seen major developments in recent years. Firstly I will elaborate on Integrand Reduction techniques for multi-loop integrals and how they can be applied to the Color-Kinematic duality and secondly how these integrals can be computed by the method of differential equations, which recently has been stimulated by new ideas. I will conclude each part by a series of examples where these techniques have been applied.

** Abstract:**
The plasma membranes of neurons are liquid crystalline aggregates that not only insulate the cells, but further play an essential role in their functioning. In particular, through fusion of spherical bilayer lipid structures (or vesicles) with the membranes, neurotransmitters are released into the extracellular space, which activate receptor proteins and thereby enable neuronal communication. Subsequently, the membrane forms new vesicles to maintain a quasi-equilibrium, i.e. a critical number for synaptic transmission. These processes show astonishing similarities with models in string theory. Membrane fusion is one of the most fundamental processes in life that occurs when two separate lipid membranes merge into a single continuous bilayer. It has recently been suggested that fusion is ultimately triggered by a membrane destabilization through extremal membrane curvature. The hereby induced stress promotes the formation of a hemifusion intermediate, opening of the fusion pore and finally convergence towards a 2-dimensional lipid bilayer surface of mean zero curvature. This process is significantly governed by proteins located at the surface of vesicles and membranes. Proteins also play a key role in vesicle formation, particularly by inducing perturbations that change the orientation of the membrane locally. A number of experimental and theoretical investigations have shed light on different aspects of this process, however less attention has been paid to its multi-scale dynamics and geometrical aspects that contain non-trivial singularities. The aim of this talk is to familiarize the audience with the mathematical foundations of possible strategies to address this question.

**Abstract: **
We discuss some recent work of others and joint work of myself on these geometric structures, which are potential singularity models for the Ricci flow.

** Abstract: **
If the quotient of a compact Hausdorff space by a suitable group action is again a compact Hausdorff space, then the C*-algebra of the quotient is simply the fixed point subalgebra of the C*-algebra of the original space. If the quotient of a compact oriented Riemannian manifold by a suitable Lie group action is again a compact oriented Riemannian manifold, what happens at the level of spectral triples? In this talk, I will discuss what it means for a compact Lie group action on a spectral triple to admit a good quotient in the form of a spectral triple, and I will give an unbounded KK-theoretic construction of a good quotient for the commutative spectral triple corresponding to a generalised Dirac operator equivariant under a free and isometric action of a compact connected Lie group. As time permits, I will then discuss applications to noncommutative principal bundles arising via Rieffel's strict deformation quantisation. This is joint work with Bram Mesland.

** Abstract:**
It is generally believed that any phase of matter with an energy gap is described at low energies by a Topological Quantum Field Theory. It is also well-known that the classification of gapped phases of matter depends on whether the fundamental degrees of freedom are bosonic or fermionic. In theories with Lorenz invariance, statistics is correlated with spin, therefore it is natural to expect that gapped fermionic phases of matter are described by spin-TQFTs, i.e. by TQFTs which depend on a spin structure. But most models of condensed matter physics are defined on a lattice which breaks Lorenz invariance, and it is not clear when the spin-statistics relation continues to hold or even makes sense in this situation. I will discuss in some detail topological lattice models with fermions in low dimensions and argue that spin structure dependence is forced on us by locality and topological invariance. From the mathematical viewpoint, I explain a combinatorial construction of equivariant spin-cobordism invariants for low-dimensional manifolds.

**Abstract:
**
We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.

** Abstract: **
This work is motivated by the study of quantum many-body systems, such as integer quantum Hall systems, topological insulators and superconductors. More generally, we try to characterize entangled quantum states of spins or electrons on an n-dimensional lattice. We may impose reasonable restrictions: the state is a ground state of some local Hamiltonian, the entanglement is also local in some sense, etc. If we consider the special case of weakly interacting fermions, which includes all examples above, then the corresponding quantum states are classified using the KO spectrum. I conjecture that in the more general setting, there are two particular Omega-spectra: F for fermionic systems and B for bosonic (or spin) systems. From the known physical examples, one can infer the topological spaces F_n, B_n in low dimensions.

** Abstract: **
In this talk a generalization of Lax pairs will be introduced. The related R-matrix ansatz, O-operators and the (modified) classical Yang-Baxter equation will be discussed. The second half of the talk will focus on the relation between an O-operator and a Post-Lie algebra and related operadic structures. This talk is based on joint works with C. Bai and L. Guo.

** Abstract: **
I explain how the scalar curvature of the conformally perturbed noncommutative 4-torus can be computed by making use of a noncommutative residue. This method justifies the remarkable cancellations that occurred when the curvature was computed previously in a joint work with M. Khalkhali, using the rearrangement lemma. Furthermore, it allows to recover the 2-variable function in the formula as the sum of a finite difference and a finite product of the 1-variable function. The simplification of the curvature formula for the dilatons associated with an arbitrary projection and an explicit computation of the gradient of the analog of the Einstein-Hilbert action will be outlined.

** Abstract: **
Noncommutative 2-torus is one of the main toy-models of noncommutative geometry, and noncommutative n-torus is a straightforward generalization of it. In 1980 Pimsner and Voiculescu described a 6-term exact sequence, which allows to compute the K-theory of non-commutative tori. It follows, that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2^(n-1) generators. The first non-trivial generator is the one of the even K-theory of the noncommutative 2-torus. It is known from 1981 as the Powers-Rieffel projector. The next one is a generator of the odd K-theory of the noncommutative 3-torus. The goal of this talk is to describe it explicitly.

**Abstract: **
The topological and SPT orders recently found in certain condensed-matter systems characterize new phases of
matter with no broken symmetries. This talk will be an introduction to the topic for mathematicians. I will first introduce
quantum Hall liquids as well as TI's; emphasis will be put on FQH liquids and the corresponding anyonic statistics, topologydependent
ground-state degeneracy, as well as parton and composite fermion constructions with emergent Chern-Simons gauge
theories. The second half of the talk will focus on the mathematical aspects of some recent theories, explaining the emergent
geometry in FQH liquids, Berry curvature, quantum metric, and NCG approach to quantum Hall states with disorder. Kitaev's
K-theoretical classification of TI will also be introduced briefly.

**Abstract: **
Periods are integrals of rational differential forms on real semialgebraic sets. Rational linear combination of Multiple zeta values are certain well studied subset of periods which form an algebra. These numbers appear in the beta function of perturbative quantum field theories but they are not enough to express all of them. Recently there has been a lot of effort to find a bigger algebra of periods which contain real numbers coming from QFT's. In this talk I will explain some recent results in this direction.

**Abstract: **
We will explain how to apply the framework of noncommutative geometry in the setting of conformal geometry. We plan to describe three main results. The first result is a reformulation of the local index formula of Atiyah-Singer in conformal geometry, i.e., in the setting of the action of a group of conformal-diffeomorphisms. The second result is the construction of new conformal invariants out of equivariant characteristic classes. The third result is a version in conformal geometry of the Vafa-Witten inequality for eigenvalues of Dirac operators. This is joint work with Hang Wang (University of Adelaide).

** Abstract: **
I'll discuss various approaches to connect subfactor theory and noncommutative geometry via planar algebras. I'll begin with the definition of a planar algebra and some basic examples. Next, we'll see how Jones found the cyclic category in the annular Temperley-Lieb category. I'll then discuss joint work with Hartglass on canonical C*-algebras associated to a planar algebra. We'll discuss their K-theory and various ways

to construct Dirac operators.

**Abstract: **
The Verlinde formula is a simple and elegant expression for the number of conformal blocks in a 2d CFT on a Riemann surface. It has interesting connections with many different subjects in mathematics, raging from number theory to geometric quantization and index theory. In this talk, I will first give an elementary introduction to Verlinde formula and its "equivariant generalization" and then discuss how the equivariant Verlinde formula is related to complex Chern-Simons theory, Hitchin moduli space and the moduli space of vortices. This talk is based on joint work with Sergei Gukov: arXiv:1501.01310.

** Abstract: **
I will discuss the vacuum structure of the standard model coupled to gravity with two spatial dimensions compactified on a two-torus or a two-sphere, as well as the case of all spatial dimensions compactified on a three-torus. I will show that for the two-dimensional compactification on a torus there exists a new standard model vacuum stabilized by Casimir energies of standard model particles for a large range of experimentally allowed neutrino masses. I will also describe how finite temperature effects modify this picture and discuss the possibility of transitions between the four-dimensional universe and the compactified spacetimes.

**Abstract: **
The AdS3/CFT2 correspondence is special because a broad class of solutions which are asymptotically AdS3 can be obtained as quotients of PSL(2, R) by discrete subgroups of its isometry group. Additionally, CFTs in 1+1 dimensions are special because the conformal group is infinite leading to highly constrained physics. The Ryu-Takayanagi conjecture provides a powerful tool for translating questions about entanglement in CFTs to computations involving extremal surfaces in the corresponding AdS dual. Using this framework, questions about the distribution of entanglement in a CFT2 are translated to computations in a corresponding moduli space, with phase transitions corresponding to points of enhanced symmetry.

** Abstract: **
The Ryu-Takayanagi formula relates the entanglement entropy in a conformal field theory to the area of a minimal surface in its holographic dual. We show that this relation can be inverted for any state in the conformal field theory to compute the bulk stress-energy tensor near the boundary of the bulk spacetime, reconstructing the local data in the bulk from the entanglement on the boundary. We also show that positivity, monotonicity, and convexity of the relative entropy for small spherical domains between the reduced density matrices of any state and of the ground state of the conformal field theory are guaranteed by positivity conditions on the bulk matter energy density. As positivity and monotonicity of the relative entropy are general properties of quantum systems, this can be interpreted as a derivation of bulk energy conditions in any holographic system with the Ryu-Takayanagi prescription applies. We discuss an information theoretical interpretation of the convexity in terms of the Fisher metric. This is joint work with Jennifer Lin, Matilde Marcolli and Hirosi Ooguri, arXiv:1412.1879.

** Abstract: **
Entanglement in quantum field theories can be thought of as encoding how far a given physical system is from classical physics, with entanglement entropy being one measure of entanglement. It has recently been argued that for conformal field theories entanglement entropy can be computed as the area of minimal surfaces in dual gravitational theories, and more generally that properties of entanglement correspond to geometric properties in the dual theory. In particular, under certain conditions the first law for entanglement entropy is equivalent to the Einstein equations, and inequalities satisfied by the entanglement entropy translate to energy conditions in the gravitational dual. This talk will overview these exciting new developments, as well as some more speculative ideas of what the future may hold.

** Abstract: **
A rigorous analytic definition of the Dirac operatoron loop spaces is a difficult open problem. When the target space is a compact Lie group, it is possible to make sense of a Dirac operator using methods from representation theory. In this talk, I will briefly review this construction and its application to twisted K-theory. I will then discuss the construction of a universal Dirac operator, which leads to a Banach Lie group with a highly non-trivial topology.

** Abstract: **
The flat geometry of noncommutative two tori can be conformally perturbed by a Weyl factor, and one can compute the local geometric invariants of these C*-algebras, such as scalar curvature, by employing Connes' pseudodifferential calculus to find explicit formulas for the heat coefficients of the Laplacian associated with the curved metric. A purely noncommutative feature is the appearance of a modular automorphism in the computations and final formulas. In this talk, I will explain my joint works with M. Khalkhali on this type of computations and the extension of the Gauss-Bonnet theorem of Connes and Tretkoff to general translation-invariant conformal structures on noncommutative two tori.