Mathematics

Micro-Pharmacology: Recognizing and Overcoming the Tissue Barriers to Drug Delivery

Speaker: 
Kasia Rejniak
Date: 
Wed, Jul 22, 2020
Location: 
Zoom
Conference: 
Mathematical Biology Seminar
Abstract: 

Systemic chemotherapy is one of the main anticancer treatments used for most kinds of clinically diagnosed tumors. However, the efficacy of these drugs can be hampered by the physical attributes of the tumor tissue, such as tortuous vasculature, dense and fibrous extracellular matrix, irregular cellular architecture, metabolic gradients, and non-uniform expression of the cell membrane receptors. This can impede the transport of therapeutic agents to tumor cells in quantities sufficient to exert the desired effect. In addition, tumor microenvironments undergo dynamic spatio-temporal changes during treatment, which can also obstruct the observed drug efficacy. To examine ways to improve drug delivery on a cell-to-tissue scale (single-cell pharmacology), we developed the microscale pharmacokinetics/pharmacodynamics modeling framework “microPKPD”. I will present how this framework can be used to design optimal schedules for various treatments and to investigate the development of drug-induced resistance.

Class: 

Something's wrong in the (cellular) neighborhood: Mechanisms of epithelial wound detection

Speaker: 
Shane Hutson
Date: 
Wed, Jul 15, 2020
Location: 
Zoom
Conference: 
Mathematical Biology Seminar
CRG: 
Abstract: 

The first response of epithelial cells to local wounds is a dramatic increase in cytosolic calcium. This increase occurs quickly – calcium floods into damaged cells within 0.1 s, moves into adjacent cells over ~20 s, and appears in a much larger set of surrounding cells via a delayed second expansion over 40-300 s – but calcium is nonetheless a reporter: cells must detect wounds even earlier. Using the calcium response as a proxy for wound detection, we have identified an upstream G-protein-coupled-receptor (GPCR) signaling pathway, including the receptor and its chemokine ligand. We present experimental and computational evidence that multiple proteases released during cell lysis/wounding serve as the instructive signal, proteolytically liberating active ligand to diffuse to GPCRs on surrounding epithelial cells. Epithelial wounds are thus detected by the activation of a protease bait. We will discuss the experimental evidence and a corresponding computational model developed to test the plausibility of these hypothesized mechanisms. The model includes calcium currents between each cell’s cytosol and its endoplasmic reticulum (ER), between cytosol and extracellular space, and between the cytosol of neighboring cells. These calcium currents are initiated in the model by cavitation-induced microtears in the plasma membranes of cells near the wound (initial influx), by flow through gap junctions into adjacent cells (first expansion), and by the activation of GPCRs via a proteolytically activated diffusible ligand (second expansion). We will discuss how the model matches experimental observations and makes experimentally testable predictions.

Supported by NIH Grant 1R01GM130130.

Class: 

Stationary measure and orbit closure classification for random walks on surfaces

Speaker: 
Ping Ngai (Brian) Chung
Date: 
Thu, Jul 16, 2020
Location: 
Zoom
Conference: 
Pacific Dynamics Seminar
West Coast Dynamics Seminar
Abstract: 

We study the problem of classifying stationary measures and orbit closures for non-abelian action on surfaces. Using a result of Brown and Rodriguez Hertz, we show that under a certain average growth condition, the orbit closures are either finite or dense. Moreover, every infinite orbit equidistributes on the surface. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homogeneous setting, and the result of Eskin-Mirzakhani in the setting of moduli spaces of translation surfaces.

We then consider the problem of verifying this growth condition in concrete settings. In particular, we apply the theorem to two settings, namely discrete perturbations of the standard map and the \Out(F_2)-action on a certain character variety. We verify the growth condition analytically in the former setting, and verify numerically in the latter setting.

Class: 
Subject: 

Quantitative weak mixing for random substitution tilings

Speaker: 
Rodrigo Treviño
Date: 
Thu, Jul 2, 2020
Location: 
Zoom
Conference: 
Pacific Dynamics Seminar
West Coast Dynamics Seminar
CRG: 
Abstract: 

"Quantitative weak mixing" is the term used to bound the dimensions of spectral measures of a measure-preserving system. This type of study has gained popularity over the last decade, led by a series of results of Bufetov and Solomyak for a large class of flows which include general one-dimensional tiling spaces as well as translation flows on flat surfaces, as well as results on quantitative weak mixing by Forni. In this talk I will present results which extend the results for flows to higher rank parabolic actions, focusing on quantitative results for a broad class of tilings in any dimension. The talk won't assume familiarity with almost anything, so I will define all objects in consideration.

Class: 
Subject: 

Unique ergodicity of horocycle flows on compact quotients of SL(2,R) following Coudène

Speaker: 
Jon Chaika
Date: 
Tue, Jun 30, 2020
Location: 
Zoom
Conference: 
University of Utah Seminar in Ergodic Theory
CRG: 
Abstract: 

Furstenberg proved that the horocycle flow on any compact quotient of SL(2,R) is uniquely ergodic. This has been generalized by many people. I will present a proof due to Yves Coudène, which I find elegant and can prove some of the generalizations of Furstenberg's theorem too.

Class: 
Subject: 

The counting formula of Eskin and McMullen

Speaker: 
Paul Apisa
Date: 
Tue, Jun 9, 2020
Location: 
Zoom
Conference: 
University of Utah Seminar in Ergodic Theory
CRG: 
Abstract: 

Given a lattice acting on the hyperbolic plane, how many orbits of a point intersect the ball the radius of r as r gets big? Similarly, given a hyperbolic surface with a geodesic gamma, how many lifts of gamma to the hyperbolic plane intersect the ball of radius r? Using the mixing of geodesic flow on hyperbolic surfaces, Eskin and McMullen found a short beautiful argument to find the asymptotics for these counting questions (and more general ones on affine symmetric spaces). The key insight is to relate the counting problems to the equidistribution of circles under geodesic flow. In this talk I will discuss how to deduce circle equidistribution and counting problem asymptotics from mixing. The talk will involve many pictures and focus on the case of hyperbolic surfaces, however, the arguments presented will be general and their application to counting on general affine symmetric spaces will be explained at the end of the talk.

Class: 
Subject: 

Specification and the measure of maximal entropy

Speaker: 
Vaughn Climenhaga
Date: 
Tue, Jun 23, 2020
Location: 
Zoom
Conference: 
University of Utah Seminar in Ergodic Theory
CRG: 
Abstract: 

There are various proofs that a transitive uniformly hyperbolic dynamical system has a unique measure of maximal entropy. I will outline a proof due to Bowen that uses the specification and expansivity properties, focusing on the example of shift spaces. If time permits, I will describe how Bowen's proof works for equilibrium states associated to nonzero potential functions.

Class: 
Subject: 

Geometric Langlands for hypergeometric sheaves

Speaker: 
Masoud Kamgarpour
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler--Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. I will report on a joint work with Lingfei Yi, where we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems. This confirms a central conjecture of the geometric Langlands program for hypergeometrics. The key tool we use is the notion of rigid automorphic data due to Zhiwei Yun. This talk is based on the preprint arXiv:2006.10870.

Class: 
Subject: 

On generalized hyperpolygons

Speaker: 
Laura Schaposnik
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan (arXiv:2001.06911). After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Time permitting, we shall conclude the talk by mentioning some partial results on current work on the construction of triple branes (in the sense of Kapustin-Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).

Class: 
Subject: 

Finding mirrors for Fano quiver flag zero loci

Speaker: 
Elana Kalashnikov
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

One interesting feature of the classification of smooth Fano varieties up to dimension three is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). Fano varieties are expected to mirror certain Laurent polynomials; given such a Fano toric complete intersection, one can produce a Laurent polynomial via the Landau-Ginzburg model. In this talk, I’ll discuss finding mirrors of four dimensional Fano quiver flag zero loci via finding degenerations of the ambient quiver flag varieties. These degenerations generalise the Gelfand-Cetlin degeneration, which in the Grassmannian case has an important role in the cluster structure of its coordinate ring.

Class: 
Subject: 

Pages