In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semiclassical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex saddle points, even when the parameters in the action are real. We find new exact complex saddles, and show that without their contribution the semi-classical expansion is in conflict with basic properties such as positive-semidefiniteness of the spectrum, and constraints of supersymmetry. Generic saddles are not only complex, but also possibly multi-valued, and even singular. This is in contrast to instanton solutions, which are real, smooth, and single-valued. The multi-valuedness of the action can be interpreted as a hidden topological angle, quantized in units of π in supersymmetric theories. The general ideas also apply to non-supersymmetric theories.

We discuss Nakamaye's Theorem and its recent extension to compact complex manifolds, together with some applications.

We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.

Most of classical theoretical ecology is based on the assumption that organisms in a community can be naturally partitioned into groups of individuals that can be treated as identical. At the same time, mounting experimental evidence from studies of microbial communities raises the intriguing question whether this intuition is an accurate description of the microbial world. This work builds on Mac Arthur’s model of competitive coexistence on multiple resources to construct a framework that does not rely on postulated existence of species as fundamental ecological variables. In one parameter regime, effective “species” with a core and accessory genome naturally appear in this model as emergent concepts. However, the same model allows a smooth transition to a highly diverse regime where the species formalism becomes inadequate. An alternative description is proposed based on the dynamical modes of population fluctuations. This approach provides a naturally hierarchical description of community dynamics which is well-defined even when the species description breaks down. The relevance of this framework for understanding the complexity of naturally observed microbial communities is discussed.

Information theory is gaining popularity as a tool to characterize performance of biological systems. However, information is commonly quantified without reference to whether or how a system could extract and use it; as a result, information-theoretic quantities are easily misinterpreted. Here we take the example of pattern-forming developmental systems which are commonly structured a cascades of sequential gene expression steps. Such a multi-tiered structure appears to constitute sub-optimal use of the positional information provided by the input morphogen because noise is added at each tier. However, the conventional theory fails to distinguish between the total information in a morphogen and information that can be usefully extracted and interpreted by downstream elements. We demonstrate that quantifying the information that is accessible to the system naturally explains the prevalence of multi-tiered network architectures as a consequence of the noise inherent to the control of gene expression. We support our argument with empirical observations from patterning along the major body axis of the fruit fly embryo. Our results exhibit the limitations of the standard information-theoretic characterization of biological signaling and illustrate how they can be resolved.

In pattern-forming developmental systems, cells commonly interpret graded input signals, known as morphogens. Morphogens often pattern tissues through cascades of sequential gene expression steps. Such a multi-tiered structure appears to constitute suboptimal use of the positional information provided by the input morphogen because noise is added at each tier. However, the conventional theory neglects the role of the format in which information is encoded. We argue that the relevant performance measure is not solely the amount of information carried by the morphogen, but the amount of information that can be accessed by the downstream network. We demonstrate that quantifying the information that is accessible to the system naturally explains the prevalence of multi-tiered network architectures as a consequence of the noise inherent to the control of gene expression. We support our argument with empirical observations from patterning along the major body axis of the fruit fly embryo.

The hallmark of a major evolutionary transition, whereby independently replicating units became grouped together into larger wholes, is the “decoupling” of fitness of the collective from the individual fitnesses of the units. It is widely believed that the key element making such decoupling possible is extensive cooperation between the lower-level entities. Here, it is demonstrated that cohesive multicellular behavior can arise in a purely competitive setting as a generic consequence of division of labor. A minimal theoretical model of competitive coexistence on multiple resources provides an explicit manifestation of fitness decoupling in a rigorous mathematical sense. This form of multicellular behavior is not vulnerable to “cheaters” and can be expected to be widespread in microbial ecosystems.

We continue our study of the chemical (graph) distance inside large critical percolation clusters in dimension two. We prove new estimates, which involve the three-arm probability, for the point-to-surface and point-to-point distances. We show that the point-to-point distance in Z 2 between two points in the same critical percolation cluster has infinite second moment. We also give quantitative versions of our previous results comparing the length of the shortest crossing to that of the lowest crossing of a box.

We consider a discrete, non-Hermitian random matrix model, which can be expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We show that, asymptotically almost surely, the real parts of the eigenvalues of the non-Hermitian matrix around any fixed index are interlaced with those of the anti-symmetric matrix. Along the way, we show that some tools recently developed to study the eigenvalue distributions of Hermitian matrices extend to the anti-symmetric setting.

We consider a discrete, non-Hermitian random matrix model, which can be expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We show that, asymptotically almost surely, the real parts of the eigenvalues of the non-Hermitian matrix around any fixed index are interlaced with those of the anti-symmetric matrix. Along the way, we show that some tools recently developed to study the eigenvalue distributions of Hermitian matrices extend to the anti-symmetric setting.