https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Andrews&feedformat=atomUW-Math Wiki - User contributions [en]2020-09-19T16:56:02ZUser contributionsMediaWiki 1.30.1https://www.math.wisc.edu/wiki/index.php?title=Colloquia/Fall18&diff=16406Colloquia/Fall182018-11-14T19:33:11Z<p>Andrews: /* Abstracts */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
The calendar for spring 2019 can be found [[Colloquia/Spring2019|here]].<br />
<br />
== Fall 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep 12 '''Room 911'''<br />
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series<br />
|[[#Sep 12: Gunther Uhlmann (Univ. of Washington)| Harry Potter's Cloak via Transformation Optics ]]<br />
| Li<br />
|<br />
|-<br />
|Sep 14 '''Room 911'''<br />
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series<br />
|[[#Sep 14: Gunther Uhlmann (Univ. of Washington) | Journey to the Center of the Earth ]]<br />
| Li<br />
|<br />
|-<br />
|Sep 21 '''Room 911'''<br />
| [http://stuart.caltech.edu/ Andrew Stuart] (Caltech) LAA lecture<br />
|[[#Sep 21: Andrew Stuart (Caltech) | The Legacy of Rudolph Kalman ]]<br />
| Jin<br />
|<br />
|-<br />
|Sep 28<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|<br />
|-<br />
|Oct 5<br />
| [http://www.personal.psu.edu/eus25/ Eyal Subag] (Penn State)<br />
|[[#Oct 5: Eyal Subag (Penn State)| Symmetries of the hydrogen atom and algebraic families ]]<br />
| Gurevich<br />
|<br />
|-<br />
|Oct 12<br />
| [https://www.math.wisc.edu/~andreic/ Andrei Caldararu] (Madison)<br />
|[[#Oct 12: Andrei Caldararu (Madison) | Mirror symmetry and derived categories ]]<br />
| ...<br />
|<br />
|-<br />
|Oct 19<br />
| [https://teitelbaum.math.uconn.edu/# Jeremy Teitelbaum] (U Connecticut)<br />
|[[#Oct 19: Jeremy Teitelbaum (U Connecticut)| Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist ]]<br />
| Boston<br />
|<br />
|-<br />
|Oct 26<br />
| [http://math.arizona.edu/~ulmer/index.html Douglas Ulmer] (Arizona)<br />
|[[#Oct 26: Douglas Ulmer (Arizona) | Rational numbers, rational functions, and rational points ]]<br />
| Yang<br />
|<br />
|-<br />
|Nov 2 '''Room 911'''<br />
| [https://sites.google.com/view/ruixiang-zhang/home?authuser=0# Ruixiang Zhang] (Madison)<br />
|[[#Nov 2: Ruixiang Zhang (Madison) | The Fourier extension operator ]]<br />
| <br />
|<br />
|-<br />
|Nov 7 '''Wednesday'''<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|<br />
|-<br />
|Nov 12 '''Monday'''<br />
| [http://www.math.tamu.edu/~annejls/ Anne Shiu] (Texas A&M)<br />
|[[#Nov 9: Anne Shiu (Texas A&M) | Dynamics of biochemical reaction systems ]]<br />
| Craciun, Stechmann<br />
|<br />
|-<br />
|Nov 19 '''Monday'''<br />
| [https://sites.google.com/site/ayomdin/ Alexander Yom Din] (Caltech) <br />
|[[#Nov 19: Alexander Yom Din (Caltech) | Passing from analysis to algebra to geometry - an example in representation theory of real groups ]]<br />
| Boston, Gurevitch<br />
|<br />
|-<br />
|Nov 20 '''Tuesday'''<br />
| [http://http://www.math.uchicago.edu/~drh/ Denis Hirschfeldt] (University of Chicago)<br />
|[[#Nov 20: Denis Hirschfeldt (University of Chicago)| Computability and Ramsey Theory ]]<br />
| Andrews<br />
|<br />
|-<br />
|Nov 30<br />
| Reserved for job talk<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|Dec 7<br />
| Reserved for job talk<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== Sep 12: Gunther Uhlmann (Univ. of Washington) ===<br />
Harry Potter's Cloak via Transformation Optics<br />
<br />
Can we make objects invisible? This has been a subject of human<br />
fascination for millennia in Greek mythology, movies, science fiction,<br />
etc. including the legend of Perseus versus Medusa and the more recent<br />
Star Trek and Harry Potter. In the last fifteen years or so there have been<br />
several scientific proposals to achieve invisibility. We will introduce in a non-technical fashion<br />
one of them, the so-called "traansformation optics"<br />
in a non-technical fashion n the so-called that has received the most attention in the<br />
scientific literature.<br />
<br />
=== Sep 14: Gunther Uhlmann (Univ. of Washington) ===<br />
Journey to the Center of the Earth<br />
<br />
We will consider the inverse problem of determining the sound<br />
speed or index of refraction of a medium by measuring the travel times of<br />
waves going through the medium. This problem arises in global seismology<br />
in an attempt to determine the inner structure of the Earth by measuring<br />
travel times of earthquakes. It has also several applications in optics<br />
and medical imaging among others.<br />
<br />
The problem can be recast as a geometric problem: Can one determine the<br />
Riemannian metric of a Riemannian manifold with boundary by measuring<br />
the distance function between boundary points? This is the boundary<br />
rigidity problem. We will also consider the problem of determining<br />
the metric from the scattering relation, the so-called lens rigidity<br />
problem. The linearization of these problems involve the integration<br />
of a tensor along geodesics, similar to the X-ray transform.<br />
<br />
We will also describe some recent results, join with Plamen Stefanov<br />
and Andras Vasy, on the partial data case, where you are making<br />
measurements on a subset of the boundary. No previous knowledge of<br />
Riemannian geometry will be assumed.<br />
<br />
=== Sep 21: Andrew Stuart (Caltech) ===<br />
<br />
The Legacy of Rudolph Kalman<br />
<br />
In 1960 Rudolph Kalman published what is arguably the first paper to develop a systematic, principled approach to the use of data to improve the predictive capability of mathematical models. As our ability to gather data grows at an enormous rate, the importance of this work continues to grow too. The lecture will describe this paper, and developments that have stemmed from it, revolutionizing fields such space-craft control, weather prediction, oceanography and oil recovery, and with potential for use in new fields such as medical imaging and artificial intelligence. Some mathematical details will be also provided, but limited to simple concepts such as optimization, and iteration; the talk is designed to be broadly accessible to anyone with an interest in quantitative science.<br />
<br />
=== Sep 28: Gautam Iyer (CMU) ===<br />
<br />
Stirring and Mixing<br />
<br />
Mixing is something one encounters often in everyday life (e.g. stirring cream into coffee). I will talk about two mathematical<br />
aspects of mixing that arise in the context of fluid dynamics:<br />
<br />
1. How efficiently can stirring "mix"?<br />
<br />
2. What is the interaction between diffusion and mixing.<br />
<br />
Both these aspects are rich in open problems whose resolution involves tools from various different areas. I present a brief survey of existing<br />
results, and talk about a few open problems.<br />
<br />
=== Oct 5: Eyal Subag (Penn State)===<br />
<br />
Symmetries of the hydrogen atom and algebraic families<br />
<br />
The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system. No prior knowledge on quantum mechanics or representation theory will be assumed.<br />
<br />
=== Oct 12: Andrei Caldararu (Madison)===<br />
<br />
Mirror symmetry and derived categories<br />
<br />
Mirror symmetry is a remarkable phenomenon, first discovered in physics. It relates two seemingly disparate areas of mathematics, symplectic and algebraic geometry. Its initial formulation was rather narrow, as a technique for computing enumerative invariants (so-called Gromov-Witten invariants) of symplectic varieties by solving certain differential equations describing the variation of Hodge structure of “mirror" varieties. Over the past 25 years this narrow view has expanded considerably, largely due to insights of M. Kontsevich who introduced techniques from derived categories into the subject. Nowadays mirror symmetry encompasses wide areas of mathematics, touching on subjects like birational geometry, number theory, homological algebra, etc.<br />
<br />
In my talk I shall survey some of the recent developments in mirror symmetry, and I will explain how my work fits in the general picture. In particular I will describe an example of derived equivalent but not birational Calabi-Yau three folds (joint work with Lev Borisov); and a recent computation of a categorical Gromov-Witten invariant of positive genus (work with my former student Junwu Tu).<br />
<br />
=== Oct 19: Jeremy Teitelbaum (U Connecticut)===<br />
Lessons Learned and New Perspectives:<br />
From Dean and Provost to aspiring Data Scientist<br />
<br />
After more than 10 years in administration, including 9 as Dean of<br />
Arts and Sciences and 1 as interim Provost at UConn, I have returned<br />
to my faculty position. I am spending a year as a visiting scientist<br />
at the Jackson Laboratory for Genomic Medicine (JAX-GM) in Farmington,<br />
Connecticut, trying to get a grip on some of the mathematical problems<br />
of interest to researchers in cancer genomics. In this talk, I will offer some personal<br />
observations about being a mathematician and a high-level administrator, talk a bit about<br />
the research environment at an independent research institute like JAX-GM, outline<br />
a few problems that I've begun to learn about, and conclude with a<br />
discussion of how these experiences have shaped my view of graduate training in mathematics.<br />
<br />
=== Oct 26: Douglas Ulmer (Arizona)===<br />
<br />
Rational numbers, rational functions, and rational points<br />
<br />
One of the central concerns of arithmetic geometry is the study of<br />
solutions of systems of polynomial equations where the solutions are<br />
required to lie in a "small" field such as the rational numbers. I<br />
will explain the landscape of expectations and conjectures in this<br />
area, focusing on curves and their Jacobians over global fields<br />
(number fields and function fields), and then survey the progress made<br />
over the last decade in the function field case. The talk is intended<br />
to be accessible to a wide audience.<br />
<br />
=== Nov 2: Ruixiang Zhang (Madison)===<br />
<br />
The Fourier extension operator<br />
<br />
I will present an integral operator that originated in the study of the Euclidean Fourier transform and is closely related to many problems in PDE, spectral theory, analytic number theory, and combinatorics. I will then introduce some recent developments in harmonic analysis concerning this operator. I will mainly focus on various new ways to "induct on scales" that played an important role in the recent solution in all dimensions to Carleson's a.e. convergence problem on free Schrödinger solutions.<br />
<br />
=== Nov 7: Luca Spolaor (MIT)===<br />
<br />
(Log)-Epiperimetric Inequality and the Regularity of Variational Problems<br />
<br />
In this talk I will present a new method for studying the regularity of minimizers to variational problems. I will start by introducing the notion of blow-up, using as a model case the so-called Obstacle problem. Then I will state the (Log)-epiperimetric inequality and explain how it is used to prove uniqueness of the blow-up and regularity results for the solution near its singular set. I will then show the flexibility of this method by describing how it can be applied to other free-boundary problems and to (almost)-area minimizing currents.<br />
Finally I will describe some future applications of this method both in regularity theory and in other settings.<br />
<br />
=== Nov 9: Anne Shiu (Texas A&M)===<br />
<br />
Dynamics of biochemical reaction systems<br />
<br />
Reaction networks taken with mass-action kinetics arise in many settings, <br />
from epidemiology to population biology to systems of chemical reactions. <br />
This talk focuses on certain biological signaling networks, namely, <br />
phosphorylation networks, and their resulting dynamical systems. For many <br />
of these systems, the set of steady states admits a rational <br />
parametrization (that is, the set is the image of a map with <br />
rational-function coordinates). We describe how such a parametrization <br />
allows us to investigate the dynamics, including the emergence of <br />
bistability in a network underlying ERK regulation, and the capacity for <br />
oscillations in a mixed processive/distributive phosphorylation network.<br />
<br />
=== Nov 19: Alexander Yom Din (Caltech)===<br />
<br />
Passing from analysis to algebra to geometry - an example in representation theory of real groups<br />
<br />
=== Nov 20: Denis Hirschfeldt (University of Chicago)===<br />
<br />
Computability and Ramsey Theory<br />
<br />
Computability theory can be seen as the study of the fine <br />
structure of definability. Much of its power relies on the deep <br />
connections between definability and computation. These connections can be seen in fundamental results such as Post's Theorem, which establishes a connection between the complexity of formulas needed to define a given set of natural numbers and its computability-theoretic strength. As has become increasingly clear, they can also be seen in the computability-theoretic analysis of objects whose definitions come from notions that arise naturally in combinatorics. The heuristic here is that <br />
computability-theoretically natural notions tend to be combinatorially <br />
natural, and vice-versa. I will discuss some results and open questions in <br />
the computability-theoretic analysis of combinatorial principles, in <br />
particular Ramsey-theoretic ones such as versions of Ramsey's Theorem for colorings of countably infinite sets, and versions of Hindman's Theorem, which states that for every coloring of the natural numbers with finitely many colors, there is an infinite set of numbers such that all nonempty sums of distinct elements of this set have the same color.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andrewshttps://www.math.wisc.edu/wiki/index.php?title=Colloquia/Fall18&diff=16405Colloquia/Fall182018-11-14T19:31:56Z<p>Andrews: /* Fall 2018 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
The calendar for spring 2019 can be found [[Colloquia/Spring2019|here]].<br />
<br />
== Fall 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep 12 '''Room 911'''<br />
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series<br />
|[[#Sep 12: Gunther Uhlmann (Univ. of Washington)| Harry Potter's Cloak via Transformation Optics ]]<br />
| Li<br />
|<br />
|-<br />
|Sep 14 '''Room 911'''<br />
| [https://sites.math.washington.edu/~gunther/ Gunther Uhlmann] (Univ. of Washington) Distinguished Lecture series<br />
|[[#Sep 14: Gunther Uhlmann (Univ. of Washington) | Journey to the Center of the Earth ]]<br />
| Li<br />
|<br />
|-<br />
|Sep 21 '''Room 911'''<br />
| [http://stuart.caltech.edu/ Andrew Stuart] (Caltech) LAA lecture<br />
|[[#Sep 21: Andrew Stuart (Caltech) | The Legacy of Rudolph Kalman ]]<br />
| Jin<br />
|<br />
|-<br />
|Sep 28<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|<br />
|-<br />
|Oct 5<br />
| [http://www.personal.psu.edu/eus25/ Eyal Subag] (Penn State)<br />
|[[#Oct 5: Eyal Subag (Penn State)| Symmetries of the hydrogen atom and algebraic families ]]<br />
| Gurevich<br />
|<br />
|-<br />
|Oct 12<br />
| [https://www.math.wisc.edu/~andreic/ Andrei Caldararu] (Madison)<br />
|[[#Oct 12: Andrei Caldararu (Madison) | Mirror symmetry and derived categories ]]<br />
| ...<br />
|<br />
|-<br />
|Oct 19<br />
| [https://teitelbaum.math.uconn.edu/# Jeremy Teitelbaum] (U Connecticut)<br />
|[[#Oct 19: Jeremy Teitelbaum (U Connecticut)| Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist ]]<br />
| Boston<br />
|<br />
|-<br />
|Oct 26<br />
| [http://math.arizona.edu/~ulmer/index.html Douglas Ulmer] (Arizona)<br />
|[[#Oct 26: Douglas Ulmer (Arizona) | Rational numbers, rational functions, and rational points ]]<br />
| Yang<br />
|<br />
|-<br />
|Nov 2 '''Room 911'''<br />
| [https://sites.google.com/view/ruixiang-zhang/home?authuser=0# Ruixiang Zhang] (Madison)<br />
|[[#Nov 2: Ruixiang Zhang (Madison) | The Fourier extension operator ]]<br />
| <br />
|<br />
|-<br />
|Nov 7 '''Wednesday'''<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|<br />
|-<br />
|Nov 12 '''Monday'''<br />
| [http://www.math.tamu.edu/~annejls/ Anne Shiu] (Texas A&M)<br />
|[[#Nov 9: Anne Shiu (Texas A&M) | Dynamics of biochemical reaction systems ]]<br />
| Craciun, Stechmann<br />
|<br />
|-<br />
|Nov 19 '''Monday'''<br />
| [https://sites.google.com/site/ayomdin/ Alexander Yom Din] (Caltech) <br />
|[[#Nov 19: Alexander Yom Din (Caltech) | Passing from analysis to algebra to geometry - an example in representation theory of real groups ]]<br />
| Boston, Gurevitch<br />
|<br />
|-<br />
|Nov 20 '''Tuesday'''<br />
| [http://http://www.math.uchicago.edu/~drh/ Denis Hirschfeldt] (University of Chicago)<br />
|[[#Nov 20: Denis Hirschfeldt (University of Chicago)| Computability and Ramsey Theory ]]<br />
| Andrews<br />
|<br />
|-<br />
|Nov 30<br />
| Reserved for job talk<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|Dec 7<br />
| Reserved for job talk<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== Sep 12: Gunther Uhlmann (Univ. of Washington) ===<br />
Harry Potter's Cloak via Transformation Optics<br />
<br />
Can we make objects invisible? This has been a subject of human<br />
fascination for millennia in Greek mythology, movies, science fiction,<br />
etc. including the legend of Perseus versus Medusa and the more recent<br />
Star Trek and Harry Potter. In the last fifteen years or so there have been<br />
several scientific proposals to achieve invisibility. We will introduce in a non-technical fashion<br />
one of them, the so-called "traansformation optics"<br />
in a non-technical fashion n the so-called that has received the most attention in the<br />
scientific literature.<br />
<br />
=== Sep 14: Gunther Uhlmann (Univ. of Washington) ===<br />
Journey to the Center of the Earth<br />
<br />
We will consider the inverse problem of determining the sound<br />
speed or index of refraction of a medium by measuring the travel times of<br />
waves going through the medium. This problem arises in global seismology<br />
in an attempt to determine the inner structure of the Earth by measuring<br />
travel times of earthquakes. It has also several applications in optics<br />
and medical imaging among others.<br />
<br />
The problem can be recast as a geometric problem: Can one determine the<br />
Riemannian metric of a Riemannian manifold with boundary by measuring<br />
the distance function between boundary points? This is the boundary<br />
rigidity problem. We will also consider the problem of determining<br />
the metric from the scattering relation, the so-called lens rigidity<br />
problem. The linearization of these problems involve the integration<br />
of a tensor along geodesics, similar to the X-ray transform.<br />
<br />
We will also describe some recent results, join with Plamen Stefanov<br />
and Andras Vasy, on the partial data case, where you are making<br />
measurements on a subset of the boundary. No previous knowledge of<br />
Riemannian geometry will be assumed.<br />
<br />
=== Sep 21: Andrew Stuart (Caltech) ===<br />
<br />
The Legacy of Rudolph Kalman<br />
<br />
In 1960 Rudolph Kalman published what is arguably the first paper to develop a systematic, principled approach to the use of data to improve the predictive capability of mathematical models. As our ability to gather data grows at an enormous rate, the importance of this work continues to grow too. The lecture will describe this paper, and developments that have stemmed from it, revolutionizing fields such space-craft control, weather prediction, oceanography and oil recovery, and with potential for use in new fields such as medical imaging and artificial intelligence. Some mathematical details will be also provided, but limited to simple concepts such as optimization, and iteration; the talk is designed to be broadly accessible to anyone with an interest in quantitative science.<br />
<br />
=== Sep 28: Gautam Iyer (CMU) ===<br />
<br />
Stirring and Mixing<br />
<br />
Mixing is something one encounters often in everyday life (e.g. stirring cream into coffee). I will talk about two mathematical<br />
aspects of mixing that arise in the context of fluid dynamics:<br />
<br />
1. How efficiently can stirring "mix"?<br />
<br />
2. What is the interaction between diffusion and mixing.<br />
<br />
Both these aspects are rich in open problems whose resolution involves tools from various different areas. I present a brief survey of existing<br />
results, and talk about a few open problems.<br />
<br />
=== Oct 5: Eyal Subag (Penn State)===<br />
<br />
Symmetries of the hydrogen atom and algebraic families<br />
<br />
The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system. No prior knowledge on quantum mechanics or representation theory will be assumed.<br />
<br />
=== Oct 12: Andrei Caldararu (Madison)===<br />
<br />
Mirror symmetry and derived categories<br />
<br />
Mirror symmetry is a remarkable phenomenon, first discovered in physics. It relates two seemingly disparate areas of mathematics, symplectic and algebraic geometry. Its initial formulation was rather narrow, as a technique for computing enumerative invariants (so-called Gromov-Witten invariants) of symplectic varieties by solving certain differential equations describing the variation of Hodge structure of “mirror" varieties. Over the past 25 years this narrow view has expanded considerably, largely due to insights of M. Kontsevich who introduced techniques from derived categories into the subject. Nowadays mirror symmetry encompasses wide areas of mathematics, touching on subjects like birational geometry, number theory, homological algebra, etc.<br />
<br />
In my talk I shall survey some of the recent developments in mirror symmetry, and I will explain how my work fits in the general picture. In particular I will describe an example of derived equivalent but not birational Calabi-Yau three folds (joint work with Lev Borisov); and a recent computation of a categorical Gromov-Witten invariant of positive genus (work with my former student Junwu Tu).<br />
<br />
=== Oct 19: Jeremy Teitelbaum (U Connecticut)===<br />
Lessons Learned and New Perspectives:<br />
From Dean and Provost to aspiring Data Scientist<br />
<br />
After more than 10 years in administration, including 9 as Dean of<br />
Arts and Sciences and 1 as interim Provost at UConn, I have returned<br />
to my faculty position. I am spending a year as a visiting scientist<br />
at the Jackson Laboratory for Genomic Medicine (JAX-GM) in Farmington,<br />
Connecticut, trying to get a grip on some of the mathematical problems<br />
of interest to researchers in cancer genomics. In this talk, I will offer some personal<br />
observations about being a mathematician and a high-level administrator, talk a bit about<br />
the research environment at an independent research institute like JAX-GM, outline<br />
a few problems that I've begun to learn about, and conclude with a<br />
discussion of how these experiences have shaped my view of graduate training in mathematics.<br />
<br />
=== Oct 26: Douglas Ulmer (Arizona)===<br />
<br />
Rational numbers, rational functions, and rational points<br />
<br />
One of the central concerns of arithmetic geometry is the study of<br />
solutions of systems of polynomial equations where the solutions are<br />
required to lie in a "small" field such as the rational numbers. I<br />
will explain the landscape of expectations and conjectures in this<br />
area, focusing on curves and their Jacobians over global fields<br />
(number fields and function fields), and then survey the progress made<br />
over the last decade in the function field case. The talk is intended<br />
to be accessible to a wide audience.<br />
<br />
=== Nov 2: Ruixiang Zhang (Madison)===<br />
<br />
The Fourier extension operator<br />
<br />
I will present an integral operator that originated in the study of the Euclidean Fourier transform and is closely related to many problems in PDE, spectral theory, analytic number theory, and combinatorics. I will then introduce some recent developments in harmonic analysis concerning this operator. I will mainly focus on various new ways to "induct on scales" that played an important role in the recent solution in all dimensions to Carleson's a.e. convergence problem on free Schrödinger solutions.<br />
<br />
=== Nov 7: Luca Spolaor (MIT)===<br />
<br />
(Log)-Epiperimetric Inequality and the Regularity of Variational Problems<br />
<br />
In this talk I will present a new method for studying the regularity of minimizers to variational problems. I will start by introducing the notion of blow-up, using as a model case the so-called Obstacle problem. Then I will state the (Log)-epiperimetric inequality and explain how it is used to prove uniqueness of the blow-up and regularity results for the solution near its singular set. I will then show the flexibility of this method by describing how it can be applied to other free-boundary problems and to (almost)-area minimizing currents.<br />
Finally I will describe some future applications of this method both in regularity theory and in other settings.<br />
<br />
=== Nov 9: Anne Shiu (Texas A&M)===<br />
<br />
Dynamics of biochemical reaction systems<br />
<br />
Reaction networks taken with mass-action kinetics arise in many settings, <br />
from epidemiology to population biology to systems of chemical reactions. <br />
This talk focuses on certain biological signaling networks, namely, <br />
phosphorylation networks, and their resulting dynamical systems. For many <br />
of these systems, the set of steady states admits a rational <br />
parametrization (that is, the set is the image of a map with <br />
rational-function coordinates). We describe how such a parametrization <br />
allows us to investigate the dynamics, including the emergence of <br />
bistability in a network underlying ERK regulation, and the capacity for <br />
oscillations in a mixed processive/distributive phosphorylation network.<br />
<br />
=== Nov 19: Alexander Yom Din (Caltech)===<br />
<br />
Passing from analysis to algebra to geometry - an example in representation theory of real groups <br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andrewshttps://www.math.wisc.edu/wiki/index.php?title=Madison_Math_Circle&diff=10264Madison Math Circle2015-09-20T17:05:36Z<p>Andrews: </p>
<hr />
<div>=LAST MINUTE LOCATION ANNOUNCEMENT=<br />
<font size="4" color = red>Due to a scheduling snafu, our Sept 14 meeting will take place in B215 Van Vleck. This is on the 2nd basement floor of Van Vleck Hall, which is right next to Ingraham Hall. (On the map below it is the "Department of Mathematics Building".)</font><br />
<br />
=Weekly Meeting=<br />
We have a weekly meeting, <b>Monday at 6pm in 120 Ingraham Hall</b>, during the school year. <b>New students are welcome at any point! </b> There is no required registration, no fee, and the talks are independent of one another, so you can just show up any week. See below for directions. <br />
<br />
If you are a student, we hope you will tell other interested students about these talks, and speak with your parents or with your teacher about organizing a car pool to the UW campus. If you are a parent or a teacher, we hope you'll tell your students about these talks and organize a car pool to the UW (all talks take place in [http://goo.gl/maps/6k5IA Ingraham Hall] room 120, on the UW-Madison campus).<br />
<br />
=What is a Math Circle?=<br />
The Madison Math Circle is a weekly series of mathematically based activities aimed at interested middle school and high school students. It is an outreach program organized by the UW Math Department. Our goal is to provide a taste of exciting ideas in math and science. In the past we've had talks about plasma and weather in outer space, video game graphics, and encryption. In the sessions, students (and parents) are often asked to explore problems on their own, with the presenter facilitating a discussion. The talks are independent of one another, so new students are welcome at any point.<br />
<br />
The level of the audience varies quite widely, including a mix of middle school and high school students, and the speakers generally address this by considering subjects that will be interesting for a wide range of students.<br />
<br />
<br />
[[Image: MathCircle_2.jpg|500px]] <br />
<br />
[[Image: MathCircle_4.jpg|500px]] <br />
<br />
<br />
After each talk we'll have pizza provided by the Mathematics Department, and students will have an opportunity to mingle and chat with the speaker and with other participants, to ask questions about some of the topics that have been discussed, and also about college, careers in science, etc.<br />
<br />
'''The Madison Math circle was featured in Wisconsin State Journal:''' http://host.madison.com/wsj/news/local/education/local_schools/school-spotlight-madison-math-circle-gives-young-students-a-taste/article_77f5c042-0b3d-11e1-ba5f-001cc4c03286.html<br />
<br />
=All right, I want to come!=<br />
==Directions and parking==<br />
Meetings are held in 120 Ingraham Hall.<br />
<br />
<div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><br />
[[File: Ingraham_Map.jpg|400px]]</div><br />
<br />
'''Parking.''' Parking on campus is rather limited. Here is as list of some options:<br />
<br />
*Directly in front of Ingraham hall, 2 metered spots (25 minute max) in [http://goo.gl/maps/HhFUm Lot 11 off of Observatory Drive].<br />
*A 0.2 mile walk to Ingraham Hall via [http://goo.gl/maps/3IFaw these directions], many spots ('''free starting 4:30pm''') [http://goo.gl/maps/Gkx1C in Lot 26 along Observatory Drive].<br />
*A 0.3 mile walk to Ingraham Hall via [http://goo.gl/maps/yFwNr these directions], many spots ('''free starting 4:30pm''') [http://goo.gl/maps/vs17X in Lot 34]. <br />
*A 0.2 mile walk to Ingraham Hall via [http://goo.gl/maps/9NNNm these directions], 2 metered spots (25 minute max) [http://goo.gl/maps/ukTcu in front of Lathrop Hall].<br />
*A 0.3 mile walk to Ingraham Hall via [http://goo.gl/maps/P156B these directions] 6 metered spots (25 minute max) around [http://goo.gl/maps/6EAnc the loop in front of Chadbourne Hall] .<br />
*For more information, see the [http://transportation.wisc.edu/parking/parking.aspx UW-Madison Parking Info website].<br />
<br />
==Email list==<br />
Sign up for our email list: https://lists.math.wisc.edu/listinfo/math-circle<br />
<br />
==Contact the organizers==<br />
If you have any questions, suggestions for topics, or so on, just email the '''organizers''' (Carolyn Abbott, Gheorghe Craciun, Daniel Erman, Lalit Jain, Ryan Julian, and Philip Matchett Wood): [mailto:math-circle-organizers@math.wisc.edu math-circle-organizers@math.wisc.edu]. We are always interested in feedback!<br />
<br />
==Report on Math Circle in 2013-14==<br />
[https://www.math.wisc.edu/wiki/images/Math_Circle_Newsletter.pdf Annual Report]<br />
<br />
==Donations==<br />
Please consider donating to the Madison Math Circle. As noted in our [https://www.math.wisc.edu/wiki/images/Math_Circle_Newsletter.pdf annual report], our main costs consist of pizza and occasional supplies for the speakers. Our costs have been covered so far by donations from the UW Math Department plus generous gifts from a private donor. But our costs are rising, primarily because this year we expect to hold more meetings than in any previous year. In fact, this year, we expect to spend at least $2500 on pizza and supplies alone.<br />
<br />
So please consider donating to support your math circle! The easiest way to donate is to go to the link:<br />
<br />
[http://www.math.wisc.edu/donate Online Donation Link]<br />
<br />
There are instructions on that page for donating to the Math Department. <b> Be sure and add a Gift Note saying that the donation is intended for the "Madison Math Circle"!</b> The money goes into the Mathematics Department Annual Fund and is routed through the University of Wisconsin Foundation, which is convenient for record-keeping, etc.<br />
<br />
Alternately, you can bring a check to one of the Math Circle Meetings. If you write a check, be sure to make it payable to the "WFAA" and add the note "Math Circle Donation" on the check. <br />
<br />
Or you can just pay in cash, and we'll give you a receipt.<br />
<br />
==Flyer==<br />
Please feel free to distribute our flyer! <br />
[https://www.math.wisc.edu/wiki/images/Flyer_MMSD.pdf Flyer]<br />
<br />
==Help us grow!==<br />
If you like Math Circle, please help us continue to grow! Students, parents, and teachers can help by:<br />
*Posting our flyer at schools or anywhere that might have interested students<br />
*Discussing the Math Circle with students, parents, teachers, administrators, and others<br />
*Making an announcement about Math Circle at PTO meetings<br />
*Donating to Math Circle<br />
Contact the organizers if you have questions or your own ideas about how to help out.<br />
<br />
<br />
=Meetings for Fall 2015 and Spring 2016=<br />
<br />
<center><br />
<br />
All talks are at '''6pm in [http://goo.gl/maps/6k5IA Ingraham Hall] room 120''', unless otherwise noted.<br />
<br />
{| style="color:black; font-size:120%" border="1" cellpadding="14" cellspacing="0"<br />
|-<br />
! colspan="3" style="background: #ffdead;" align="center" | Fall 2015 <br />
|-<br />
! Date !! Speaker !! Topic<br />
|-<br />
| September 14, 2015 || David Sondak || [[#Sondak | How to SEE Sound]] <br />
|-<br />
| September 21, 2015 || Uri Andrews|| [[#Andrews | Guarding Mona Lisa]] <br />
|-<br />
| September 28, 2015 || Eva Elduque|| [[#TBA | Abstract]] <br />
|-<br />
| October 5, 2015 || Jessica Lin|| [[#Lin | Abstract]] <br />
|-<br />
| October 12, 2015 || Ryan Julian || [[#TBA | Abstract]] <br />
|-<br />
| October 19, 2015 || Keith Rush|| [[#TBA | Abstract]] <br />
|-<br />
| October 26, 2015 || Megan Maguire || [[#TBA | Abstract]] <br />
|-<br />
| November 2, 2015 || Marko Budisic|| [[#TBA | Abstract]] <br />
|-<br />
| November 9, 2015 || Tess Anderson || [[#TBA | Abstract]] <br />
|-<br />
| November 16, 2015 || DJ Bruce || [[#TBA | Abstract]] <br />
|-<br />
| November 23, 2015 || Tullia Dymarz (Last meeting of fall) || [[#TBA | Abstract]] <br />
|-<br />
! colspan="3" style="background: #ffdead;" align="center" | Spring 2016<br />
|-<br />
! Date !! Speaker !! Topic<br />
|-<br />
| February 1, 2016 || Will Mitchell || [[#Mitchell | Are these networks the same?]] <br />
|-<br />
| February 8, 2016 || TBA || [[#TBA | Abstract]] <br />
|-<br />
| February 15, 2016 || Jordan Ellenberg || [[#TBA | Abstract]] <br />
|-<br />
| February 22, 2016 || TBA || [[#TBA | Abstract]] <br />
|-<br />
| February 29, 2016 || TBA || [[#TBA | Abstract]] <br />
|-<br />
| March 7, 2016 || TBA || [[#TBA | Abstract]] <br />
|-<br />
| March 14, 2016 || TBA || [[#TBA | Abstract]] <br />
|-<br />
| March 21, 2016 || No Meeting (Spring Break) || [[#TBA | Abstract]] <br />
|-<br />
| March 28, 2016 || No Meeting (Spring Break) || [[#TBA | Abstract]] <br />
|-<br />
| April 4, 2016 || TBA || [[#TBA | Abstract]] <br />
|-<br />
| April 11, 2016 || Andrew Kidd || [[#TBA | Abstract]] <br />
|-<br />
| April 18, 2016 || TBA || [[#TBA | Abstract]] <br />
|-<br />
| April 25, 2016 || TBA || [[#TBA | Abstract]] <br />
|-<br />
| May 2, 2016 || TBA || [[#TBA | Abstract]] <br />
|}<br />
<br />
</center><br />
<br />
=High school meetings for Fall 2015=<br />
<br />
We are experimenting with holding some Math Circle meetings directly at local high schools. Our schedule for the fall is below. If you are interesting in having us come to your high school, please contact us!<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="1" cellpadding="14" cellspacing="0"<br />
|-<br />
! colspan="5" style="background: #ffdead;" align="center" | Fall 2015<br />
|-<br />
|-<br />
! Date !! Location !! Speaker !! Topic !! Link for more info<br />
|-<br />
| September 28, 2015 || 2:45pm East High || TBA || [[#TBA | Abstract]] ||<br />
|-<br />
| October 19, 2015 || 2:45pm East High || Uri Andrews || [[#TBA | Abstract]] ||<br />
|-<br />
|}<br />
<br />
</center><br />
== Abstracts ==<br />
<br />
===David Sondak=== <br />
''How to SEE Sound''<br />
<br />
The idea is to give a simple overview of sound waves by introducing sines and cosines and some of their basic anatomy (amplitude and frequency). We will then have a computational component where the students create their own sound waves by fiddling with parameters in the sines and cosines (again, amplitude, frequency and different superpositions of the sines and cosines). They will actually be able to see plots of their waves AND listen to their waves. Finally, if time permits, the students will use their own sound waves to make Oobleck dance. This will bring the exercise full circle in that they will be able to see their very own sound waves in action.<br />
<br />
===Uri Andrews===<br />
"Guarding Mona Lisa"<br />
<br />
You have gotten a tip that a famous art thief is going to steal something from the Louvre. It is your task to organize a security team that can watch all the works of art. The problem is that the Louvre is really big and has a strange layout. Where do you put your guards? And how many do you need?<br />
<br />
<br />
===Will Mitchell=== <br />
''Are these networks the same?''<br />
<br />
The question of deciding whether two things are the same comes up in many different places in math. In this session we'll consider the problem of deciding if two networks or "graphs" are the same. This leads to some entertaining and challenging puzzles. We will also learn a bit about how people try to solve similar problems using computers. This problem has applications in the design of electronic circuits and in searching for organic chemical compounds within large databases.<br />
<br />
=Contact Information Form=<br />
[https://fs18.formsite.com/crabbott/form1/index.html Link to Contact Information Form]<br />
<br />
==[[Archived Math Circle Material]]==<br />
[[Archived Math Circle Material]]<br />
<br />
=Link for presenters (in progress)=<br />
[https://www.math.wisc.edu/wiki/index.php/Math_Circle_Presentations https://www.math.wisc.edu/wiki/index.php/Math_Circle_Presentations]</div>Andrewshttps://www.math.wisc.edu/wiki/index.php?title=Cookie_seminar&diff=3482Cookie seminar2012-02-12T22:37:58Z<p>Andrews: </p>
<hr />
<div>'''General Information''': Cookie seminar will take place on Mondays at 3:30 in the 9th floor lounge area. Talks should be of interest to the general math community, and generally will not run longer then 20 minutes. Everyone is welcome to talk, please just sign up on this page. Alternatively I will also sign interested people up at the seminar itself. As one would expect from the title there will generally be cookies provided, although the snack may vary from week to week. To sign up to bring snacks one week please visit the [[Cookie_Sign-up|Cookie Sign-up]]<br />
<br />
<br />
To sign up please provide your name and a title. Abstracts are welcome but optional.<br />
<br />
'''Seminar talks''':<br />
<br />
January 30<br />
{|border="2"<br />
|Speaker || George Craciun <br />
|-<br />
|Title || Persistence in biological networks<br />
|-<br />
|Abstract || I will describe some open problems in mathematical biology, having to do with existence of invariant regions for nonlinear dynamical systems. There is NSF grant funding (RA support) to work on some of these problems.<br />
|}<br />
<br />
February 6<br />
{|border="2"<br />
|Speaker || Leland Jefferis<br />
|-<br />
|Title || Intuitive computational methods<br />
|}<br />
<br />
February 13<br />
{|border="2"<br />
|Speaker || <br />
|-<br />
|Title || <br />
|-<br />
|Abstract || <br />
|}<br />
<br />
February 20<br />
{|border="2"<br />
|Speaker || Uri Andrews<br />
|-<br />
|Title || Hercules and the Hydra<br />
|-<br />
|Abstract || We will talk about important techniques of self-defense against an invading Hydra. The following, from Pausanias (Description of Greece, 2.37.4) describes the beginning of the battle of Hercules against the Lernaean hydra:<br />
<br />
As a second labour he ordered him to kill the Lernaean hydra.<br />
That creature, bred in the swamp of Lerna,<br />
used to go forth into the plain<br />
and ravage both the cattle and the country.<br />
Now the hydra had a huge body, with nine heads,<br />
eight mortal, but the middle one immortal. . . .<br />
By pelting it with fiery shafts he forced it to come out,<br />
and in the act of doing so he seized and held it fast.<br />
But the hydra wound itself about one of his feet and clung to him.<br />
Nor could he effect anything by smashing its heads with his club,<br />
for as fast as one head was smashed there grew up two.<br />
|}</div>Andrewshttps://www.math.wisc.edu/wiki/index.php?title=Cookie_seminar&diff=3481Cookie seminar2012-02-12T22:36:40Z<p>Andrews: </p>
<hr />
<div>'''General Information''': Cookie seminar will take place on Mondays at 3:30 in the 9th floor lounge area. Talks should be of interest to the general math community, and generally will not run longer then 20 minutes. Everyone is welcome to talk, please just sign up on this page. Alternatively I will also sign interested people up at the seminar itself. As one would expect from the title there will generally be cookies provided, although the snack may vary from week to week. To sign up to bring snacks one week please visit the [[Cookie_Sign-up|Cookie Sign-up]]<br />
<br />
<br />
To sign up please provide your name and a title. Abstracts are welcome but optional.<br />
<br />
'''Seminar talks''':<br />
<br />
January 30<br />
{|border="2"<br />
|Speaker || George Craciun <br />
|-<br />
|Title || Persistence in biological networks<br />
|-<br />
|Abstract || I will describe some open problems in mathematical biology, having to do with existence of invariant regions for nonlinear dynamical systems. There is NSF grant funding (RA support) to work on some of these problems.<br />
|}<br />
<br />
February 6<br />
{|border="2"<br />
|Speaker || Leland Jefferis<br />
|-<br />
|Title || Intuitive computational methods<br />
|}<br />
<br />
February 13<br />
{|border="2"<br />
|Speaker || <br />
|-<br />
|Title || <br />
|-<br />
|Abstract || <br />
|}<br />
<br />
February 20<br />
{|border="2"<br />
|Speaker || Uri Andrews<br />
|-<br />
|Title || Hercules and the Hydra<br />
|-<br />
|Abstract || We will talk about important techniques of self-defense<br />
against an invading Hydra. The following, from Pausanias (Description<br />
of Greece, 2.37.4) describes the beginning of the battle of Hercules<br />
against the Lernaean hydra:<br />
As a second labour he ordered him to kill the Lernaean hydra.<br />
That creature, bred in the swamp of Lerna,<br />
used to go forth into the plain<br />
and ravage both the cattle and the country.<br />
Now the hydra had a huge body, with nine heads,<br />
eight mortal, but the middle one immortal. . . .<br />
By pelting it with fiery shafts he forced it to come out,<br />
and in the act of doing so he seized and held it fast.<br />
But the hydra wound itself about one of his feet and clung to him.<br />
Nor could he effect anything by smashing its heads with his club,<br />
for as fast as one head was smashed there grew up two.<br />
|}</div>Andrewshttps://www.math.wisc.edu/wiki/index.php?title=Cookie_seminar&diff=3480Cookie seminar2012-02-12T22:35:18Z<p>Andrews: </p>
<hr />
<div>'''General Information''': Cookie seminar will take place on Mondays at 3:30 in the 9th floor lounge area. Talks should be of interest to the general math community, and generally will not run longer then 20 minutes. Everyone is welcome to talk, please just sign up on this page. Alternatively I will also sign interested people up at the seminar itself. As one would expect from the title there will generally be cookies provided, although the snack may vary from week to week. To sign up to bring snacks one week please visit the [[Cookie_Sign-up|Cookie Sign-up]]<br />
<br />
<br />
To sign up please provide your name and a title. Abstracts are welcome but optional.<br />
<br />
'''Seminar talks''':<br />
<br />
January 30<br />
{|border="2"<br />
|Speaker || George Craciun <br />
|-<br />
|Title || Persistence in biological networks<br />
|-<br />
|Abstract || I will describe some open problems in mathematical biology, having to do with existence of invariant regions for nonlinear dynamical systems. There is NSF grant funding (RA support) to work on some of these problems.<br />
|}<br />
<br />
February 6<br />
{|border="2"<br />
|Speaker || Leland Jefferis<br />
|-<br />
|Title || Intuitive computational methods<br />
|}<br />
<br />
February 13<br />
{|border="2"<br />
|Speaker || <br />
|-<br />
|Title || <br />
|-<br />
|Abstract || <br />
|}<br />
<br />
February 20<br />
{|border="2"<br />
|Speaker || Uri Andrews<br />
|-<br />
|Title || Hercules and the Hydra<br />
|-<br />
|Abstract || We will talk about important techniques of self-defense<br />
against an invading Hydra. The following, from Pausanias (Description<br />
of Greece, 2.37.4) describes the beginning of the battle of Hercules<br />
against the Lernaean hydra:<br />
As a second labour he ordered him to kill the Lernaean hydra.<br />
That creature, bred in the swamp of Lerna,<br />
used to go forth into the plain<br />
and ravage both the cattle and the country.<br />
Now the hydra had a huge body, with nine heads,<br />
eight mortal, but the middle one immortal. . . .<br />
By pelting it with ery shafts he forced it to come out,<br />
and in the act of doing so he seized and held it fast.<br />
But the hydra wound itself about one of his feet and clung to him.<br />
Nor could he e ect anything by smashing its heads with his club,<br />
for as fast as one head was smashed there grew up two.<br />
|}</div>Andrews