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January 5: Danna Zhang
Title: High dimensional testing for non-Gaussian data Abstract: High dimensional non-Gaussian data are increasingly encountered in a wide range of applications. It poses new challenges to traditional statistical tools. In this talk, we will present some recent development on methodologies and theories for the analysis of fat-tailed data as well as some high dimensional estimation and inference problems. Slides |
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January 12: David Stapleton
Title: Projective space, hypersurfaces, and algebraic geometry Abstract: 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). |
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January 19: Zhouli Xu
Title: Generalized Poincare Conjecture, Homotopy Groups of Spheres, and the Motivic Adams spectral sequence Abstract: I will introduce and discuss some recent development of a fundamental problem in topology - the classification of continuous maps between spheres up to homotopy. These mathematical objects are called homotopy groups of spheres. I will start with some geometric background - its connection to the Generalized Poincare Conjecture for example. I will then introduce some classical and new methods of doing such computations, using certain spectral sequences. If time permits, I will discuss some recent development using motivic homotopy theory, a theory that was designed to use algebraic topology to study algebraic geometry, but has now been applied successfully in the reverse direction. Old and new open problems will be mentioned along the discussion. |
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January 26: Ioana Dumitriu
Title: Random matrices, random graphs, and applications to machine learning Abstract: The last decade has seen tremendous progress in applying random matrix methods to adjacency matrices or Laplacians of random graphs, in order to understand their spectra and be able to apply the new results to algorithms in machine learning, coding theory, data science, etc. Nevertheless, many problems remain. I will present some of the most interesting tools and new results and mention some (still) open problems. |
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February 2: Xiaochuan Tian
Title: An invitation to nonlocal models Abstract: There has been a growing interest in the study of nonlocal models as more general and sometimes more realistic alternatives to the conventional PDE models. We will give an introduction to nonlocal models in this talk. In particular, we will focus on the nonlocal models with a finite range of nonlocal interactions, which serve as bridges connecting the classical PDEs, nonlocal discrete models and the fractional differential equations. This talk will cover topics including nonlocal modeling, nonlocal calculus and numerical analysis for the nonlocal models. |
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February 9: Aaron Pollack
Title: Modular forms and sums of four squares Abstract: How many ways can a positive integer be written as the sum of four squares? There is a simple formula for the number of ways, which goes back to Jacobi. I'll introduce modular forms and sketch how they provide an answer to this question. |
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February 16: Brandon Seward
Title: Bernoulli shifts and entropy theory Abstract: In ergodic theory, one often studies measure-preserving actions of countable groups on probability spaces. Bernoulli shifts are a class of such actions that are particularly simple to define, but despite several decades of study some elementary questions about them still remain open, such as how they are classified up to isomorphism. Progress in understanding Bernoulli shifts has historically gone hand-in-hand with the development of a tool known as entropy. In this talk, I will review classical concepts and results, which apply in the case where the acting group is amenable, and then I will discuss recent developments that are beginning to illuminate the case of non-amenable groups. |
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February 23: Jonathan Novak
Title: HCIZ, BGW, and other capital letters Abstract: This talk will be about a pair of related matrix integrals, the Harish-Chandra/Itzykson-Zuber integral and the Brezin-Gross-Witten integral, which play an important role in random matrix theory, representation theory, and mathematical physics. While these integrals cannot be exactly evaluated, an old conjecture says that they admit asymptotic expansions whose coefficients are themselves generating functions for some unspecified combinatorial invariants of compact Riemann surfaces (or smooth projective curves). |
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March 2: Luca Spolaor
Title: Regularity of the free-boundary for the Obstacle Problem Abstract: In this talk I will discuss the so-called Obstacle Problem, describing where it originates from, the regularity of its solutions and of their free-boundary. |
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March 9: Amir Mohammadi
Title: Dynamics on homogeneous spaces and applications Abstract: We will discuss, using explicit examples, how dynamical systems can be used to study certain problems in number theory and geometry. |