Mathematics (Faculty of)http://hdl.handle.net/10012/99242024-07-18T08:54:41Z2024-07-18T08:54:41ZHorospherical geometry: combinatorial algebraic stacks and approximating rational pointsMonahan, Seanhttp://hdl.handle.net/10012/207302024-07-18T02:31:11Z2024-07-17T00:00:00ZHorospherical geometry: combinatorial algebraic stacks and approximating rational points
Monahan, Sean
The purpose of this thesis is to explore and develop several aspects of the theory of
horospherical geometry. Horospherical varieties are equipped with the action of a reductive
algebraic group such that there is an open orbit whose points are stabilized by maximal
unipotent subgroups. This includes the well-known classes of toric varieties and flag
varieties. Using this orbit structure and representation-theoretic condition on the stabilizer,
one can classify horospherical varieties using combinatorial objects called coloured fans.
We give an overview of the main features of this classification through a new, accessible
notational framework.
There are two main research themes in this thesis. The first is the development of a
combinatorial theory for horospherical stacks, vastly generalizing that for horospherical
varieties. We classify horospherical stacks using combinatorial objects called stacky coloured
fans, extending the theory of coloured fans. As part of this classification, we describe the
morphisms of horospherical stacks in terms of maps between the stacky coloured fans, we
completely describe the good moduli space of a horospherical stack, and we introduce a
special, hands-on class of horospherical stacks called coloured fantastacks.
The second major theme is using horospherical varieties to probe a conjecture in
arithmetic geometry. In 2007, McKinnon conjectured that, for a given point on a projective
variety, there is a sequence, lying on a curve, which best approximates this point. We
verify a version of this conjecture for horospherical varieties, contingent on Vojta’s Main
Conjecture, which says that there is a sequence, lying on a curve, which approximates the
given point better than any Zariski dense sequence.
2024-07-17T00:00:00ZEfficient Memory Allocator for Restricting Use-After-Free ExploitationsWang, Ruizhehttp://hdl.handle.net/10012/207282024-07-18T02:31:05Z2024-07-17T00:00:00ZEfficient Memory Allocator for Restricting Use-After-Free Exploitations
Wang, Ruizhe
Attacks on heap memory, encompassing memory overflow, double and invalid free, use-after-free (UAF), and various heap-spraying techniques are ever-increasing. Existing secure memory allocators can be generally classified as complete UAF-mitigating allocators that focus on detecting and stopping UAF attacks, type-based allocators that limit type confusion, and entropy-based allocators that provide statistical defenses against virtually all of these attack vectors. In this thesis, I introduce two novel approaches, SEMalloc and S2Malloc, for type- and entropy-based allocation, respectively. Both allocators are designed to restrict, but not to fully eliminate, the attacker's ability, using allocation strategies. They can significantly increase the security level without introducing excessive overheads.
SEMalloc proposes a new notion of thread-, context-, and flow-sensitive 'type', SemaType, to capture the semantics and prototype a SemaType-based allocator that aims for the best trade-off amongst the impossible trinity. In SEMalloc, only heap objects allocated from the same call site and via the same function call stack can possibly share a virtual memory address, which effectively stops type-confusion attacks and make UAF vulnerabilities harder to exploit.
S2Malloc aims to enhance UAF-attempt detection without compromising other security guarantees or introducing significant overhead. We use three innovative constructs in secure allocator design: free block canaries (FBC) to detect UAF attempts, random in-block offset (RIO) to stop the attacker from accurately overwriting the victim object, and random bag layout (RBL) to impede attackers from estimating the block size based on its address.
This thesis demonstrates the importance of memory security and highlights the potential of more secure and efficient memory allocation by constraining attacker actions.
2024-07-17T00:00:00ZOptimization, model uncertainty, and testing in risk and insuranceJiao, Zhanyihttp://hdl.handle.net/10012/207182024-07-12T02:31:42Z2024-07-11T00:00:00ZOptimization, model uncertainty, and testing in risk and insurance
Jiao, Zhanyi
This thesis focuses on three important topics in quantitative risk management and actuarial science: risk optimization, risk sharing, and statistical hypothesis testing in risk.
For the risk optimization, we concentrate on risk optimization under model uncertainty where only partial information about the underlying distribution is available. One key highlight, detailed in Chapter 2, is the development of a novel formula named the reverse Expected Shortfall (ES) optimization formula. This formula is derived to better facilitate the calculation of the worst-case mean excess loss under two commonly used model uncertainty sets – moment-based and distance-based (Wasserstein) uncertainty sets. Further exploration reveals that the reverse ES optimization formula is closely related to the Fenchel-Legendre transforms, and our formulas are generalized from ES to optimized certainty equivalents, a popular class of convex risk measures. Chapter 3 considers a different approach to derive the closed-form worst-case target semi-variance by including distributional shape information, crucial for finance (symmetry) and insurance (non-negativity) applications. We demonstrate that all results are applicable to robust portfolio selection, where the closed-form formulas greatly simplify the calculations for optimal robust portfolio selections, either through explicit forms or via easily solvable optimization problems.
Risk sharing focuses on the redistribution of total risk among agents in a specific way. In contrast to the traditional risk sharing rules, Chapter 4 introduces a new risk sharing framework - anonymized risk sharing, which requires no information on preferences, identities, private operations, and realized losses from the individual agents. We establish an axiomatic theory based on four axioms of fairness and anonymity within the context of anonymized risk sharing. The development of this theory provides a solid foundation for further explorations on decentralized and digital economy including peer-to-peer (P2P) insurance, revenue sharing of digital contents and blockchain mining pools.
Hypothesis testing plays a vital role not only in statistical inference but also in risk management, particularly in the backtesting of risk measures. In Chapter 5, we address the problem of testing conditional mean and conditional variance for non-stationary data using the recent emerging concept of e-statistics. We build e-values and p-values for four types of non-parametric composite hypotheses with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality, and their combination. Using the obtained e-values and p-values, we construct tests via e-processes, also known as testing by betting, as well as some tests based on combining p-values for comparison. To demonstrate the practical application of these methodologies, empirical studies using financial data are conducted under several settings.
2024-07-11T00:00:00ZTechnology Design Recommendations Informed by Observations of Videos of Popular Musicians Teaching and Learning Songs by EarLiscio, Christopherhttp://hdl.handle.net/10012/207172024-07-12T02:31:49Z2024-07-11T00:00:00ZTechnology Design Recommendations Informed by Observations of Videos of Popular Musicians Teaching and Learning Songs by Ear
Liscio, Christopher
Instrumentalists who play popular music often learn songs by ear, using recordings in lieu of sheet music or tablature. This practice was made possible by technology that allows musicians to control playback events. Until now, researchers have not studied the human-recording interactions of musicians attempting to learn pop songs by ear. Through a pair of studies analyzing the content of online videos from YouTube, we generate hypotheses and seek a better understanding of by-ear learning from a recording. Combined with results from neuroscience studies of tonal working memory and aural imagery, our findings reveal a model of by-ear learning that highlights note-finding as a core activity. Using what we learned, we discuss opportunities for designers to create a set of novel human-recording interactions, and to provide assistive technology for those who lack the baseline skills to engage in the foundational note-finding activity.
2024-07-11T00:00:00Z