Abstract: In this talk, we study the Ahlfors regularity of planar self-affine sets under natural conditions: strong separation condition, strong irreducibility and proximality. Not surprisingly, if the dimension is strictly larger than 1, the set is never Ahlfors regular. In case if the dimension is less than or equal to 1 under the extra condition of dominated splitting, we show that the Ahfors regularity is equivalent to the positive proper dimensional Hausdorff measure and to positive proper dimensional Hausdorff measure of the projections in every Furstenberg direction.
Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions, and a temperature fixing boundary, described by Dirichlet boundary conditions. I have recently started to explore the statistics of these Robin eigenvalues for planar domains, and the fluctuations of the gaps between the Robin and Neumann spectrum, in part driven by numerical experimentation. Read more about Cancelled - Basic Notions: Zeev Rudnik "The Robin eigenvalue problem: statistics and arithmetic"
Abstract: Thestudy of geometric stability begins with Mumford's geometric invariant theory.The Kempf-Ness theorem establishes a connection between geometric invarianttheory and symplectic quotients. An infinite dimensional analog of theKempf-Ness theorem leads to a deep connection between algebraic geometricstability and special metric geometries. Examples of this connection includethe work of Donaldson and Uhlenbeck-Yau on the Kobayashi-Hitchin correspondenceand work of Yau, Tian, Donaldson and many others on extremal Kahler metrics. Read more about Basic Notions: Jake Solomon "Geometric stability"
Abstract: Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions, and a temperature fixing boundary, described by Dirichlet boundary conditions. I have recently started to explore the statistics of these Robin eigenvalues for planar domains, and the fluctuations of the gaps between the Robin and Neumann spectrum, in part driven by numerical experimentation.
ABSTRACT:
We study sufficient conditions for the absence of positive eigenvalues of magnetic Schroedinger operators in R^n. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller-Simon shows that our result is sharp as far as the decay of the magnetic field is concerned.
The talk is based on a joint work with Silvana Avramska-Lukarska and Dirk Hundertmark.
There is a rich history of studying dynamical systems through the lens of operator algebras, and particularly through C*-algebras. For instance, in the work of Giordano, Matui, Putnam and Skau, C*-algebras were used as a key tool for classifying Cantor minimal $\mathbb{Z}^d$ systems up to various notions of orbit equivalence. Another successful study was conducted by Cuntz and Krieger, where subshifts of finite type (SFTs) are interpreted through C*-algebras of directed graphs, and invariants studied in symbolic dynamics naturally arise from these C*-algebras. Read more about Analysis Seminar: Adam Dor-On (Copenhagen) "Operator algebras for subshifts and random walks"
Abstract: The study of geometric stability begins with Mumford's geometric invariant theory. The Kempf-Ness theorem establishes a connection between geometric invariant theory and symplectic quotients. An infinite-dimensional analog of the Kempf-Ness theorem leads to a deep connection between algebraic-geometric stability and special metric geometries. Examples of this connection include the work of Donaldson and Uhlenbeck-Yau on the Kobayashi-Hitchin correspondence and work of Yau, Tian, Donaldson, and many others on extremal Kahler metrics. Read more about Basic Notions: Jake Solomon "Geometric stability"
Repeats every week every Sunday until Sun Jan 17 2021 .
4:00pm to 6:00pm
4:00pm to 6:00pm
4:00pm to 6:00pm
4:00pm to 6:00pm
4:00pm to 6:00pm
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Abstract: We will discuss stability conditions on triangulated categories following the work of Douglas and Bridgeland. Concrete examples of stability conditions will be given from symplectic and algebraic geometry, which will also illustrate mirror symmetry. An effort will be made to give a gentle introduction to the relevant background material from category theory, symplectic geometry and algebraic geometry.
Abstract: Diophantine approximation quantifies the density of the rational numbers in the real line. The extension of this theory to algebraic numbers raises many natural questions. I will focus on a dynamical resolution to Davenport's problem and show that there are uncountably many badly approximable pairs on the parabola. The proof uses the Kleinbock--Margulis uniform estimate for nondivergence of nondegenerate curves in the space of lattices and a variant of Schmidt's game.
Abstract: We show that some examples of type-III:1 Bernoulli shifts on two symbols have a factor that is equivalent to an independent and identically distributed system and prove that there are type-III:1 Bernoulli shifts of every possible ergodic index. The latter implies that the classification of type III Bernoulli shifts according to metric isomorphism is more subtle than its classical counterpart (Ornstein theory).
Monadic stability and growth rates of ω-categorical structures.
Abstract: We will present the following work by Samuel Braunfeld.
For M ω-categorical and stable, we investigate the growth rate of M, i.e. the number of orbits of Aut(M) on n-sets, or equivalently the number of n-substructures of M after performing quantifier elimination. We show that monadic stability corresponds to a gap in the spectrum of growth rates, from slower than exponential to faster
Charged domain walls are a type of transition layers in thin ferromagnetic films which appear due to global topological constraints. The underlying micromagnetic energy is determined by a competition between a diffuse interface energy and the long-range magnetostatic interaction. The underlying model is non-convex and vectorial. In the macroscopic limit we show that the energy Γ-converges to a limit model where jump discontinuities of the magnetization are penalized anisotropically. In particular, we identify a supercritical regime which allows for tangential variation of the domain walls.
Repeats every week every Sunday until Sat Oct 24 2020 .
4:00pm to 6:00pm
Abstract: We will discuss stability conditions on triangulated categories following the work of Douglas and Bridgeland. Concrete examples of stability conditions will be given from symplectic and algebraic geometry, which will also illustrate mirror symmetry. An effort will be made to give a gentle introduction to the relevant background material from category theory, symplectic geometry and algebraic geometry.
Repeats every week every Sunday until Sun Jan 17 2021 .
11:00am to 1:00pm
11:00am to 1:00pm
11:00am to 1:00pm
11:00am to 1:00pm
11:00am to 1:00pm
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11:00am to 1:00pm
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11:00am to 1:00pm
11:00am to 1:00pm
Abstract: A fundamental lemma is an identity relating p-adic integrals on two different groups. These pretty identities fit into a larger story of trace formulas and special values of L-functions. Our goal is to present recent work of Beuzart-Plessis on the Jacquet-Rallis fundamental lemma, comparing integrals on GL(n) and U(n).
Repeats every week every Sunday until Sun Jan 17 2021 .
2:00pm to 4:00pm
2:00pm to 4:00pm
2:00pm to 4:00pm
2:00pm to 4:00pm
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2:00pm to 4:00pm
Abstract: Given a smooth and proper curve X and a reductive group G one can consider the stack Bun_{G,X} of principal G-bundles on X. This stack has an important role in Algebraic Geometry and Representation Theory especially with regard to the Langlands program. We shall study the geometry of Bun_{G,X} and the category D(G,X) of constructible sheaves on Bun_{G,X}. We shall be especially interested in the subcategory D_{nil}(G,X) of sheaves with nilpotent singular support.
