Springer theory for reductive algebraic groups plays an important role in determining irreducible characters of finite groups of Lie type. We discuss its generalisation to the setting of graded Lie algebras. We explain how level-rank dualities arise from unipotent irreducible characters and their connections with the graded Springer theory. If time permits, we discuss a conjectural realisation of these dualities using affine Springer fibers.
In many scientific computing and machine learning problems, we expend the majority of our computational resources in solving large-scale linear algebra problems, typically linear systems Ax = b, eigenvalue problems Ax = λx, or the singular value decomposition A = USV'. Numerical linear algebra (NLA) is a research field that attempts to devise practical algorithms for solving these problems. Broadly, methods in NLA can be divided into three categories: direct, iterative, and randomized. In this talk I will give a whistle-stop tour of these classes of methods, highlighting the incredible robustness of classical (direct) methods, and the exciting speed and advances in randomized methods.
In this talk, we explore the expected topology (measured via homology) and local geometry of two different models of random subcomplexes of the regular cubical grid: percolation clusters, and the Eden Cell Growth model. We will also compare the expected topology that these average structures exhibit with the topology of the extremal structures that it is possible to obtain in the entire set of these cubical complexes. You can look at some of these random structures here (https://skfb.ly/6VINC) and start making some guesses about their topological behavior.
We describe a couple of centrality measures on graphs that can be obtained from certain Markov chain models associated to them, and their computation with methods taken from numerical linear algebra. The Kemeny constant is a quantity that measures the connectedness of a Markov chain by studying certain properties of the random walk associated to it. The variation in the Kemeny constant can be used to identify edges whose removal would alter the connectivity of a network; this is useful information, for instance, in planning urban and regional road networks. For problems based on "spreading" on a graph, such as news propagation and infectious disease modelling, instead models based on a single random walker fall short: they are unable to capture characteristics of the model such as the time to saturation. We study this phenomenon, and propose an alternative way to treat computationally the full model, which can be interpreted as another Markov chain with an exponential number of states. The resulting metric can once again be interpreted as a measure of the centrality of the vertices / agents in the network in the propagation.
Forced displacement is a global problem that requires planning for the relocation and integration of displaced people. Most studies focus on conflict-driven forced displacement, and hence the refugee resettlement problem. These studies generally focus on short-term planning and assume that demand within the fixed time interval is given. However, forced displacement, including environmental displacement as well as conflict-driven displacement, is not a one-time event. On the contrary, it is an ongoing and long-term process with dynamic parameters. We are interested in the long-term displacement problem, especially for climate-driven cases in which people will be forced to leave uninhabitable regions in to escape slow-onset climate change impacts such as water stress, crop failure, and sea level rise. To reflect the long-term planning requirements of the climate-driven displacement problem in the parameters and the model, we propose a two-stage stochastic program where demand uncertainty is represented with various demand scenarios, demand and capacity are managed dynamically, and integration outcomes and related costs are optimized.
Euler didn't conceive his notorious theorem as an efficient manner to cipher messages, but two centuries later, his result backs the omnipresent RSA cryptosystem. Neither Abel, nor Poincaré were specially concerned on how to communicate messages in a secure manner when they tackled elliptic integrals and still, elliptic curves are at the basis of the SSL and TLS Internet protocols. With the frantic development of quantum computing (IBM announced Osprey three months ago, a 433 qbits processor, beating its already commercialised 21 qbits QSystem1 ), we must set ourselves en guard as soon as possible. This is the reason why the NIST launched a public contest to standardise postquantum cryptographic primitives in 2017, recently resolved in July 2022. However, the mathematical tools backing these new proposals are, if no more complicated, at least more challenging than the previous ones. The goal of my talk is to mention my research lines developed since 2011 until now, a journey which started in Barcelona with such ethereal topics as Shimura curves, modularity and p-adic L-functions and led me to questions as designing efficient codes, crypto-analysing postquantum primitives while still working in more mystic maths in my free time.
Most real-world optimization problems contain parameters which are not known at the time a decision is to be made. In robust optimization one specifies the uncertainty in a scenario set and tries to hedge against the worst case. Classical robust optimization aims at finding a solution which is best in the worst-case scenario. It is a well-studied concept but it is known to be very conservative: A robust solution comes with a high price in its nominal objective function value. This motivated researchers to introduce less conservative robustness concepts in the last decade. Moreover, many real-world problems involve not only one, but multiple criteria. While robust single-objective optimization has been investigated for 25 years, robust multi-objective optimization is a new field in which already the definition of "robust" is a challenge. In the talk, several robustness concepts will be discussed and illustrated at applications from public transport.
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