Gergen Lectures, Spring_2019
https://services.math.duke.edu/mcal?listgroup-6
Gergen Lectures Upcoming Seminarsen-us2024-03-28T21:46:48-04:00https://services.math.duke.edu/mcal2024-01-01T12:00:00-05:002dailyA Variational Perspective on Wrinkling Patterns in Thin Elastic Sheets
https://services.math.duke.edu/mcal?abstract-10695
The wrinkling of thin elastic sheets is very familiar: our skin
wrinkles, drapes have coarsening folds, and a sheet stretched
over a round surface must wrinkle or fold.
<br/><br/>
What kind of mathematics is relevant? The stable configurations of a
sheet are local minima of a variational problem with a rather special
structure, involving a nonconvex membrane term (which favors isometry)
and a higher-order bending term (which penalizes curvature). The bending
term is a singular perturbation; its small coefficient is the sheet
thickness squared. The patterns seen in thin sheets arise from energy
minimization -- but not in the same way that minimal surfaces arise
from area minimization. Rather, the analysis of wrinkling is an example
of "energy-driven pattern formation," in which our goal is to understand
the asymptotic character of the minimizers in a suitable limit (as the
nondimensionalized sheet thickness tends to zero).
<br/><br/>
What kind of understanding is feasible? It has been fruitful to focus
on how the minimum energy scales with sheet thickness, i.e. the "energy
scaling law." This approach entails proving upper bounds and
lower bounds that scale the same way. The upper bounds tend to be
easier, since nature gives us a hint. The lower bounds are more subtle,
since they must be ansatz-free; in many cases, the arguments used to
prove the lower bounds help explain "why" we see particular patterns.
A related but more ambitious goal is to identify the prefactor as well
as the scaling law; Ian Tobasco's striking recent work on geometry-driven
wrinkling has this character.<br/><br/>
Lecture 1 will provide an overview of this topic (assuming no background
in elasticity, thin sheets, or the calculus of variations). Lecture 2 will
discuss some examples of tensile wrinkling, where identification of the
energy scaling law is intimately linked to understanding the local
length scale of the wrinkles. Lecture 3 will discuss our emerging
undertanding of geometry-driven wrinkling, where (as Tobasco has
shown) it is the prefactor not the scaling law that explains the
patterns seen experimentally.Robert V. Kohn (New York University, Courant Institute of Mathematical Sciences)2019-03-19T15:15:00-04:0010695Gergen LecturesGergen Lectures SeminarSpring, 2019Tue, 19 Mar 2019 15:15:00 EDTGross Hall 103Tue, 19 Mar 2019 16:15:00 EDTTuesday, March 19, 2019, 3:15pmhttps://services.math.duke.edu/mcal_files/<div class="frb"><img src="/mcal_files/10795.jpg" /></div>
A Variational Perspective on Wrinkling Patterns in Thin Elastic Sheets: What sets the local length scale of tensile wrinkling?
https://services.math.duke.edu/mcal?abstract-10696
The wrinkling of thin elastic sheets is very familiar: our skin
wrinkles, drapes have coarsening folds, and a sheet stretched
over a round surface must wrinkle or fold.
<br/><br/>
What kind of mathematics is relevant? The stable configurations of a
sheet are local minima of a variational problem with a rather special
structure, involving a nonconvex membrane term (which favors isometry)
and a higher-order bending term (which penalizes curvature). The bending
term is a singular perturbation; its small coefficient is the sheet
thickness squared. The patterns seen in thin sheets arise from energy
minimization -- but not in the same way that minimal surfaces arise
from area minimization. Rather, the analysis of wrinkling is an example
of "energy-driven pattern formation," in which our goal is to understand
the asymptotic character of the minimizers in a suitable limit (as the
nondimensionalized sheet thickness tends to zero).
<br/><br/>
What kind of understanding is feasible? It has been fruitful to focus
on how the minimum energy scales with sheet thickness, i.e. the "energy
scaling law." This approach entails proving upper bounds and
lower bounds that scale the same way. The upper bounds tend to be
easier, since nature gives us a hint. The lower bounds are more subtle,
since they must be ansatz-free; in many cases, the arguments used to
prove the lower bounds help explain "why" we see particular patterns.
A related but more ambitious goal is to identify the prefactor as well
as the scaling law; Ian Tobasco's striking recent work on geometry-driven
wrinkling has this character.<br/><br/>
Lecture 1 will provide an overview of this topic (assuming no background
in elasticity, thin sheets, or the calculus of variations). Lecture 2 will
discuss some examples of tensile wrinkling, where identification of the
energy scaling law is intimately linked to understanding the local
length scale of the wrinkles. Lecture 3 will discuss our emerging
undertanding of geometry-driven wrinkling, where (as Tobasco has
shown) it is the prefactor not the scaling law that explains the
patterns seen experimentally.Robert V. Kohn (New York University, Courant Institute)2019-03-20T12:00:00-04:0010696Gergen LecturesGergen Lectures SeminarWed, 20 Mar 2019 12:00:00 EDTSpring, 2019Wednesday, March 20, 2019, 12:00pm119 PhysicsWed, 20 Mar 2019 13:00:00 EDTA Variational Perspective on Wrinkling Patterns in Thin Elastic Sheets: What sets the patterns seen in geometry-driven wrinkling?
https://services.math.duke.edu/mcal?abstract-10697
The wrinkling of thin elastic sheets is very familiar: our skin
wrinkles, drapes have coarsening folds, and a sheet stretched
over a round surface must wrinkle or fold.
<br/><br/>
What kind of mathematics is relevant? The stable configurations of a
sheet are local minima of a variational problem with a rather special
structure, involving a nonconvex membrane term (which favors isometry)
and a higher-order bending term (which penalizes curvature). The bending
term is a singular perturbation; its small coefficient is the sheet
thickness squared. The patterns seen in thin sheets arise from energy
minimization -- but not in the same way that minimal surfaces arise
from area minimization. Rather, the analysis of wrinkling is an example
of "energy-driven pattern formation," in which our goal is to understand
the asymptotic character of the minimizers in a suitable limit (as the
nondimensionalized sheet thickness tends to zero).
<br/><br/>
What kind of understanding is feasible? It has been fruitful to focus
on how the minimum energy scales with sheet thickness, i.e. the "energy
scaling law." This approach entails proving upper bounds and
lower bounds that scale the same way. The upper bounds tend to be
easier, since nature gives us a hint. The lower bounds are more subtle,
since they must be ansatz-free; in many cases, the arguments used to
prove the lower bounds help explain "why" we see particular patterns.
A related but more ambitious goal is to identify the prefactor as well
as the scaling law; Ian Tobasco's striking recent work on geometry-driven
wrinkling has this character.<br/><br/>
Lecture 1 will provide an overview of this topic (assuming no background
in elasticity, thin sheets, or the calculus of variations). Lecture 2 will
discuss some examples of tensile wrinkling, where identification of the
energy scaling law is intimately linked to understanding the local
length scale of the wrinkles. Lecture 3 will discuss our emerging
undertanding of geometry-driven wrinkling, where (as Tobasco has
shown) it is the prefactor not the scaling law that explains the
patterns seen experimentally.Robert V. Kohn (New York University, Courant Institute)2019-03-21T15:15:00-04:0010697Gergen LecturesGergen Lectures SeminarThu, 21 Mar 2019 15:15:00 EDTSpring, 2019Thursday, March 21, 2019, 3:15pm119 PhysicsThu, 21 Mar 2019 16:15:00 EDTOn the geometry of algebraic varieties
https://services.math.duke.edu/mcal?abstract-10713
Algebraic varieties are geometric objects defined by polynomial equations. The minimal model program (MMP) is an ambitious program that aims to classify algebraic varieties. According to the MMP, there are 3 building blocks: Fano varieties, Calabi-Yau varieties and varieties of general type which are higher dimensional analogs of Riemann Surfaces of genus 0,1 or at least 2 respectively. In this talk I will recall the general features of the MMP and discuss recent advances in our understanding of varieties of general type.Christopher Hacon (University of Utah, Mathematics)2019-04-25T15:30:00-04:0010713Gergen LecturesGergen Lectures SeminarThursday, April 25, 2019, 3:30pmhttps://services.math.duke.edu/mcal_files/<div class="frb"><img src="/mcal_files/10794.jpg" /></div>
Thu, 25 Apr 2019 16:30:00 EDTPhysics 130Thu, 25 Apr 2019 15:30:00 EDTSpring, 2019Birational geometry in characteristic $p>5$
https://services.math.duke.edu/mcal?abstract-10714
After the recent exciting progress in understanding the geometry of algebraic varieties over the complex numbers, it is natural to try to understand the geometry of varieties over an algebraically closed field of characteristic $p>0$. Many technical issues arise in this context. Nevertheless, there has been much recent progress. In particular, the MMP was established for 3-folds in characteristic $p>5$ by work of Birkar, Hacon, Xu and others. In this talk we will discuss some of the challenges and recent progress in this active area.Christopher Hacon (University of Utah, Mathematics)2019-04-26T12:00:00-04:0010714Gergen LecturesGergen Lectures SeminarFri, 26 Apr 2019 13:00:00 EDTPhysics 119https://services.math.duke.edu/mcal_files/<div class="frb"><img src="/mcal_files/10794.jpg" /></div>
Friday, April 26, 2019, 12:00pmSpring, 2019Fri, 26 Apr 2019 12:00:00 EDT