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Abstract: For a real reductive group G, the set of unipotent representations is a finite set of irreducible representations that are supposed to be “building blocks” for all unitary representations of G. Moreover, unipotent representations are expected to be indexed by nilpotent G-orbits, possibly with some additional data. There were several attempts to give a definition of a unipotent representation, most notably the set of special unipotent representations, due to Barbasch-Vogan and Arthur. However, this notion does not include some interesting unitary representations, such as a metaplectic representation. In joint work with Ivan Losev and Lucas Mason-Brown, we propose a new definition of unipotent representations of a complex semisimple Lie group that extends the set of special unipotent ones. Using these definitions we are able to parameterize the unipotent representations and prove their key properties, including some that were not known for special unipotent representations before. In this talk, I will explain the definition and the parametrization of unipotent representations. Time permitting I will sketch what we expect to be a definition of a unipotent representation of a real semisimple Lie group, and our approach to studying them. The talk is based on arXiv:2108.03453.

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Abstract: Over the last decade, algorithms and systems based on deep learning and other data-driven methods have contributed to the reemergence of Artificial Intelligence-based systems with applications in national security, defense, medicine, intelligent transportation, and many other domains. However, another AI winter may be lurking around the corner if challenges due to bias, domain shift and lack of robustness to adversarial attacks are not considered while designing the AI systems. In this talk, I will present our approach to bias mitigation and designing AI systems that are robust to domain shift and a variety of adversarial attacks.
Zoom Link for November 22nd Webinar:

https://umd.zoom.us/j/93733745978?pwd=UWVSOG5lblVvNThPS2RoVFVSWjRGZz09

Meeting ID: 937 3374 5978
Meeting Passcode: FFT

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Abstract: We discuss operators of the form:
Af(x)=\int\int b(x,y,\xi)f(y)exp[2pi*i\psi(x,y,\xi)] dyd\xi, where b is a symbol and \psi is a phase function.
The main result reads as follows. Assume A is a linear operator on L^2(R^d) of the form above, where p \in [1,2], and c is a second class Fourier integral operator slice permutation. For certain values of p_1,...,p_6d, If the kernel of A, b*exp[2pi*i\psi(x,y,\xi)] \in M(c)^{p_1,...,p_6d}, the mixed modulation space, then A belongs to the Schatten p-class.
The proof of this theorem is based on Shannon Bishop’s PhD thesis and one corollary of that thesis.

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Abstract: I'll present a few results around the general problem of understanding "uniform" constructions of linear orders. It turns out that the same flavor of question - "when are there guaranteed to be 'incomparable' levels of complexity, and where can they arise?" - leads to interesting problems in quite different topics; we'll see some local computable structure theory, some forcing and large countable ordinals, and some thoughts on counterexamples to Vaught's conjecture (and potential "near-misses" to same). This talk will not assume any computability theory background.

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Abstract: Join Zoom Meeting
https://umd.zoom.us/j/93066642791?pwd=bC9vcUQyTXUrdjdkWkJJbmgwRnd4QT09

Meeting ID: 930 6664 2791
Passcode: DLRIT

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Abstract: We will continue the march towards showing the index of stability $\alpha$ of a stable distribution satisfies $0 < \alpha \le 2$. The strategy is to show that, for any stable random variable, the norming constants in the definition of stability take the form $a_n = n^{1/\alpha}$. For this, we follow John Nolan's approach in his book Univariate Stable Distributions. Since any stable law is infinitely divisible, the Lévy-Khintchine formula specifies the form of a stable law's characteristic function. Then, using the uniqueness of the characteristic triplet in the Lévy-Khintchine formula and the definition of stability, we can argue that the index $\alpha$ must satisfy the proposed bounds, $0 < \alpha \le 2$. For this, we follow Ken-iti Sato's book Lévy Processes and Infinitely Divisible Distributions.