Department of Mathematics Colloquium 2020-2021

September 8

Pablo Stinga (Iowa State University)

Title: Nonlocal fractional Monge—Ampère equations

Abstract: The Monge—Ampère equation is one of the most important partial differential equations as it plays a crucial role in several areas of analysis, geometry, and applied mathematics. We will present recent developments on the open question of finding an appropriate notion of a nonlocal, fractional Monge—Ampère equation.

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September 15

Rana Parshad (Iowa State University)

Title: Some recent results on the control of invasive species

Abstract: The spread and control of invasive species/pests is a major societal, ecological and economic issue. In this talk I will discuss various results that focus on methods in biological control, such as the use of introduced predators to keep pest populations in check. I will also briefly delve into two species competitive systems, with some applications to refuge design.

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September 22

Jonas T. Hartwig (Iowa State University)

Title: Gelfand-Tsetlin representations

Abstract: In 1925, Cartan obtained a classification of finite-dimensional irreducible representations of an arbitrary finite-dimensional complex reductive Lie algebra. We will review recent progress on the problem of classifying not necessarily finite-dimensional irreducible representations of Lie algebras and related algebras. Focus will be on the general linear Lie algebra (consisting of all square matrices) which is the most studied case, and for which notions can be made concrete. Towards the end we mention ongoing generalizations of the theory to quantum groups and beyond.

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September 29

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October 6

Maria Chudnovsky (Princeton University)

Title: Induced subgraphs and tree decompositions

Abstract: Tree decompositions are a powerful tool in structural graph
theory, that is  traditionally used in the context of forbidden graph minors.
Connecting tree decompositions and forbidden induced subgraphs has so far
remained out of reach. Recently we obtained several results in this direction;
the talk will be a survey of these results.

Link: 

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October 13

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October 20

Mahamadi Warma (George Mason University)

Title: CONTROLLABILITY PROPERTY OF MULTI-D FRACTIONAL HEAT EQUATIONS

Abstract: We analyze the controllability of the non-local multi-dimensional heat equation involving the fractional Laplace operator in a bounded open set of R^N (N ≥ 1). We prove that the system is null- controllable in any time horizon T > 0, when the control acts on a neighborhood of the boundary, in the range of exponents s ∈ (1/2, 1). The proof employs and combines several tools and ingredients. We start by considering the corresponding fractional wave equation and, using a fractional version of Pohozaev’s identity, we establish a partial null controllability result of projections into low frequency eigenmodes. This is achieved when the control is active in a neighborhood of the boundary ∂Ω, in a time horizon that grows as the number of controlled frequencies increases. We then apply transmutation techniques to transfer this result into the parabolic setting, thus obtaining a frequency-dependent null controllability result for the fractional heat equation, for s ∈ [1/4, 1). Finally, we employ an iterative strategy to deal with the high- frequency components of the fractional heat equation, exploiting its dissipativity properties, and prove the full null controllability result for finite energy solutions for all s ∈ (1/2, 1).

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October 27

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November 3

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November 10

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November 17

Kristin Lauter (Microsoft Corporation)

Title: Private AI—Machine Learning on Encrypted Data

Abstract: As the world adopts Artificial Intelligence, the privacy risks are many.  AI can improve our lives, but may leak or misuse our private data.  Private AI is based on Homomorphic Encryption (HE), a new encryption paradigm which allows the cloud to operate on private data in encrypted form, without ever decrypting it, enabling private training and private prediction.  Our 2016 ICML CryptoNets paper showed for the first time that it was possible to evaluate neural nets on homomorphically encrypted data, and opened new research directions combining machine learning and cryptography. The security of Homomorphic Encryption is based on hard problems in mathematics involving lattices, a candidate for post-quantum cryptography.  Cyclotomic number rings are a good source of the lattices used in practice, which leads to new interesting problems in number theory.  This talk will explain Homomorphic Encryption, Private AI, and show demos of HE in action.

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