<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 5.0 Transitional//EN" "http://www.w3.org/TR/REC-html50/loose.dtd"><HTML> <HEAD> <LINK REL="SHORTCUT ICON" HREF="ksda2.ico"> <TITLE>KSDA - Graduate Center series</TITLE> <META NAME="keywords" CONTENT="Differential Algebra, Kolchin, Differential Galois Theory, Graduate Center, Hunter College, Differential Equations, Differential Algebraic Group"> <META NAME="descriptions" CONTENT="Welcome to KSDA: Kolchin Seminar in Differential Algebra, City University of New York"> <!-- do not use underline in links --> <STYLE type="text/css"> <!-- A{text-decoration:none} A:hover{text-decoration:underline!important} --> </STYLE> <!-- caligraphic type --> <style type="text/css">span.cal{font-family:"cursive"}</style> <!-- horizontal backspace --> <style type="text/css">span.bksp10{margin-left: -10px}</style> <!--Example use, successful- this is<span class="bksp10"></span>a test<br> You can use backspace to make overlapped symbols: <font size="4">&macr;</font><span class="bksp10"></span><I>A</I> is $\bar A$.--> <style type="text/css">span.bksp5{margin-left: -5px}</style><!-- Example use successful: displays product with upper and lower limits aligned: Note the use of doubled sup and sub. &Product;<sup><sup>1</sup></sup><span class="bksp5"><sub><sub>1</sup></sub></span>--> <!--Example for integral with limits: <B>&int;</B><I><sup><sup>x</sup></sup><span class="bksp5"><sub><sub>a</sub></sub></span></I>&nbsp;<I>f</I>(<I>t</I>)<I>dt</I>--> <!-- to have blinking text: adjust width=scrollamount, and scrolldelay and use sparingly Source: http://www.wikihow.com/Make-Blinking-Text-Without-the-Text-Tag-or-JavaScript https://developer.mozilla.org/en-US/docs/Web/HTML/Element/marquee width is about 6 (in px) per character --> <!-- Using marquee to blink 3 times (see i:\html\marquee.htm) scrollamount and width should be width of text Use box (style="border:1px solid red") to measure the width of the text by setting width=scrollamount="textwidth" ; the text should blink 3 times and stay at same position of the box. For "Tuesday&nbsp;&nbsp;", width is 62px.--> <!-- Success examples: <marquee behavior="slide" direction="right" loop="3" width="62" scrollamount="62" scrolldelay="500"> Tuesday&nbsp;&nbsp; </marquee>(Friday Schedule)<br> <b><FONT color="000000" size="4"> <font color="red"><marquee behavior="slide" direction="right" loop="3" width="72" scrollamount="72" scrolldelay="500" style="border:1px solid red"> Tuesday&nbsp;&nbsp; </marquee>(Friday Schedule)</font></FONT></b> --> <!-- Math ML example: <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>&#x222B;<!-- +" --></mo> <mi>a</mi> <mi>x</mi> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </math> --> <!-- use the following to allow math fonts like \(\mathfrak g\) in text --> <script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML" async></script> </HEAD> <BODY BGCOLOR="#CCCCFF" TEXT="#000000" LINK="#0000FF" VLINK="#0000FF" ALINK="#0000FF"> <FONT face="Arial, Times Roman, Lucida Console, Arial"><!-- 001 Face only--> <!-- KSDA and CUNY logo --> <table width="100%" bgcolor="#003399"> <tr><td> <FONT SIZE=6 COLOR="#FFFFFF">Kolchin Seminar in Differential Algebra</FONT></td> <td><a href="http://www.cuny.edu"><img src="cuny.bmp" border=0></a></td> </tr></table> <!-- KSDA Links and GC address --> <table> <tr><td> <!-- KSDA Links --> <table> <tr><td width="150">&bull; <a href="index.html">KSDA Home</a> </td><td width="30">&nbsp;&nbsp;&nbsp;&nbsp;</td><td width="300">&bull; <a href="gradcenter.html">Graduate Center Series</a> </td><td width="30">&nbsp;&nbsp;&nbsp;&nbsp;</td><td width="250">&bull; <a href="hunter.html">Hunter College Series</a></td> </tr><tr> <td>&bull; <a href="people.html">People</a> </td><td width="30">&nbsp;&nbsp;&nbsp;&nbsp;</td><td width="150">&bull; <a href="posted.html">Posted Papers</a></td> </td><td width="30">&nbsp;&nbsp;&nbsp;&nbsp;</td><td width="150">&bull; <a href="conference.html">Conferences</a> </td></tr> </table> </td> <td width="50"></td> <!-- End of KSDA Links, Start of GC address --> <td width="400">The Graduate Center<br> 365 Fifth Avenue, New York, NY 10016-4309<br> General Telephone: 1-212-817-7000<br> </td></tr></table> <!-- End of KSDA Links and GC address --> <HR SIZE="20" width="100%" color="#003399"> <table width="100%"> <tr><td width="110" height="96"VALIGN=BOTTOM> <a href="http://math.gc.cuny.edu"><img src="gc_logo.gif" border=0></a></td> <td><FONT size="6" color="000000"> <b>Academic Year 2019&ndash;2020</b></font> <p><font size="4">Last updated on January 28, 2020<br>Other years: <p> <a href="gradcenter2005.html">2005&ndash;2006</a>&nbsp;&nbsp; <a href="gradcenter2006.html">2006&ndash;2007</a>&nbsp;&nbsp; <a href="gradcenter2007.html">2007&ndash;2008</a>&nbsp;&nbsp; <a href="gradcenter2008.html">2008&ndash;2009</a>&nbsp;&nbsp; <a href="gradcenter2009.html">2009&ndash;2010</a>&nbsp;&nbsp; <a href="gradcenter2010.html">2010&ndash;2011</a>&nbsp;&nbsp; <a href="gradcenter2011.html">2011&ndash;2012</a>&nbsp;&nbsp; <a href="gradcenter2012.html">2012&ndash;2013</a>&nbsp;&nbsp; <a href="gradcenter2013.html">2013&ndash;2014</a>&nbsp;&nbsp; <a href="gradcenter2014.html">2014&ndash;2015</a>&nbsp;&nbsp; <a href="gradcenter2015.html">2015&ndash;2016</a>&nbsp;&nbsp; <a href="gradcenter2016.html">2016&ndash;2017</a>&nbsp;&nbsp; <a href="gradcenter2017.html">2017&ndash;2018</a>&nbsp;&nbsp; <a href="gradcenter2018.html">2018&ndash;2019</a>&nbsp;&nbsp; </font> </td></tr></table> <HR SIZE="6" width="100%" color="#003399"><a name="past"></a> <B><font size="5" color="003399">Past Lectures, Fall, 2019</font></B><br> <p><p><FONT color="000000" size="4"> <b>Friday, Aug 30, 2019 and Friday, Sept 6, 2019: No seminar.</b></FONT> <p><p><FONT color="000000" size="4"> <b>Friday, Sept 13, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b>Daniel Robertz, University of Plymouth</b><br>Algorithmic Approach to Strong Consistency Analysis of Finite Difference Approximations to PDE Systems</FONT> <p>The most common numerical method for solving partial differential equations is the finite difference method. Consistency of a finite difference scheme with a given PDE is a basic requirement for this method. Earlier work by V. P. Gerdt and the speaker introduced the notion of strong consistency that takes into account the differential ideal and the difference ideal associated with the PDE system and the approximating difference system, respectively. We present an algorithmic approach to strong consistency for polynomially nonlinear PDE systems based on a new decomposition technique for nonlinear partial difference systems that is analogous to the differential Thomas decomposition. <br>This is joint work with Vladimir P. Gerdt (JINR, Dubna) <p>For a copy of the slides, please click <a href="PostedPapers/Robertz091319.pdf">slides</a>.<br> For a review of the lecture, please click <a href="https://www.youtube.com/watch?v=_MKL19s43ko">video</a>. </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Sept 20, 201, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE>Free Discussions </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Sept 27, 201, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE>Free Discussions </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Oct 4, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b>Anand Pillay, University of Notre Dame</b><br>Finiteness Theorems for Kolchin's Constrained Cohomology</FONT> <p>This is joint work with Omar Leon Sanchez. Working under a certain general assumption on the differential field <I>K</I> (which includes the case where <I>K</I> is a closed order differential field in the sense of Michael Singer), we prove finiteness of the "constrained cohomology sets" <I>H</I><sup><sup>1</sup></sup><span class="bksp5"><sub><sub>&part;</sup></sub></span>(<I>K</I>, <I>G</I>), for any linear differential algebraic group <I>G</I> over <I>K</I>. I will define everything and touch on some applications. <p>For a review of the lecture, please click <a href="https://www.youtube.com/watch?v=83YL75LeFhM">video</a>. </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Oct 11, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b>Yi Zhou, Florida State University</b><br>Algorithms on <I>p</I>-Curvatures of Linear Difference Operators</FONT> <p>In the study of factoring linear difference operators, we have found <I>p</I>-curvature a powerful tool. I will talk about algorithms for computing <I>p</I>-curvatures and the math behind them. <p>For a copy of the slides, please click <a href="PostedPapers/Zhou101119.pdf">slides</a>.<br> For a review of the lecture, please click <a href="https://youtu.be/2A6cO359tDQ">video</a>. </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Oct 18, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b>Omar Leon Sanchez, University of Manchester</b><br> Differentially Large Fields</FONT> <p>Recall that a field <I>K</I> is <I>large</I> if it is existentially closed in the field of Laurent series <I>K</I>((<I>t</I>)). Examples of such fields are the complex, the real, and the <I>p</I>-adic numbers. This class of fields has been exploited significantly by F.&nbsp;Pop and others in inverse Galois-theoretic problems. In recent work with Tressl, we introduced and explored a differential analogue of largeness, that we conveniently call <I>differentially large</I>. I will present some properties of such fields and characterize them using formal Laurent series and even construct  natural examples (which ultimately yield examples of DCFs and CODFs... acronyms that will be explained in the talk). Time permitting I will mention some applications to Parameterized Picard-Vessiot theory." <p>For a review of the lecture, please click <a href="https://www.youtube.com/watch?v=BW0NN4cings">video</a>. </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Oct 25, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b>Fabian Immler, Carnegie Mellon University</b><br>Formal Mathematics and a Proof of Chaos</FONT> <p>Formal proof has been successfully applied to the verification of hardware and software systems. But formal proof is also applicable to mathematics: proofs can be checked with ultimate rigor and one can build libraries of computer-searchable, formalized mathematics.<br> <p>I will talk about formalization of mathematics and my formalization of ordinary differential equations in the Isabelle/HOL theorem prover. This underpins the formal verification of the computer-assisted part of Tucker's proof of Smale's 14th problem, a proof that relies on numerical bounds to certify chaos for the Lorenz system of ordinary differential equations. <p>For a copy of the slides, please click <a href="PostedPapers/Immler102519.pdf">slides</a>.<br><!--For a review of the lecture, please click <a href="https://www.youtube.com/link">video</a>.--> </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Nov 1, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b>Carsten Schneider, Johannes Kepler University</b><br>An Algorithmic Difference Ring Theory for Symbolic Summation</FONT> <p>Inspired by Karr's pioneering work (1981), we developed an algorithmic difference ring theory for symbolic summation that enables one to rephrase indefinite nested sums and products in formal difference rings. An important outcome of this representation is that one obtains a simplified expression where the arising sums and products are algebraically independent among each other. In this talk the main ideas of these algorithmic constructions and crucial features of the underlying difference ring theory are presented. Combining such optimal representations in combination with definite summation algorithms, like creative telescoping and recurrence solving in the setting of difference rings, yield a strong summation toolbox for practical problem solving. We will demonstrate this machinery implemented in the summation package Sigma by concrete examples coming from particle physics. <p>For a copy of the slides, please click <a href="PostedPapers/Schneider110119.pdf">slides</a>.<br>For a review of the lecture, please click <a href="https://www.youtube.com/watch?v=mVdJV9pe5VQ">video</a>. </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Nov 8, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b> Lo Jimenez, University of Notre Dame</b><br>Strengthenings of <I>C</I>-Algebraicity in Differentially Closed Fields of Characteristic Zero</FONT> <p>In model theory, the notion of internality to a fixed family of types plays an important role. During this talk, I will focus on one of its differential algebraic manifestations: being <I>C</I>-algebraic, where <I>C</I> is the field of constants of a differentially closed field. An irreducible differential-algebraic variety is <I>C</I>-algebraic if it is, roughly speaking, differentially birational to an algebraic variety in <I>C</I>. I will discuss a new property called uniform <I>C</I>-internality, and discuss examples, non-examples, and applications. <p>For a copy of the slides, please click <a href="PostedPapers/Jimenez110819.pdf">slides</a>. <p>For a review of the lecture, please click <a href="https://www.youtube.com/watch?v=0JqtpgVCx2U">video</a>. </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Nov 15, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b>Jonathan Kirby, University of East Anglia</b><br>Local Definability of Holomorphic Functions (remote presentation)</FONT> <p>Given a collection <I>F</I> of complex or real analytic functions, one can ask what other functions are obtainable from them by finitary algebraic operations. If we just mean polynomial operations we get some field of functions. If we include as algebraic operations such things as taking implicit functions, maybe in several variables, we get a much more interesting framework, which is closely related to the theory of local definability in an o-minimal setting, starting with suitable restrictions of the functions in <I>F</I>. O-minimality is a setting for tame topology of real- or complex-analytic functions which does not allow for "bad" singularities. However some more tame singularities can occur. In this talk I will explain work showing what singularities we have to consider to get a characterisation of the locally definable functions in terms of complex analytic operations. Ax s theorem on the differential algebra version of Schanuel s conjecture is important to give one counterexample, and also for some applications to exponential and elliptic functions. <p>This is joint work with Gareth Jones, Olivier Le Gal, and Tamara Servi. <p>For a copy of the slides, please click <a href="PostedPapers/Kirby111519.pdf">slides</a>.<br>For a review of the lecture, please click <a href="https://www.youtube.com/watch?v=8D1yVb4OStU">video</a>. </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Nov 22, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b>Thomas Dreyfus, Universit&eacute; de Strasbourg</b><br> Differential Transcendence of Solutions of Difference Equations (remote presentation)</FONT> <p>A function is said to be <I>differentially algebraic</I> if it satisfies a non trivial algebraic differential equation. It is said to be <I>differentially transcendent</I> otherwise. Example of differentially transcendent functions are known, for instance, the Gamma function, or the generating series of automatic sequences. All these functions have in common to satisfy a linear functional equation. In this framework, the difference Galois theory provides tools to prove the differential transcendence of the functions. This strategy has given many recent papers presenting results that get more and more general. In this talk we are going to present a new result for which the hypotheses are very minimal. This is a joint work with B. Adamczewki and C. Hardouin. <p>For a copy of the slides, please click <a href="PostedPapers/Dreyfus112219.pdf">slides</a>.<br> For a review of the lecture, please click <a href="https://www.youtube.com/watch?v=owGwanutVU0">video</a>. </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Nov 29, 2019, Thanksgiving Weekend, No Seminar.</b> </p></FONT> <p><p><FONT color="000000" size="4"> <b>Friday, Dec 6, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b>Sam Coogan, Georgia Tech</b><br> Probabilistic Guarantees for Autonomous Systems</FONT> <p> For complex autonomous systems subject to stochastic dynamics, providing absolute assurances of performance may not be possible. Instead, probabilistic guarantees that assure, for example, desirable performance with high probability are often more appropriate. In this talk, we first describe how interval-valued Markov Decision Processes (IMDP) are able to model stochastic dynamical systems. Unlike classical Markov Decision Processes, IMDPs allow for a range of transition intervals between any two states. We then show that such IMDPs arise naturally when computing finite state abstractions of discrete-time, nonlinear stochastic dynamics. In general, computing such IMDP abstractions can be computationally challenging. However, we present a class of mixed monotone systems for which such abstractions can be efficiently computed. Mixed monotonicity extends the classical notion of monotonicity for dynamical systems to allow for dynamics that have cooperative and competitive effects among the state variables. <p>For a copy of the slides, please click <a href="PostedPapers/Coogan120619.pdf">slides</a>.<br> For a review of the lecture, please click <a href="https://www.youtube.com/watch?v=rZ-IAjcqf64">video</a>. </BLOCKQUOTE> <p><p><FONT color="000000" size="4"> <b>Friday, Dec 13, 2019, 10:15&ndash;11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT> <BLOCKQUOTE> <FONT color="000000" size="4"><b>Yi Zhang, University of Texas at Dallas</b><br>Apparent Singularities of D-Finite Systems</FONT> <p>We generalize the notions of ordinary points and singularities from linear ordinary differential equations to D-finite systems. Ordinary points and apparent singularities of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities. This is joint work with Shaoshi Chen, Manuel Kauers, and Ziming Li. <p>For a copy of the slides, please click <a href="PostedPapers/Zhang121319.pdf">slides</a>.<br>For a review of the lecture, please click <a href="https://www.youtube.com/watch?v=LYEkILiFAYY">video</a>. <HR SIZE="6" width="100%" color="#003399"> <a name="year"></a> <B>Other Academic Years</B> <br><p> &nbsp;&nbsp; <a href="gradcenter2005.html">2005&ndash;2006</a> &nbsp;&nbsp; <a href="gradcenter2006.html">2006&ndash;2007</a> &nbsp;&nbsp; <a href="gradcenter2007.html">2007&ndash;2008</a> &nbsp;&nbsp; <a href="gradcenter2008.html">2008&ndash;2009</a> &nbsp;&nbsp; <a href="gradcenter2009.html">2009&ndash;2010</a> &nbsp;&nbsp; <a href="gradcenter2010.html">2010&ndash;2011</a> &nbsp;&nbsp; <a href="gradcenter2011.html">2011&ndash;2012</a> &nbsp;&nbsp; <a href="gradcenter2012.html">2012&ndash;2013</a> &nbsp;&nbsp; <a href="gradcenter2013.html">2013&ndash;2014</a> &nbsp;&nbsp; <a href="gradcenter2014.html">2014&ndash;2015</a> &nbsp;&nbsp; <a href="gradcenter2015.html">2015&ndash;2016</a> &nbsp;&nbsp; <a href="gradcenter2016.html">2016&ndash;2017</a>&nbsp;&nbsp; <a href="gradcenter2017.html">2017&ndash;2018</a>&nbsp;&nbsp; <a href="gradcenter2018.html">2018&ndash;2019</a>&nbsp;&nbsp; </font> <br><br> <P><HR SIZE="6" width="100%" color="#003399"><center> <table width="100%" height="80"> <tr> <td><FONT color="#003399">Hosted by</FONT></td> <td><a href="http://www.sci.ccny.cuny.edu"><img src="scilogo.jpg" border="0"></a></td> <td bgcolor="#000000"></td> <td> <table><tr ALIGN=LEFT><tr ALIGN="LEFT"><td><FONT color="#003399">Created by the KSDA Organizing Committee</FONT></td></tr> <tr ALIGN=LEFT><td><FONT color="#003399">Please submit web page problems to </td></tr> <tr ALIGN="LEFT"><td>William Sit <!--&#x6d;&#x61;&#x69;&#x6c;&#x74;&#x6f;&#x3a;--><&#x77;&#x73;&#x69;&#x74;&#x40;&#x63;&#x63;&#x6e;&#x79;&#x2e;&#x63;&#x75;&#x6e;&#x79;&#x2e;&#x65;&#x64;&#x75;> </td></tr></table> </td></tr></table></center> </body> </html>