Let $\Gamma$ be an Anosov subgroup of a connected semisimple real linear Lie group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on $\Gamma \backslash G$, up to proportionality, is homeomorphic to $\mathbb{R}^{\mathrm{rank} (G)-1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup which normalizes $N$.
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This is joint work with Hee Oh.

Kronheimer-Mrowka recently proved that the Dehn twist along a 3-sphere in the neck of $K3\#K3$ is not smoothly isotopic to the identity. This provides a new example of self-diffeomorphisms on 4-manifolds that are isotopic to the identity in the topological category but not smoothly so. (The first such examples were given by Ruberman.)
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In this talk, we study the Bauer-Furuta invariant as an element in the Pin(2)-equivariant stable homotopy group of spheres. We use it to show that this Dehn twist is not smoothly isotopic to the identity even after a single stabilization (connected summing with the identity map on S2 cross S2). This gives the first example of exotic phenomena on simply-connected smooth 4-manifolds that do not disappear after a single stabilization. In particular, it implies that one stabilization is not enough in the diffeomorphism isotopy problem for 4-manifolds. It gives an interesting comparison with Auckly-Kim-Melvin-Ruberman-Schwartz's theorem that one stabilization is enough in the surface isotopy problem.

In this talk things will get extreme, but not with number fields or schemes. Instead we'll focus on combinatorics, though we won't discuss clever tricks. We speak only of glorious theorems, as well as some notorious problems. And for all those that attend, there is a surprise at the end!

We give a quick and friendly introduction to projective space and then introduce and explore some of the most elementary and fundamental examples in algebraic geometry: hypersurfaces in projective space (especially cubic hypersurfaces).

Anisotropic energies are functionals defined by integrating over a generalized surface (such as a current or a varifold) an integrand depending on the tangent plane to the surface. In the case of a constant positive integrand, one obtains the area functional, and hence one can see anisotropic energies as a generalization of it. A long standing question in geometric measure theory is to establish regularity properties of critical points to such functionals. In this talk, I will discuss some recent developments on this theory, addressing in particular the question of rectifiability of stationary points and regularity of stationary Lipschitz graphs.
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The talk is based on joint work with Antonio De Rosa.

The inhomogeneous l-TASEP is an interacting particle process wherein particles stochastically enter, unidirectionally traverse, and finally exit a one-dimensional lattice segment at rates that may depend on a particle's location within the lattice. Its homogeneous version is known to exhibit various phase transitions in macroscopic observables like particle density and current, with fluctuations governed by what is known as the KPZ equation. In this talk, we begin to extend such results to the inhomogeneous setting by developing the so-called hydrodynamic limit, which governs the system dynamics on an LLN-type scale. If time permits, we apply our results to elucidate the key determinants of protein synthesis, which motivated the introduction of TASEP fifty years ago.
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This is based on joint work with Khanh Dao Duc and Yun S. Song.

Locally analytic representations of $p$-adic analytic groups
have played a crucial role in many areas of arithmetic and
representation theory (including in $p$-adic local Langlands program)
since their introduction by Schneider and Teitelbaum. In this talk we
will briefly review some aspects of the theory of locally analytic
representations. Then, for a locally analytic representation $V$ of
$GL_2(F)$ we will construct a coefficient system attached to the
Bruhat-Tits tree of $Gl_2(F)$. Finally we will use this coefficient
system to construct a resolution for locally analytic principal series
of $GL_2(F)$.

In neuroscience, learning and memory are usually associated to long-term changes of neuronal connectivity. Synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. Spike-Timing Dependent Plasticity (STDP) is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons.
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In this talk we present a new, general, mathematical framework to study synaptic plasticity associated to different STDP rules. The system composed of two neurons connected by a single synapse is investigated and a stochastic process describing its dynamical behavior is presented and analyzed. We show that a large number of STDP rules from neuroscience and physics can be represented by this formalism. Several aspects of these models are discussed and compared to canonical models of computational neuroscience. An important sub-class of plasticity kernels with a Markovian formulation is also defined and investigated via averaging principles.
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Joint work with Gaetan Vignoud

This will be the continuation to Calum's talk. The plan,
building on what Calum explained, is to discuss some recent work
building towards the birational classification of holomorphic foliations
on projective varieties (particularly 3folds) in the spirit of the
Minimal Model program.
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We will explain some applications of these ideas to the study of the
dynamics and geometry of foliations and foliation singularities.
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Features works of C. Spicer and P. Cascini and joint work with C. Spicer.