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An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to the receptive fields of place cells. Lastly we describe several purely geometric results related to neural codes.

Essay 1: The Determinants of the Convexity in the Flow-Performance Relationship There is substantial evidence that the flow-performance relationship of mutual funds is convex. In this work, I empirically investigate the determinants of such convexity. In particular, I study the impact that fund fees (marketing and non-marketing fees) and the uncertainty related to the replacement option of fund production factors (managerial ability and investment strategy) have on the convexity of the flow-performance relationship. I also analyze the impact of the priors about managerial ability and idiosyncratic risk on such convexity. The evidence suggests that marketing fees are positively related to the convexity of the flow-performance relationship. In addition, non-marketing fees do not have a negative impact on this convexity. The evidence associated with the value of the managerial and investment replacement option is mixed. Consistent with investment restrictions being relevant in explaining investors’ allocation decisions, sector, index, and hedge funds exhibit lower convexity in their flow-performance relationship than respectively diversified, non-index, and mutual funds. Finally, the dispersion of the priors about managerial ability and idiosyncratic risk are positively related to the convexity in the flow-performance relationship. Essay 2: Implicit Incentives and Tournament Behavior in the Mutual Fund Industry The convexity of the flow-performance relationship in the mutual fund industry produces implicit incentives for mutual fund managers to modify risk-taking behavior as a function of their prior performance (Brown, Harlow, and Starks (1996)). Rather than focusing only on tournament behavior, I investigate the link between the determinants of the convexity in the flow-performance relationship and the inter-temporal risk-shifting behavior of a fund’s manager. Hence, I examine how the sources of implicit compensation incentives shape tournament behavior. The evidence indicates that the relationship between changes in managers’ relative risk choices and mid-year performance is non-monotonic (U-shaped). Higher convexity in the flow-performance relationship increases the convexity of the U-shaped tournament behavior. For extreme performers, an increase in the convexity of the flow-performance relationship directly translates into higher risk-taking incentives. For average performers, the incentive to increase risk produced by the convexity in the compensation schedule is counterbalanced by an increase in the risk of termination. I find that the uncertainty about managerial ability, marketing efforts, and the size of family complexes affect the convexity of the U-shaped tournament behavior. These results are robust to the consideration of termination risks due to funds’ organizational form, investment objectives, or past performance. My results suggest that the risk strategies of younger funds, funds spending more on marketing, funds belonging to smaller families, sector funds, funds that are team-managed, or funds that have experienced consistent poor performance are more sensitive to intermediate performance.

In the early 1980s a landmark result was obtained by Atiyah and independently Guillemin and Sternberg: the image of the momentum map for a torus action on a compact symplectic manifold is a convex polyhedron. Atiyah's proof makes use of the fact that level sets of the momentum map are connected. These proofs work in the setting of finite-dimensional compact symplectic manifolds. One can ask how these results generalize. A well-known example of an infinite-dimensional symplectic manifold with a finite-dimensional torus action is the based loop group. Atiyah and Pressley proved convexity for this example, but not connectedness of level sets. A proof of connectedness of level sets for the based loop group was provided by Harada, Holm, Jeffrey and Mare in 2006. In this thesis we study Hilbert manifolds equipped with a strong symplectic structure and a finite-dimensional group action preserving the strong symplectic structure. We prove connectedness of regular generic level sets of the momentum map. We use this to prove convexity of the image of the momentum map.

In the early 1980s a landmark result was obtained by Atiyah and independently Guillemin and Sternberg: the image of the momentum map for a torus action on a compact symplectic manifold is a convex polyhedron. Atiyah's proof makes use of the fact that level sets of the momentum map are connected. These proofs work in the setting of finite-dimensional compact symplectic manifolds. One can ask how these results generalize. A well-known example of an infinite-dimensional symplectic manifold with a finite-dimensional torus action is the based loop group. Atiyah and Pressley proved convexity for this example, but not connectedness of level sets. A proof of connectedness of level sets for the based loop group was provided by Harada, Holm, Jeffrey and Mare in 2006. In this thesis we study Hilbert manifolds equipped with a strong symplectic structure and a finite-dimensional group action preserving the strong symplectic structure. We prove connectedness of regular generic level sets of the momentum map. We use this to prove convexity of the image of the momentum map.

This thesis deals with the topic of min-max formulations of robust model predictive control problems. The sets involved in guaranteeing robust feasibility of the min-max program in the presence of state constraints are of particular interest, and expanding the applicability of well understood solvers of linearly constrained quadratic min-max programs is the main focus. To this end, a generalisation for the set of uncertainty is considered: instead of fixed bounds on the uncertainty, state- and input-dependent bounds are used. To deal with state- and input dependent constraint sets a framework for a particular class of set-valued maps is utilised, namely parametrically convex set-valued maps. Relevant properties and operations are developed to accommodate parametrically convex set-valued maps in the context of robust model predictive control. A quintessential part of this work is the study of fundamental properties of piecewise polyhedral set-valued maps which are parametrically convex, we show that one particular property is that their combinatorial structure is constant. The study of polytopic maps with a rigid combinatorial structure allows the use of an optimisation based approach of robustifying constrained control problems with probabilistic constraints. Auxiliary polytopic constraint sets, used to replace probabilistic constraints by deterministic ones, can be optimised to minimise the conservatism introduced while guaranteeing constraint satisfaction of the original probabilistic constraint. We furthermore study the behaviour of the maximal robust positive invariant set for the case of scaled uncertainty and show that this set is continuously polytopic up to a critical scaling factor, which we can approximate a-priori with an arbitrary degree of accuracy. Relevant theoretical statements are developed, discussed and illustrated with examples.

Maximum likelihood approach for independent but not identically distributed observations is studied. In the first part of the thesis, conditions for consistency and asymptotic normality of the maximum likelihood estimates for this case are stated. Uniform integrability has a major role in proving the desired properties. K-sample problem serves as an example for using the described method. The second part is focused on estimates obtained by minimizing convex functions. Convexity is a key for showing the consistency and asymptotic normality of the estimates in this case. The results can be used for maximum likelihood when observations with logconcave densities are involved. Finally, normal linear model, logistic regression and Poisson regression examples are provided to present the application of the method.

A spatial logic is any formal language with geometric interpretation. Research on region-based spatial logics, where variables are set to range over certain subsets of geometric space, have been investigated recently within the qualitative spatial reasoning paradigm in AI. We axiomatised the theory of (ROQ(R 2), conv, ≤) , where ROQ(R 2) is the set of regular open rational polygons of the real plane; conv is the convexity property and ≤ is the inclusion relation. We proved soundness and completeness theorems. We also proved several expressiveness results. Additionally, we provide a historical and philosophical overview of the topic and present contemporary results relating to affine spatial logics.

In this thesis, we consider the planning problem for first-order mean-field games (MFG). These games degenerate into optimal transport when there is no coupling between players. Our aim is to extend the concept of displacement convexity from optimal transport to MFGs. This extension gives new estimates for solutions of MFGs. First, we introduce the Monge-Kantorovich problem and examine related results on rearrangement maps. Next, we present the concept of displacement convexity. Then, we derive first-order MFGs, which are given by a system of a Hamilton-Jacobi equation coupled with a transport equation. Finally, we identify a large class of functions, that depend on solutions of MFGs, which are convex in time. Among these, we find several norms. This convexity gives bounds for the density of solutions of the planning problem.

Let M_k^2 be the complete, simply connected, Riemannian 2-manifold of constant curvature k ± 0. Let E be a closed, simply connected subspace of M_k^2 with the property that every two points in E are connected by a rectifi able path in E. We show that E is CAT(k) under the induced path metric.