Chaos and emergent computation in biological dynamics

Speaker: 

William Gilpin

Institution: 

Harvard Quantitative Biology Initiative

Time: 

Friday, January 15, 2021 - 2:00pm to 3:00pm

Host: 

Location: 

ZOOM

Experimental measurements of biological systems often have a limited number of independent channels, hindering the construction of interpretable models of living processes. Dynamical systems theory provides a rich set of tools for inferring underlying mathematical structure from partial observations, yet translating these insights to real-world biological datasets remains challenging. In this talk, I will overview my recent work at the intersection of nonlinear dynamics, chaos, and biology. I will first focus on my recent work on developing physics-informed machine learning algorithms that extract dynamical models directly from raw experimental data. I will present a general technique for inferring strange attractors directly from diverse biological time series, including gene expression, patient electrocardiograms, fitness trackers, and neural spiking. Next, I will next discuss my efforts to apply concepts from dynamical systems theory to understand particular biological phenomena. I will describe my work on biological fluid dynamics, and the discovery of a beautiful vortex crystal formed by the swimming strokes of early-diverging animals—which enables a novel feeding strategy based on chaotic mixing of the local microenvironment. I will relate this work to broader questions at the intersection of nonlinear dynamics and organismal behavior. I will conclude by discussing how these insights open up several exciting new avenues at the intersection of dynamical systems theory, systems biology, and machine learning.

 

 

Traveling waves and patterns in multiple-timescale dynamical systems

Speaker: 

Paul Carter

Institution: 

University of Minnesota

Time: 

Friday, October 9, 2020 - 4:00am to 5:00am

Host: 

Location: 

ZOOM

Systems with multiple timescales, in which the dynamics separate into slow and fast components, occur ubiquitously in models of physical, biological and ecological processes. In this talk I will focus on two example applications: the dynamics of vegetation patterns in water-limited regions and the propagation of impulses along nerve fibers. These processes can be modeled by reaction diffusion PDEs in which the patterns arise as traveling wave solutions. The slow/fast timescale separation induces a geometry on the underlying equations which can be exploited to gain insight into the structure and stability of these patterns.

 

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Invariant theory meets P vs NP

Speaker: 

Visu Makam

Institution: 

Princeton University

Time: 

Friday, January 31, 2020 - 4:00pm to 4:50pm

The study of symmetries in the setting of group actions via “invariant” polynomials is called invariant theory. Throughout the 20th century, invariant theory has had a profound influence in several fields in mathematics, most notably those that fall under the broad purview of algebra. Over the last two decades, new directions in invariant theory have emerged out of connections to computational complexity. Particularly exciting is the Geometric Complexity Theory (GCT) program that has uncovered connections between invariant theory and foundational problems in complexity such as identity testing and the celebrated P vs NP problem.

In this talk, I will discuss recent advances in invariant theory concerning matrix invariants and semi-invariants, (non-commutative) identity testing, null cones and orbit closures. Towards the end, I will discuss some very promising directions for the future in this rapidly expanding field. Based on several joint works with Harm Derksen, Ankit Garg, Rafael Oliveira, and Avi Wigderson.

The Birch and Swinnerton-Dyer Conjecture, one prime at a time

Speaker: 

Florian Sprung

Institution: 

Arizona State University

Time: 

Wednesday, January 29, 2020 - 4:00pm to 4:50pm

Location: 

RH 306

Elliptic curves are simple-looking polynomial equations in two variables whose solutions are still a mystery. The Birch and Swinnerton-Dyer Conjecture (a millennium problem) relates these solutions to a complex function. The conjecture is deep because it connects algebra with analysis. After explaining the conjecture, we discuss some recent results towards it, along with strategies of proving it one prime at a time.

Unraveling local cohomology

Speaker: 

Emily Witt

Institution: 

University of Kansas

Time: 

Monday, January 27, 2020 - 3:00pm to 3:50pm

Location: 

RH 306

Local cohomology modules are fundamental tools in commutative algebra, due to the algebraic and geometric information they carry. For instance, they can help determine the number of equations necessary to define an affine variety. Unfortunately, however, the application of local cohomology is limited by the fact that these modules are typically very large (e.g., not finitely generated), and can be difficult to determine explicitly. In this talk, we discuss new techniques developed to understand the structure of local cohomology (e.g., coming from invariant theory).  We also describe recently- discovered "connectedness properties" of spectra that local cohomology encodes.

Moments in families of L-functions

Speaker: 

Alexandra Florea

Institution: 

Columbia University

Time: 

Friday, January 24, 2020 - 3:00pm to 3:50pm

Location: 

RH 306

The moments of the Riemann zeta function were introduced by Hardy and Littlewood more than 100 years ago, in an attempt to prove the Lindelöf hypothesis, which provides a strong upper bound on the size of the Riemann zeta function on the critical line. Since then, moments became central objects of study in number theory. I will give an overview of the problem of computing moments in different families of L-functions, and I will discuss some of the applications. For example, I will explain how one can extract information about the values of L-functions at special points by computing moments of the L-functions in question.

Compactifying moduli spaces in algebraic geometry

Speaker: 

Ken Ascher

Institution: 

Princeton

Time: 

Tuesday, January 21, 2020 - 3:00pm to 3:50pm

Location: 

306 RH

Algebraic geometry is concerned with algebraic varieties, which can be understood as solution sets of polynomial equations. At the heart of research is the classification of algebraic varieties, and a geometric solution is provided in the form of a moduli space. Roughly speaking, a moduli space is itself an algebro-geometric object whose points represent equivalence classes of algebraic varieties of a fixed type. This talk begins with the moduli space of curves, which parametrizes equivalence classes of complex algebraic curves (i.e. Riemann surfaces) of a fixed genus. This moduli space, like most moduli spaces appearing in algebraic geometry, is not a compact space. A celebrated result of Deligne and Mumford provides a geometric way to compactify this space. The goal of this talk is to discuss recent progress towards compactifying moduli spaces of higher dimensional complex algebraic varieties (e.g. complex algebraic surfaces).

Searching for the spectrum of a non-commutative algebra.

Speaker: 

Manuel Reyes

Institution: 

Bowdoin College

Time: 

Wednesday, January 8, 2020 - 4:00pm to 4:50pm

Location: 

RH 306

The spectrum of a commutative algebra is a geometric or topological space on which the algebra may be viewed as a ring of functions. Spectrum constructions for various classes of commutative algebras famously provide a bridge between algebra and geometry, serving as one of the primary inspirations for various flavors of noncommutative geometry. But what kind of object should fill the role of the spectrum of a noncommutative algebra? In recent years, a number of results have ruled out both naive and subtle attempts to resolve this problem. This suggests that we frame the problem in light of a more fundamental question: What objects should serve as quantum discrete spaces in noncommutative geometry? In this talk, I will survey various obstructions and partial progress on these problems in both ring theory and operator algebra.

Rigid subalgebras of s-malleable deformations

Speaker: 

Rolando de Santiago

Institution: 

UCLA

Time: 

Thursday, January 9, 2020 - 4:00pm to 5:00pm

Location: 

RH 306

The works of F. Murray and J. von Neumann outlined a procedure to associate a von Neumann algebra to a group. Since then, an active area of research investigates which structural aspects of the group are detectable in its von Neumann algebra.  The difficulty of this problem is best illustrated by Conne's landmark result which states all countable ICC amenable groups give rise to isomorphic objects.  In essence, standard group invariants alone are typically too weak to be detectable the resulting von Neumann algebra.  However, when the group is non-amenable the situation may be strikingly different.

 

This talk surveys advances made in this area, with an emphasis on the results stemming from Popa's deformation/rigidity theory.  I present several instances where elementary structural features of a group, such as the direct product, can be recovered from the algebra.  We will then discuss recent progress made by Ben Hayes, Dan Hoff, Thomas Sinclair and myself on the analysis of s-malleable deformations, in the sense of Popa,  of tracial von Neumann algebras and its relationship to $\ell^2$ cohomology theory of groups.    Finally, we will detail the applications of our work which may resolve open conjectures of Peterson and Thom for von Neumann algebra of the free group $\mathbb{F}_2$.

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