The locust olfactory system interfaces with the external world through antennal receptor neurons (ORNs), which represent odors in a distributed, combinatorial manner. ORN axons bundle together to form the antennal nerve, which relays sensory information centrally to the antennal lobe (AL). Within the AL, an odor produces a stimulus-specific temporal progression of neuronal spiking, inspiring the hypothesis that the AL encodes odors through dynamically evolving ensembles of active cells. Such a coding strategy, however, requires higher olfactory centers to integrate a prolonged dynamic profile of AL signals prior to stimulus assessment, a process that is likely to be slow and inconsistent with the generation of quick behavioral responses. Our modeling work has led us to propose an alternate hypothesis: the dynamical interplay of fast and slow inhibition within the locust AL induces transient correlations in the spiking activity of an odor-dependent neural subset, giving rise to a temporal binding code and allowing rapid stimulus detection by downstream elements.

A K3 surface is a 4-dimensional shape that appears in many areas of geometry and mathematical physics. I will define it and explore many of its amazing properties.

Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, bounded Petri nets, and agent-based models. Simulation is a common practice for analyzing discrete models, but many systems are far too large to capture all the relevant dynamical features through simulation alone. We convert discrete models into algebraic models and apply tools from computational algebra to analyze their dynamics. The key feature of biological systems that is exploited by our algorithms is their sparsity: while the number of nodes in a biological network may be quite large, each node is affected only by a small number of other nodes. In our experience with models arising in systems biology and random models, this structure leads to fast computations when using algebraic models, and thus efficient analysis. All algorithms and methods are available in our package Analysis of Dynamic Algebraic Models (ADAM), a user friendly web-interface that allows for fast analysis of large models, without requiring understanding of the underlying mathematics or any software installation. ADAM is available as a web tool, so it runs platform independent on all systems.

Cancer emerges due to an evolutionary process in somatic tissue. The fundamental laws of evolution can best be formulated as exact mathematical equations. Therefore, the process of cancer initiation and progression is amenable to mathematical investigation. Of special importance are changes that occur early during malignant transformation because they may result in oncogene addiction and represent promising targets for therapeutic intervention. Here we describe a mathematical approach, called Retracing the Evolutionary Steps in Cancer (RESIC), to deduce the temporal sequence of genetic events during tumorigenesis from crosssectional genomic data of tumors at their fully transformed stage. When applied to a dataset of 70 advanced colorectal cancers, our algorithm accurately predicts the sequence of APC, KRAS, and TP53 mutations previously defined by analyzing tumors at different stages of colon cancer formation. We further validate the method with glioblastoma and leukemia sample data and then apply it to complex integrated genomics databases, finding that high-level EGFR amplification appears to be a late event in primary glioblastomas. RESIC represents the first evolutionary mathematical approach to identify the temporal sequence of mutations driving tumorigenesis and may be useful to guide the validation of candidate genes emerging from cancer genome surveys.

In this talk I will discuss a branching process model developed to study intra-tumor diversity (i.e. the variation amongst the cells within a single tumor). This variation arises due to heritable (epi)genetic alterations which can confer changes in cellular fitness and drug response. In the asymptotic (t-> infinity) regime, we study the growth rates of the population as well as some ecological measures of diversity in the tumor. In the second half of the talk I will discuss applications of this model to studying the emergence of drug resistant populations in Chronic Myeloid Leukemia (CML). (Joint work w/K. Leder, J. Mayberry, R. Durrett, F. Michor)

Cochlear implants are neural prostheses that restore a sense of hearing to individuals with severe to profound deafness. Two fundamental theoretical questions that we face are: How does the auditory nerve respond to electrical stimulation? And how is sound information represented in the spike trains of auditory nerve fibers? We will discuss model-based efforts to investigate these questions. I will focus on the development of reduced models that incorporate essential biological features of this complicated system, and remain useful tools for analyzing neural coding.

Using a point process model of the auditory nerve, I simulate amplitude modulation detection, a common test of temporal resolution. I find that the temporal information in the simulated spike trains does not limit modulation sensitivity in cochlear implant users, and discuss how the point process framework can be extended to include additional biophysical mechanisms. Next, I illustrate how spatial spread of excitation and neural degeneration can lead to of within- and across-patient variability in listening outcomes. This points toward an important goal of computational modeling: to develop patient-specific models that can be used to optimize stimulation strategies for individual cochlear implant users.

The prevailing numerical methods for solving the reaction-diffusion systems in models of cardiac electrical activity currently use second-order adaptive mesh refinement, refining the spatial and temporal meshes only near the traveling wavefront(s). However, in two and three spatial dimensions under biologically relevant initial conditions and forcing, these wavefronts can constitute a relatively high percentage of the computational domain, limiting the effectiveness of the scheme. In this talk, I will present a numerical scheme based on higher order finite elements and spectral deferred correction designed to improve the efficiency in computing for domains of cardiac cells.

Gels formed from mixtures of polymers and solvent are ubiquitous in physiological systems. The structure and properties of a gel can change dramatically in response to chemical modulation. Two examples of the influence of chemistry on gel properties will be discussed. The structure of fibrin gels that form during blood clotting is strongly influenced by the concentration of the enzyme thrombin that produces the fibrin monomers from which the gel is built. Presumably at higher thrombin concentrations, fibrin monomers are produced more rapidly from the precursor fibrinogen molecules. I will present an analysis of a possible mechanism of fibrin branching that can explain the sensitivity of fibrin structure to the rate of supply of monomers. Mucin gel is released from vesicles in goblet cells. During this exocytotic process, the polyelectrolyte mucin gel swells to many times its original volume at a very rapid rate. I will present a model in which this swelling is triggered by an exchange of divalent calcium ions in the vesicle and monovalent sodium ions in the extracellular space, and in which the ion concentrations and the rheological properties of the mucin gel determine its equilibrium size and the dynamics of its swelling.

Many pulse-coupled dynamical systems possess synchronous attracting states. Even stochastically driven model networks of Integrate and Fire neurons demonstrate synchrony over a large range of parameters. We study the interplay between fluctuations which de-synchronize and synaptic coupling along the network connections that synchronize the network by calculating the probability to see repeated cascading total firing events, during which all the neurons in the network fire at once. Using this characterization of synchrony we investigate the significance of the local network topology and of more physiological additions to the model on the model neuronal networks ability to synchronize. The mean time between total firing events characterizes the perfectly synchronous state, and we compute this from a first-passage time problem in terms of a Fokker-Planck equation for a single neuron.

At first glance, knot theory and representation theory seem to be unrelated fields of mathematics. In fact, this is not the case: in the early 90's, Reshetikhin and Turaev proved that knot invariants (and 3-manifold invariants) can be derived via the representation theory of quantum groups. The key link (no pun intended) between these areas is the observation that both the category of tangles and the category of representations share many similar structural features. In this talk we will explore these ideas, and if time permits, their categorified counterparts. If things like categories scare you, fear not; as the title suggests, all categories (and constructions on them) we encounter will have pictorial descriptions. In fact, no knowledge of category theory or representation theory is assumed. At the same time, if you have indeed taken Math 253, then this talk will provide context for the material in that course.

We introduce a new measure constructed from high-frequency financial data which we call the Realized Laplace Transform of volatility. The statistic provides a nonparametric estimate for the empirical Laplace transform of the latent stochastic volatility process over a given interval of time. When a long span of data is used, i.e., under joint long-span and fill-in asymptotics, it is an estimate of the volatility Laplace transform. The asymptotic behavior of the statistic depends on the small scale behavior of the driving martingale. We derive the asymptotics both in the case when the latter is known and when it needs to be inferred from the data. When the underlying process is a jump-diffusion our statistic is robust to jumps and when the process is pure-jump it is robust to presence of less active jumps. We apply our results to simulated and real financial data.

I'll describe a new combinatorial method for computing the delta-graded knot Floer homology of a link in S^3. Our construction comes from iterating an unoriented skein exact triangle discovered by Manolescu, and yields a chain complex for knot Floer homology which is reminiscent of that of Khovanov homology, but is generated (roughly) by spanning trees of the black graph of the link. This is joint work with Adam Levine.

The nature of dynamical processes occuring on a directed network can depend qualitatively on the logic implemented at each node. In large random networks where nodes act as Boolean gates, there is a phase transition from quiescent to chaotic behavior as the average degree and/or the probabilities of assigning the different logic functions are varied. To understand the behavior at the transition, we are led to a special type of percolation problem in which the relevant question is not whether a cascade spans the system, but whether it covers the entire system. I will introduce the problem of exhaustive percolation, outline a method for solving it, and describe its application to random Boolean networks.

This work was done in close collaboration with M. Khovanov. The Witten-Reshetikhin-Turaev invariant Z(M,L;r) of a link L in a 3-manifold M is a seemingly random function of an integer r. However, for a small class of 3-manifolds constructed by identical gluing of two handlebodies (e.g., for S3 and for S2 x S1) and for sufficiently large values of r the ratio Z(M,L;r)/Z(M;r) is equal to a rational function J(M,L;q) of q evaluated at the first 2r-th root of unity. If M = S3, then J is the Jones polynomial. Khovanov categorified J(S3,L), that is, to a link L in S3 he assigned a homology H(L) with an extra Z-grading such that its graded Euler characteristic equals J(S3,L). We extend Khovanov's construction to links in S2 x S1 thus categorifying J(S2xS1,L). In his work on categorification of the Jones polynomial, Khovanov introduced special algebras H_n and assigned a H_m x H_n module to every (2m,2n)-tangle. We show that if a link L in S2 x S1 is presented as a closure of a (2n,2n)-tangle, then the Hochschild homology of its H_n bimodule is determined by the link itself and serves as a categorificaiton of J(S2xS1,L). Moreover, we show that this Hochschild homology can be approximated by Khovanov homology of the circular closure of the tangle within S3 by a high twist torus braid, thus providing a practical method of its computation.

I will discuss various approaches to the question: When does a vector bundle admit a holomorphic structure? I will explore applications of Yang-Mills theory, geometric quantization, and discrete dynamics to this problem.

Taylor-Couette flow provides one of the pre-eminent examples of bifurcation in fluid dynamics. This phrase refers to the flow between concentric rotating cylinders. If the rotation speed is sufficiently rapid, the primary rotary flow around the axis becomes unstable, leading to a steady secondary flow in approximately periodic cells. Assuming infinite cylinders and exact periodicity in his theory, Taylor obtained remarkable agreement with experiment for the onset of instability, agreement that remains unsurpassed in fluid mechanics to this day. This talk is concerned with incorporating the effect of finite-length cylinders into the theory, an issue whose importance was emphasized by Benjamin. Numerous experiments and simulations of the Navier Stokes equations all support to the following, seemingly paradoxical, observations: No matter how long the apparatus, finite-length effects greatly perturb many of the bifurcating flows but, provided the cylinders are long, hardly perturb others. We understand this paradox as a result of symmetry breaking. The relevant symmetry, which is only approximate, is a symmetry between two normal-mode flows with large, and nearly equal, numbers of cells.

Although originally formed as an esoteric field of study, in the last few decades Algebraic Topology has emerged as a vastly applicable field. In this talk we will discuss the basics of Computational Topology and an application to one such coverage problem in sensor networks which even involves a little probability. This talk will be accessible to anyone who enjoys doing math via lots and lots of pictures.

A remarkable model of stochastic coalescence arises from considering shock statistics in scalar conservation laws with random initial data. While originally rooted in the study of Burgers turbulence, the model has deep connections to statistics, kinetic theory, random matrices, and completely integrable systems. The evolution takes the form of a Lax pair which, in addition to yielding interesting conserved quantities, admits some rather intriguing exact solutions. We will describe several distinct derivations for the evolution equation and, time-permitting, discuss properties of the corresponding kinetic system. This talk consists of joint work with Govind Menon (Brown).

In the early 1960s Atiyah and Singer derived a cohomological formula that computes the Fredholm index of an elliptic differential operator. The subsequent development of analytic K-theory of noncommutative C*-algebras greatly clarified the proof of the index formula, leading to many further generalizations. As a recent application of these techniques I will discuss the solution of the index problem for certain hypoelliptic operators on contact manifolds, first proposed by Epstein and Melrose. The final topological formula is quite easy to state, but the proof relies heavily on noncommutative techniques.

Beyond Krull dimension, rings and modules have various "dimensions": depth, height, projective/injective dimensions, flat dimension, global dimension, weak dimension, among others. These notions are defined homologically in terms of lengths of resolutions, and Ext and Tor functors provide a way to measure them. I'll talk about how they are related with each other. I'll start from geometic interpretation of Krull dimension and height, followed by regular sequences and depth. Then I'll introduce Cohen-Macaulay modules and Gorenstein modules as modules having particularly nice homological properties. Just as in the case of completion where analysis is introduced to algebra to prove some highly nontrivial results, homological techniques have proved to be very powerful in modern commutative algebra, producing such surprising results as homological characterization of regular rings (Serre, Auslander, Buchsbaum). I'll briefly introduce the notion of canonical modules and the question of finiteness of injective resolution. Finally, I'll talk about how these notions can be globalized to scheme and sheaves, which makes geometry "kind of equivalent" to algebra.

The interface between the mathematical sciences and the biosciences is two-way and may be summarized as "math -> bio" and "bio -> math." This talk will have two parts. First, I will describe previous work on gaits of four-legged animals (based on distinguishing gaits, such as walk, trot, and pace, by their spatio-temporal symmetries). Second, I will discuss how the application to gaits has led to results about phase-shift synchrony in periodic solutions of coupled systems of differential equations.

Consider a connected network and suppose each edge in the network has a random positive edge weight. Understanding the structure and weight of the shortest path between nodes in the network is one of the most fundamental problems studied in modern probability theory and goes under the name first passage percolation. It arises as a fundamental building block in many interacting particle system models such as the spread of epidemics on networks. To a large extent such problems have been only studied in the context of the n-dimensional lattice. In the modern context these problems take on an additional significance with the minimal weight measuring the cost of sending information while the number of edges on the optimal path (hopcount) representing the actual time for messages to get between vertices in the network. Given general models of random graphs with random edge costs, can one develop techniques to analyze asymptotics of functionals of interest which are robust to the model formulation? The aim of this talk is to describe a heuristic based on continuous time branching processes which gives very easily, a wide array of asymptotic results for random network models in terms of the Malthusian rate of growth and the stable age distribution of associated branching process. These techniques allow us to solve not only first passage percolation problems rigorously but also understand functionals such as the degree distribution of shortest path trees, congestion across edges as well as asymptotics for “betweeness centrality” a concept of crucial interest in social networks, in terms of Cox processes and extreme value distributions. These techniques also allow one to exactly solve models of “weak disorder” in the context of the stochastic mean field model of distance, a model of great interest in probabilistic combinatorial optimization.

We discuss a HOMFLY polynomial invariant for links in lens spaces. We then show how this polynomial is related to the contact invariants of Legendrian and transverse links in lens spaces having a certain tight contact structure. In fact, we generalize a result of Ng, casting Bennequin-type inequalities in such contact lens spaces into a general framework.

My group studies colloidal suspensions, which are solid micron-sized particles in a liquid. We use an optical confocal microscope to view the motion of these colloidal particles in three dimensions. In some experiments, these particles arrange into a crystalline lattice, and thus the sample is analogous to a traditional solid. We study the interface between colloidal crystals and colloidal liquids, and find that this interface is quite sharply defined. In other experiments, the sample is analogous to a glass, with particles randomly packed together. The particles correspond to individual molecules in a traditional glass, and the sample exhibits glassy behavior when the particle concentration is high. This allows us to directly study the microscopic behavior responsible for the macroscopic viscosity divergence of glasses.

Kick off this year's flu season with a better understanding of within-host influenza dynamics. Influenza A is a rapidly-evolving RNA virus that typically escapes herd immunity through the generation of new antigenic variants every 3 to 8 years. An important part of this antigenic evolution is believed to occur at the intrahost level. I will present two competing models of intrahost dynamics and compare their predictions to empirical observations.

Biology is a tremendously diverse field covering systems operating at vastly different scales, with differing levels of interaction between these. Much of the effort in mathematical biology has been driven by the desire to develop the general principles by which biological systems are organized and operate. Yet at the same time there are demands for answers to quite specific questions to better manage natural systems, to enhance human health, and to plan for the future impacts of human actions. I will give a variety of examples of projects in mathematical ecology that lie at the interface between theory and practice, providing some indication of the utility of quantitative methods to elucidate general patterns of natural system response to management actions. These will include applications of optimal control methods to problems in wildlife management and disease ecology as well as a discussion of individual-based models. An objective of these approaches is to develop hypothetical "best" methods to manage a system, and use this as a template to compare and contrast management scenarios arising from the differing view points of diverse stakeholders in a relative assessment framework.

We consider first-passage percolation on the cylinder graph of length $n$ and width $h_n$ in the d-dimensional square lattice where each edge has an i.i.d.~nonnegative weight. The passage time for a path is defined as the sum of weights of all the edges in that path and the first-passage time between two vertices is defined as the minimum passage time over all paths joining the two vertices. We show that the first-passage time $T_n$ between the origin and the vertex $(n,0,\ldots,0)$ satisfies a Gaussian CLT as long as $h_n=o(n^{1/(d+1)})$. The proof is based on moment estimates, a decomposition of $T_n$ as an approximate sum of independent random variables and a renormalization argument. We conjecture that the CLT holds upto $h_n=o(n^{2/3})$ for $d=2$ and provide support for that. Based on joint work with Sourav Chatterjee.

Using Calabi's method, I will construct rotationally symmetric Kahler- Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao and Feldman-Ilmanen-Knopf.

Vaccines protect their recipients by inducing long-term structural changes in populations of immune cells. Part of that restructuring is exactly analogous to Darwinian Selection. New antibody molecules are created by somatic mutation of existing antibody genes. Subsequently, the immune cell populations that possess these mutated receptors overtake the "wild-type" immune cells due  to the selective advantage they have acquired. Thus the immune system is vastly better prepared to recognize and eliminate the eliciting pathogen the next time around.

New sequencing and biosynthesis technologies, together with mathematical and computational tools, now allow us to investigate this fascinating and important phenomenon more deeply than ever before. I will illustrate this development with examples from the immune response to HIV infection.