12 research outputs found

### Fredholm theory for elliptic operators on quasi-asymptotically conical spaces

We consider the mapping properties of generalized Laplace-type operators
${\mathcal L} = \nabla^* \nabla + {\mathcal R}$ on the class of
quasi-asymptotically conical (QAC) spaces, which provide a Riemannian
generalization of the QALE manifolds considered by Joyce. Our main result gives
conditions under which such operators are Fredholm when between certain
weighted Sobolev or weighted H\"older spaces. These are generalizations of
well-known theorems in the asymptotically conical (or asymptotically Euclidean)
setting, and also sharpen and extend corresponding theorems by Joyce. The
methods here are based on heat kernel estimates originating from old ideas of
Moser and Nash, as developed further by Grigor'yan and Saloff-Coste. As
demonstrated by Joyce's work, the QAC spaces here contain many examples of
gravitational instantons, and this work is motivated by various applications to
manifolds with special holonomy

### Flexible Memory Networks

Networks of neurons in some brain areas are flexible enough to encode new
memories quickly. Using a standard firing rate model of recurrent networks, we
develop a theory of flexible memory networks. Our main results characterize
networks having the maximal number of flexible memory patterns, given a
constraint graph on the network's connectivity matrix. Modulo a mild
topological condition, we find a close connection between maximally flexible
networks and rank 1 matrices. The topological condition is H_1(X;Z)=0, where X
is the clique complex associated to the network's constraint graph; this
condition is generically satisfied for large random networks that are not
overly sparse. In order to prove our main results, we develop some
matrix-theoretic tools and present them in a self-contained section independent
of the neuroscience context.Comment: Accepted to Bulletin of Mathematical Biology, 11 July 201

### Friendly giant meets pointlike instantons? On a new conjecture by John McKay

A new conjecture due to John McKay claims that there exists a link between (1) the conjugacy classes of the Monster sporadic group and its offspring, and (2) the Picard groups of bases in certain elliptically fibered Calabi-Yau threefolds. These Calabi-Yau spaces arise as F-theory duals of point-like instantons on ADE type quotient singularities. We believe that this conjecture, may it be true or false, connects the Monster with a fascinating area of mathematical physics which is yet to be fully explored and exploited by mathematicians. This article aims to clarify the statement of McKay's conjecture and to embed it into the mathematical context of heterotic/F-theory string-string dualities

### Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three

For $G$ a finite subgroup of ${\rm SL}(3,{\mathbb C})$ acting freely on
${\mathbb C}^3{\setminus} \{0\}$ a crepant resolution of the Calabi-Yau
orbifold ${\mathbb C}^3\!/G$ always exists and has the geometry of an ALE
non-compact manifold. We show that the tautological bundles on these crepant
resolutions admit rigid Hermitian-Yang-Mills connections. For this we use
analytical information extracted from the derived category McKay correspondence
of Bridgeland, King, and Reid [J. Amer. Math. Soc. 14 (2001), 535-554]. As a
consequence we rederive multiplicative cohomological identities on the crepant
resolution using the Atiyah-Patodi-Singer index theorem. These results are
dimension three analogues of Kronheimer and Nakajima's results [Math. Ann. 288
(1990), 263-307] in dimension two

### Diversity of emergent dynamics in competitive threshold-linear networks: a preliminary report

Threshold-linear networks consist of simple units interacting in the presence
of a threshold nonlinearity. Competitive threshold-linear networks have long
been known to exhibit multistability, where the activity of the network settles
into one of potentially many steady states. In this work, we find conditions
that guarantee the absence of steady states, while maintaining bounded
activity. These conditions lead us to define a combinatorial family of
competitive threshold-linear networks, parametrized by a simple directed graph.
By exploring this family, we discover that threshold-linear networks are
capable of displaying a surprisingly rich variety of nonlinear dynamics,
including limit cycles, quasiperiodic attractors, and chaos. In particular,
several types of nonlinear behaviors can co-exist in the same network. Our
mathematical results also enable us to engineer networks with multiple dynamic
patterns. Taken together, these theoretical and computational findings suggest
that threshold-linear networks may be a valuable tool for understanding the
relationship between network connectivity and emergent dynamics.Comment: 12 pages, 9 figures. Preliminary repor

### QAC spaces

Non UBCUnreviewedAuthor affiliation: University of FreiburgPostdoctora

### Encoding binary neural codes in networks of threshold-linear neurons

Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship between network connectivity and encoded patterns is still poorly understood. Here we consider this problem for networks of threshold-linear neurons whose computational function is to learn and store a set of binary patterns (e.g., a neural code) as “permitted sets” of the network. We introduce a simple Encoding Rule that selectively turns “on” synapses between neurons that co-appear in one or more patterns. The rule uses synapses that are binary, in the sense of having only two states (“on” or “off ), but also heterogeneous, with weights drawn from an underlying synaptic strength matrix S. Our main results precisely describe the stored patterns that result from the Encoding Rule - including unintended “spurious states - and give an explicit characterization of the dependence on S. In particular, we find that binary patterns are successfully stored in these networks when the excitatory connections between neurons are geometrically balanced - i.e., they satisfy a set of geometric constraints. Furthermore, we find that certain types of neural codes are natural in the context of these networks, meaning that the full code can be accurately learned from a highly undersampled set of patterns. Interestingly, many commonly observed neural codes in cortical and hippocampal areas are natural in this sense. As an application, we construct networks that encode hippocampal place field codes nearly exactly, following presentation of only a small fraction of patterns. To obtain our results, we prove new theorems using classical ideas from convex and distance geometry, such as Cayley-Menger determinants, revealing a novel connection between these areas of mathematics and coding properties of neural networks

### Eta-invariants and Molien series for unimodular group

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.Includes bibliographical references (p. 57-58).We look at the singularity Cn/[Gamma], for [Gamma] finite subgroup of SU(n), from two perspectives. From a geometrical point of view, Cn/[Gamma] is an orbifold with boundary S2n-1/[Gamma]. We define and compute the corresponding orbifold [eta]-invariant. From an algebraic point of view, we look at the algebraic variety Cn/[Gamma] and we analyze the associated Molien series. The main result is formula which relates the two notions: [eta]-invariant and Molien series. Along the way computations of the spectrum of the Dirac operator on the sphere are performed.by Anda Degeratu.Ph.D