| Talk_id | Date | Speaker | Title |
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26849
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Tuesday 9/1
4:00 PM
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Dmitry Chelkak, École Normale Supérieure
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Bipartite dimer model and minimal surfaces in the Minkowski space
- Dmitry Chelkak, École Normale Supérieure
- Bipartite dimer model and minimal surfaces in the Minkowski space
- 09/01/2020
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
We discuss a new approach to the convergence of height fluctuations in the bipartite dimer model considered on big planar graphs. This viewpoint is based upon special embeddings of weighted planar graphs into the complex plane known under the name Coulomb gauges or, equivalently, t-embeddings. The long-term motivation comes from trying to understand fluctuations on irregular graphs, notably on random planar maps equipped with the dimer (or, similarly, the critical Ising) model.
When the dimer model is considered on subgraphs of refining lattices, a classical conjecture due to Kenyon--Okounkov predicts the convergence of fluctuations to the Gaussian Free Field in a certain conformal structure. However, the latter is defined via a lattice-dependent entropy functional, which makes the analysis of irregular graphs highly problematic. To overcome this difficulty, we introduce a notion of 'perfect t-embeddings' of abstract weighted bipartite graphs and develop new discrete complex analysis techniques to handle correlation functions of the dimer model on t-embeddings. Though in full generality the existence of perfect embeddings remains an open question, we prove that - at least in some concrete cases - they reveal the relevant conformal structure in a lattice-independent way: as that of a related Lorentz-minimal surface in the Minkowski space.
Based upon joint works with Benoît Laslier, Sanjay Ramassamy and Marianna Russkikh.
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26855
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Tuesday 9/15
4:00 PM
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Martin Hairer, Imperial College London
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Taming infinities
- Martin Hairer, Imperial College London
- Taming infinities
- 09/15/2020
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! What's worse, this doesn't just happen for some exotic theories, but in the standard theories describing some of the most fundamental aspects of nature. Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will dip our toes into some of the conceptual and mathematical aspects of these techniques and we will see how they have recently been used in probability theory to study equations whose meaning was not even clear until now.
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26856
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Tuesday 9/22
4:00 PM
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John Lesieutre, Penn State University
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Polynomial interpolation is harder than it sounds
- John Lesieutre, Penn State University
- Polynomial interpolation is harder than it sounds
- 09/22/2020
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
Suppose that $(x_1,y_1),\ldots,(x_r,y_r)$ is a set of points in the plane. Given a degree $d$ and multiplicities $m_i$, does there a nonzero polynomial in two variables of degree at most $d$ which vanishes to order at least $m_i$ at $(x_i,y_i)$? What is the dimension of the space of such polynomials, and how does it vary with the parameters? I will explain some of the basic results and conjectures and show how this problem is connected to some questions of current interest in algebraic geometry.
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26869
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Tuesday 9/29
4:00 PM
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Ruixiang Zhang, IAS
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Kakeya type problems and analysis
- Ruixiang Zhang, IAS
- Kakeya type problems and analysis
- 09/29/2020
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
Informally, Kakeya type problems ask whether tubes with different positions and directions can overlap a lot. One usually expects the answer to be no in an appropriate sense. Thanks to the uncertainty principle, such a quantified non-overlapping theorem would often see powerful applications in analysis problems that have Fourier aspects. Perhaps the most well-known Kakeya type problem is the Kakeya conjecture. It remains widely open in $\Bbb{R}^n (n>2)$ as of today. Nevertheless, in the recent few decades people have been able to prove new Kakeya type theorems that led to improvements or complete solutions to analysis problems that appeared out of reach before. I will give an introduction to Kakeya type problems/theorems and analysis problems that see their applications. Potentially reporting some recent progress joint with Du, Guo, Guth, Hickman, Iosevich, Ou, Rogers, Wang and Wilson.
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26870
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Tuesday 10/6
4:00 PM
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Colin McLarty, Case Western Reserve University
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Grothendieck's personal idea of a topos as a space
- Colin McLarty, Case Western Reserve University
- Grothendieck's personal idea of a topos as a space
- 10/06/2020
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
In 33 hours of tape recordings in 1973 Grothendieck described his view of topos beyond what is in the collective volume Théorie des topos et cohomologie étale (SGA 4). In particular, this shows how Grothendieck got his idea of a "generalized topological space" simultaneously with what became etale cohomology during a 1958 talk by Jean-Pierre Serre.
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26871
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Tuesday 10/20
4:00 PM
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Alexander Volberg, Michigan State University
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Metric properties of Banach spaces, Enflo's problem, Pisier's inequality and quantum random variables
- Alexander Volberg, Michigan State University
- Metric properties of Banach spaces, Enflo's problem, Pisier's inequality and quantum random variables
- 10/20/2020
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure.
In the joint paper with Paata Ivanisvili and Ramon Van Handel we prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier's inequality on the discrete cube, which, in its turn, is based on a certain formula that we used before in improving the constants in the scalar Poincaré inequality on the Hamming cube. I will also show several extensions of Pisier's inequality with ultimate assumptions on a Banach space structure.
Some of our results use approach via quantum random variables.
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