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.

Fuchsian groups are discrete subgroups of isometries of the hyperbolic plane. This thesis will primarily work with the upper half-plane model, though we will provide an example in the disk model. We will define Fuchsian groups and examine their properties geometrically and algebraically. We will also discuss the relationships between fundamental regions, Dirichlet regions and Ford regions. The goal is to see how a Ford region can be constructed with isometric circles.

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One of the methods used for the construction of the classical Alexander horned sphere leads naturally to generalization to horned spheres of higher order. Let M2, denote the Alexander horned sphere. This is a 2-horned sphere of order 2. Denote by M 3 and M4, two 2-horned spheres of orders 3 and 4, respectively, constructed by such a generalization. The fundamental groups of the complements of M2, M3, and M4 are derived, and representations of these groups onto the Alternating Group, A5, are found. The form of the presentations of these fundamental groups leads to a more general class of groups, denoted by Gk, k ≥ 2. A set of homomorphisms ϴkl : Gk, k ≥ l ≥ 2 is found, which has a clear geometric meaning as applied to the groups G2, G3, and G4. Two theorems relating to direct systems of non-abelian groups are proved and applied to the groups Gk. The implication of these theorems is that the groups Gk, k≥2 are all free groups of countably infinite rank and that the embeddings of M2, M3, and M4 in E3 cannot be distinguished by means of fundamental groups. *33 pages)

We begin by recalling the notion of a coarse space as defined by John Roe. We show that metrizability of coarse spaces is a coarse invariant. The concepts of bounded geometry, asymptotic dimension, and Guoliang Yu's Property A are investigated in the setting of coarse spaces. In particular, we show that bounded geometry is a coarse invariant, and we give a proof that finite asymptotic dimension implies Property A in this general setting. The notion of a metric approximation is introduced, and a characterization theorem is proved regarding bounded geometry. Chapter 7 presents a discussion of coarse structures on the minimal uncountable ordinal. We show that it is a nonmetrizable coarse space not of bounded geometry. Moreover, we show that this space has asymptotic dimension 0; hence, it has Property A.Finally, Chapter 8 regards coarse structures on products of coarse spaces. All of the previous concepts above are considered with regard to 3 different coarse structures analogous to the 3 different topologies on products in topology. In particular, we see that an arbitrary product of spaces with any of the 3 coarse structures with asymptotic dimension 0 has asymptotic dimension 0.

The topic of this thesis is the study of moduli spaces of zero-dimensional geometricobjects. The thesis consists of three articles each focusing on a particular moduli space.The first article concerns the Hilbert scheme Hilb(X). This moduli space parametrizesclosed subschemes of a fixed ambient scheme X. It has been known implicitly for sometime that the Hilbert scheme does not behave well when the scheme X is not separated.The article shows that the separation hypothesis is necessary in the sense thatthe component Hilb1(X) of Hilb(X) parametrizing subschemes of dimension zero andlength 1 does not exist if X is not separated.Article number two deals with the Chow scheme Chow 0,n(X) parametrizing zerodimensionaleffective cycles of length n on the given scheme X. There is a relatedconstruction, the Symmetric product Symn(X), defined as the quotient of the n-foldproduct X ×. . .×X of X by the natural action of the symmetric group Sn permutingthe factors. There is a canonical map Symn(X) " Chow0,n(X) that, set-theoretically,maps a tuple (x1, . . . , xn) to the cycle!nk=1 xk. In many cases this canonical map is anisomorphism. We explore in this paper some examples where it is not an isomorphism.This will also lead to some results concerning the question whether the symmetricproduct commutes with base change.The third article is related to the Fulton-MacPherson compactification of the configurationspace of points. Here we begin by considering the configuration space F(X, n)parametrizing n-tuples of distinct ordered points on a smooth scheme X. The schemeF(X, n) has a compactification X[n] which is obtained from the product Xn by a sequenceof blowups. Thus X[n] is itself not defined as a moduli space, but the pointson the boundary of X[n] may be interpreted as geometric objects called stable degenerations.It is then natural to ask if X[n] can be defined as a moduli space of stabledegenerations instead of as a blowup. In the third article we begin work towards ananswer to this question in the case where X = P2. We define a very general modulistack Xpv2 parametrizing projective schemes whose structure sheaf has vanishing secondcohomology. We then use Artin’s criteria to show that this stack is algebraic. Onemay define a stack SDX,n of stable degenerations of X and the goal is then to provealgebraicity of the stack SDX,n by using Xpv2. / QC 20100729

This thesis is divided into three part, the first part concerns metric spaces and specically length spaces where the existence of shortest path between points is the main focus. In the second part, an example of a length space, the Riemannian geometry will be given. Here both a classical approach to Riemannian geometry will be given together with specic results when considered as a metric space. In the third part, the Finsler geometry will be examined both with a classical approach and trying to deal with it as a metric space.

The results of this thesis concern folding of one-dimensional objects in two dimensions: planar linkages. More precisely, a planar linkage consists of a collection of rigid bars (line segments) connected at their endpoints. Foldings of such a linkage must preserve the connections at endpoints, preserve the bar lengths, and (in our context) prevent bars from crossing. The main result of this thesis is that a planar linkage forming a collection of polygonal arcs and cycles can be folded so that all outermost arcs (not enclosed by other cycles) become straight and all outermost cycles become convex. A complementary result of this thesis is that once a cycle becomes convex, it can be folded into any other convex cycle with the same counterclockwise sequence of bar lengths. Together, these results show that the configuration space of all possible foldings of a planar arc or cycle linkage is connected. These results fall into the broader context of folding and unfolding <I>k</I>-dimensional objects in <i>n</i>-dimensional space, <I>k</I> less than or equal to <I>n</I>. Another contribution of this thesis is a survey of research in this field. The survey revolves around three principal aspects that have received extensive study: linkages in arbitrary dimensions (folding one-dimensional objects in two or more dimensions, including protein folding), paper folding (normally, folding two-dimensional objects in three dimensions), and folding and unfolding polyhedra (two-dimensional objects embedded in three-dimensional space).

In this thesis, we study local mixture models with a Hilbert space structure. First, we consider the fibre bundle structure of local mixture models in a Hilbert space. Next, the spectral decomposition is introduced in order to construct local mixture models. We analyze the approximation error asymptotically in the Hilbert space. After that, we will discuss the convexity structure of local mixture models. There are two forms of convexity conditions to consider, first due to positivity in the $-1$-affine structure and the second by points having to lie inside the convex hull of a parametric family. It is shown that the set of mixture densities is located inside the intersection of the sets defined by these two convexities. Finally, we discuss the impact of the approximation error in the Hilbert space when the domain of mixing variable changes.

The results of this thesis concern folding of one-dimensional objects in two dimensions: planar linkages. More precisely, a planar linkage consists of a collection of rigid bars (line segments) connected at their endpoints. Foldings of such a linkage must preserve the connections at endpoints, preserve the bar lengths, and (in our context) prevent bars from crossing. The main result of this thesis is that a planar linkage forming a collection of polygonal arcs and cycles can be folded so that all outermost arcs (not enclosed by other cycles) become straight and all outermost cycles become convex. A complementary result of this thesis is that once a cycle becomes convex, it can be folded into any other convex cycle with the same counterclockwise sequence of bar lengths. Together, these results show that the configuration space of all possible foldings of a planar arc or cycle linkage is connected. These results fall into the broader context of folding and unfolding <I>k</I>-dimensional objects in <i>n</i>-dimensional space, <I>k</I> less than or equal to <I>n</I>. Another contribution of this thesis is a survey of research in this field. The survey revolves around three principal aspects that have received extensive study: linkages in arbitrary dimensions (folding one-dimensional objects in two or more dimensions, including protein folding), paper folding (normally, folding two-dimensional objects in three dimensions), and folding and unfolding polyhedra (two-dimensional objects embedded in three-dimensional space).