In 1917, Soichi Kakeya posed the question: What is the smallest amount of area required to continuously rotate a unit line segment in the plane by a full rotation? Inpsired by this, what is the smallest measure of a set in $\mathbb{R}^n$ that contains a unit line segment in every direction? Such sets are called Kakeya sets, and can be shown to have arbitrarily small measure w.r.t. n-dimensional Lebesgue measure [and in fact, measure zero]. The Kakeya conjecture asserts that the Hausdorff and Minkowski dimension of these sets in $\mathbb{R}^n$ is $n$. In this talk, I will introduce at a very elementary level the machinery necessary to understand what the Kakeya conjecture is asking, and how the Kakeya conjecture has consequences for fields diverse as multidimensional Fourier summation, wave equations, Dirichlet series in analytic number theory, and random number generation. I'll also touch on how tools from various mathematical disciplines from additive combinatorics and algebraic geometry to multiscale analysis and heat flow can be used to obtain partial results to this problem. The talk will be geared towards a general audience.

In this talk I will discuss joint work with J. Achter and C. Vial showing that the image of the Abel--Jacobi map on algebraically trivial cycles descends to the field of definition for smooth projective varieties defined over subfields of the complex numbers. The main focus will be on applications to topics such as: descending cohomology geometrically, a conjecture of Orlov regarding the derived category and Hodge theory, and motivated admissible normal functions.

Tyrosine kinase inhibitors (TKIs), such as imatinib (IM), have significantly improved treatment of chronic myelogenous leukemia (CML). However, the majority of patients are not cured for undetermined reasons. It turns out that many patients who otherwise responded well to IM therapy still show variations in their BCR-ABL transcripts. In this talk we will overview mathematical models for leukemia, drug resistance, and stem cells. Our main focus will be on our recent results concerning mathematical models that integrate CML and an autologous immune response.  This is a joint work with G. Clapp, T. Lepoutre, and F. Nicolini.

By studying universal motivic unipotent representations of fundamental group of varieties and comparing their different realizations, we combine Kim's recent method in Diophantine geometry with Deligne-Goncharov's theory of motivic fundamental groups to develop a machinery for approaching Diophantine problems concerning integral points.

A link to the paper can be found on her web page. In solid tumors, targeted treatments can lead to dramatic regressions, but responses are often short-lived because resistant cancer cells arise. The major strategy proposed for overcoming resistance is combination therapy. We present a mathematical model describing the evolutionary dynamics of lesions in response to treatment. We first studied 20 melanoma patients receiving vemurafenib. We then applied our model to an independent set of pancreatic, colorectal, and melanoma cancer patients with metastatic disease. We find that dual therapy results in long-term disease control for most patients, if there are no single mutations that cause cross-resistance to both drugs; in patients with large disease burden, triple therapy is needed. We also find that simultaneous therapy with two drugs is much more effective than sequential therapy. Our results provide realistic expectations for the efficacy of new drug combinations and inform the design of trials for new cancer therapeutics.

More than 95% of the present day curvature of the universe is not a result of regular baryonic matter represented by the periodic table of elements. About 73% is well described by a geometrically natural cosmological constant, also referred to as dark energy, which results in a very small amount of curvature uniformly spread throughout the universe. We will explore the possibility that the remaining 23%, commonly referred to as dark mater, could also be explained very naturally from a geometric point of view.

Essential dimension is an invariant introduced by Buhler and Reichstein to measure how many parameters are needed to define an algebraic object such as a field extension or an algebraic curve over a field. I will describe joint work with Vistoli and Reichstein which studies essential dimension in the case where the algebraic objects are represented by a stack. I will also give examples of applications in the theory of quadratic forms.

We present an exact construction of the D0-brane as a soliton on the worldvolume of a BPS D2-brane in the presence of a large B field. We match the mass and spectrum of fluctuations of this soliton to the D0-D2 conformal field theory. This spectrum includes a tachyon: the endpoint of tachyon condensation represents the ``disappearance'' of the D0-brane, and all modes associated with it, as it dissolves in the D2-brane. This construction thus sheds some light on issues related to tachyon condensation.

Starting from the Poisson summation formula, I discuss spectral summation formulae on GL(2) and GL(3) and present a variety of applications to automorphic forms, analytic number theory, and arithmetic.

We propose a framework for modelling stochastic systems which satisfy detailed balance (or in other terminology, time-reversibility). Rather than specifying the dynamics through a state-dependent drift and diffusion coefficients, we specify an equilibrium probability density and a state-dependent diffusion coefficient. We argue that our framework is more natural from the modelling point of view and has a distinct advantage in situations where either the equilibrium probability density or the diffusion coefficient is discontinuous. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and elegantly handles discontinuities in the coefficients. This is joint work with Xin Yang.

We will discuss techniques for approximating a point in high-dimensional Euclidean space which is close to a known low-dimensional compact submanifold when only a compressed linear sketch of the point is available. More specifically, given a point, x, close to a known submanifold of R^D, we will consider linear measurement operators, M: R^D -> R^m, which have associated nonlinear inverses, A: R^m -> R^D, so that || x - A(Mx) || is small even when m << D. Both the design of good linear operators, M, and the design of stable nonlinear inverses, A, will be discussed. An algorithmic implementation of a particular nonlinear inverse will be presented, along with related stability bounds for the approximation of manifold data.

TBA

A surprising anomaly in diffeomorphisms acting on the normal bundle of the M-theory fivebrane remained a puzzle for almost two years. We present a cancellation mechanism based on a careful treatment of the fivebrane by smoothing it out through coupling to gravity which results in a relation between antisymmetric tensor gauge transformations and diffeomorphisms and leads to a complete cancellation of the anomalies. This formalism is used for derivation of the R-symmetry anomalies of the A_N (0,2) tensor-multiplet theories and for understanding the role of anomalies in the AdS/CFT correspondence. The Kaluza-Klein origin of Chern-Simons terms in lower-dimensional supergravity theories is displayed via modification of the Freund-Rubin ansatz. The results imply the existence of interesting 1/N corrections in the AdS/CFT correspondence. A simple derivation of black hole entropy in d=4, N=2 compactifications of M-theory is presented.

We exhibit examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact 1-forms \lambda_1^* on rational homology spheres which admit irreducible solutions together with a version of the Selberg trace formula relating the spectrum of the Laplacian on coexact 1-forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise certified numerical bounds on \lambda_1^* for several hyperbolic rational homology spheres.

A major feature of biological sciences in the 21st Century is their transition from phenomenological and descriptive disciplines to quantitative and predictive ones. However, the emergence of complexity in self-organizing biological systems poses fabulous challenges to their quantitative description because of the excessively high dimensionality. A crucial question is how to reduce the number of degrees of freedom, while preserving the fundamental physics in complex biological systems. We discuss a multiscale multiphysics and multidomain paradigm for biomolecular systems. We describe macromolecular system, such as protein, DNA, ion channel, membrane, molecular motors etc., by a number of approaches, including macroscopic electrostatics and elasticity and/or microscopic molecular mechanics and quantum mechanics; while treating the aqueous environment as a dielectric continuum or electrolytic fluids. We use differential geometry theory of surfaces to couple various microscopic and macroscopic domains on an equal footing. Based on the variational principle, we derive the coupled Poisson-Boltzmann, Nernst-Planck, Kohn-Sham, Laplace-Beltrami, Newton, elasticity and/or Navier-Stokes equations for the structure, function, dynamics and transport of protein, protein-ligand binding and ion-channel systems.

Chemotaxis is a movement of micro-organisms or cells towards the areas of high concentration of a certain chemical, which attracts the cells and may be either produced or consumed by them. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. It is well-known that solutions of such systems may develop spiky structures or even blow up in finite time provided the total number of cells exceeds a certain threshold. This makes development of numerical methods for chemotaxissystems extremely delicate and challenging task. In this talk, I will present a new family of high-order finite-volume finite-difference methods for the Keller-Segel chemotaxis system and several related models. Applications of the proposed methods to the classical Patlak-Keller-Segel model, its extensions to the two-species case as well as to the coupled chemotaxis–fluid system will also be discussed.

We discuss one approach to determining the hazard threat to a locale due to a large volcanic avalanche. The methodology employed includes large-scale numerical simulations, field data reporting the volume and runout of flow events, and a detailed statistical analysis of uncertainties in the modeling and data. The probability of a catastrophic event impacting a locale is calculated, together with a estimate of the uncertainty in that calculation. By a careful use of simulations, a hazard map for an entire region can be determined. The calculation can be turned around quickly, and the methodology can be applied to other hazard scenarios.

Many algebraic objects are notorious for being easy to define, but hard to find explicitly. However, certain algebraic objects, when viewed with the "correct" combinatorial framework, become much easier to actually find. This allows us to compute much larger examples by hand, and often gives us insight into the object's underlying structure. In this talk, we will define irreducible decompositions of ideals, and explore their underlying combinatorial structure in the special case of monomial ideals in polynomial rings. As time permits, we will look at recent results in the case of binomial ideals. This talk will be accessible to anyone who has taken a course in Abstract Algebra.

We present an exact construction of the D0-brane as a soliton on the worldvolume of a BPS D2-brane in the presence of a large B field. We match the mass and spectrum of fluctuations of this soliton to the D0-D2 conformal field theory. This spectrum includes a tachyon: the endpoint of tachyon condensation represents the ``disappearance'' of the D0-brane, and all modes associated with it, as it dissolves in the D2-brane. This construction thus sheds some light on issues related to tachyon condensation.

Fluid-structure interaction problems with composite structures arise in many applications. One example is the interaction between blood flow and arterial walls. Arterial walls are composed of several layers, each with different mechanical characteristics and thickness. No mathematical results exist so far that analyze existence of solutions to nonlinear, fluid-structure interaction problems in which the structure is composed of several layers. In this talk we will summarize the main difficulties in studying this class of problems, and present a computational scheme based on which a proof of the existence of a weak solution was obtained. Our results reveal a new physical regularizing mechanism in FSI problems: inertia of the thin fluid-structure interface with mass regularizes evolution of FSI solutions. Implications of our theoretical results on modeling the human cardiovascular system will be discussed. This is a joint work with Boris Muha (University of Zagreb, Croatia), Martina Bukac (U of Notre Dame, US) and Roland Glowinski (UH). Numerical results with vascular stents were obtained with S. Deparis and D. Forti (EPFL, Switzerland), and with A. Quaini (UH). Collaboration with medical doctors Dr. S. Little (Methodist Hospital Houston) and Dr. Z. Krajcer (Texas Heart Institute) is also acknowledged.

Complex interactions occur between tumor and host immune system during cancer development and treatment, and a weak systemic immune response can be prognostic of poor patient outcomes. We strive to not only better understand the dynamic behavior of circulating immune cell populations before and during cancer therapy, but also to monitor these dynamic changes to facilitate real-time prediction of patient outcomes and potentially therapy adaptation. I will provide examples of both theoretical (mathematical) and data-driven (epidemiological) approaches to incorporating established systemic immune markers into clinical decision-making. First, coupling models of local tumor-immune dynamics and systemic T cell trafficking allows us to simulate the evolution of tumor and immune cell populations in anatomically distant sites following local therapy, in turn identifying the optimal treatment target for maximum reduction of global tumor burden. Second, improved understanding of how circulating immune markers vary both within and between individual patients can allow more accurate risk stratification at diagnosis, and personalized prediction of patient response to therapy. The importance of multi-disciplinary collaborations in making predictive and prognostic models clinically relevant will be discussed.

I will explain some attempts to develop a theory of Schubert calculus on the affine Grassmannian. I will begin with the different descriptions of the (co)homology rings due to Bott, Kostant and Kumar, and Ginzburg. Then I will discuss the problems of finding polynomial representatives for Schubert classes and the explicit determination of structure constants in (co)homology.