# Upcoming Seminars & Events

## Primary tabs

##### Stochastic homogenization: renormalization, regularity, and quantitative estimates

There has been a lot of work in recent years on the problem of understanding the behavior of solutions of PDEs with random coefficients, with most of the work focused on linear elliptic equations in divergence form. I will give an overview of recent joint works with Smart, Kuusi and Mourrat in which we introduce a ``renormalization group" method, which leads to a very precise, quantitative description of the solutions.

##### TBA-Yu-Shen Lin

##### Two problems involving breakup of a liquid film

Understanding the breakup of a liquid film is complicated by the fact that there is no obvious instability driving breakup: surface tension favors a film of uniform thickness over a deformed one. Here, we identify two mechanisms driving a film toward (infinite time) pinch-off. In the first problem, we show how the rise of a bubble is arrested in a narrow tube, on account of the lubricating film pinching off. In the second problem, breakup of a free liquid film is driven by a strong temperature gradient across the pinch region.

##### The Hilbert scheme of points on singular surfaces.

The Hilbert scheme of points on a quasi-projective variety parametrizes its zero-dimensional subschemes. When the variety is a singular surface, the geometry of the Hilbert scheme reflects the singularity of the underlying surface. I will present some work in progress on the Hilbert schemes of points on surfaces of embedding dimension three and four. Part of the talk will be on a joint work with Lawrence Ein.

##### TBA - Dr. Efi Efrati

##### Shimura curves and new abc bounds

Existing unconditional progress on the abc conjecture and Szpiro's conjecture is rather limited and coming from essentially only two approaches: The theory of linear forms in p-adic logarithms, and bounds for the degree of modular parametrizations of elliptic curves by using congruences of modular forms. In this talk I will discuss a new approach as well as some unconditional results that it yields.

##### Data-driven analysis of neuronal activity

Recent advances in experimental methods in neuroscience enable the acquisition of large-scale, high-dimensional and high-resolution datasets. These complex and rich datasets call for the development of advanced data analysis tools, as commonly used techniques do not suffice to capture the spatiotemporal network complexity. In this talk I will present new data-driven methods based on global and local spectral embeddings for the processing and organization of high-dimensional datasets, and demonstrate their application to the analysis of neuronal measurements.

##### TBA-Timo Seppalainen

##### Regularity of area-minimizing surfaces in higher codimension: old and new

The theory of integral currents, developed by Federer and Fleming in the 60s, gives a powerful framework to solve the Plateau's problem in every dimension and codimension. The interior and boundary regularity theory for the codimension

one case is rather well understood, thanks to the work of several mathematicians in the 60es, 70es and 80es.

##### TBA-Amina Abdurrahman

##### TBA-Joel Moreira

##### Multiplicative structure of the cohomology of real toric spaces

A real toric space is a topological space which admits a well-behaved \Z_2^k-action.

Real moment-angle complexes, real toric varieties and small covers are typical examples of real toric spaces.

A real toric space is determined by the pair of a simplicial complex K and a characteristic matrix \Lambda.

In this talk, we discuss an explicit -cohomology ring formula of a real toric space in terms of K and \Lambda, where R is a commutative ring with unity in which 2 is a unit. Interestingly, it has a natural (\Z \oplus \row \Lambda)-grading.

##### Singularity formation in the contour dynamics for 2d Euler equation on the plane.

We will study 2d Euler dynamics of centrally symmetric pair of patches on the plane. In the presence of exterior regular velocity, we will show that these patches can merge so fast that the distance between them allows double-exponential upper bound which is known to be sharp. The formation of the 90 degree corners on the interface and the self-similarity analysis of this process will be discussed. For a model equation, we will prove existence of the curve of smooth stationary solutions that originates at singular stationary steady state.

##### Locally symmetric spaces: p-adic aspects

p-adic period spaces have been introduced by Rapoport and Zink as a generaliza- tion of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. The proof consists in a thorough study of modifications of G-bundles on the curve.

##### Irreducible SL(2,C)-representations of integer homology 3-spheres

We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation.

##### TBA-Maria Colombo

##### Attractors: "when inertial waves meet topography..."

##### Bulk-Edge Duality and Complete Localization for Chiral Chains

We study one dimensional insulators obeying a chiral symmetry in the single-particle picture where the Fermi energy is assumed to lie within a mobility gap. Topological invariants are defined for infinite (bulk) or half-infinite (edge) systems, and it is shown that for a given bulk system with nearest neighbor hopping, the invariant is equal to the induced-edge-system's invariant.

##### Foliated surfaces and their stable reduction.

In 1977 Bogomolov proved that on surfaces of general type with c_1^2>c_2, curves of a given genus form a bounded family. The role played by foliations in his proof was further investigated by McQuillan, who in 1998 proved the Green-Griffiths conjecture for surfaces of general type with c_1^2>2c_2.