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<!-- 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"
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<table> <tr><td width="150">• <a href="index.html">KSDA Home</a> </td><td width="30"> </td><td width="300">• <a href="gradcenter.html">Graduate Center Series</a> </td><td width="30"> </td><td width="250">• <a href="hunter.html">Hunter College Series</a></td> </tr><tr> <td>• <a href="people.html">People</a> </td><td width="30"> </td><td width="150">• <a href="posted.html">Posted Papers</a></td> </td><td width="30"> </td><td width="150">• <a href="conference.html">Conferences</a> </td></tr> </table> </td> <td width="50"></td>
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<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>
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border=0></a></td> <td><FONT size="6" color="000000"> <b>Academic Year 2018–2019</b></font> <p><font size="4">Last updated on January 31, 2020<br>Other years:
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<HR SIZE="6" width="100%" color="#003399">S
<p><p><FONT color="000000" size="4"> <b>Friday, September 7, 2018, 10:00–11:00 a.m.<a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Stephen Melczer, University of Pennsylvania</b><br>Symbolic Computation and Analytic Combinatorics in Several Variables</FONT>
<p>The field of analytic combinatorics in several variables (ACSV) is at the forefront of computational combinatorics—a subject dedicated to computability and complexity questions in enumeration. Drawing from singularity theory, computational topology, and algebraic geometry, ACSV raises a wide range of interesting computer algebra questions. This talk will survey ACSV from a computer algebra viewpoint, discuss current work, and highlight remaining open problems and generalizations.
<p>For a review of the lecture, please click <a href="https://youtu.be/wl9jGy9Audo">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, September 14, 2018, 10:00–11:00 a.m.<a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Joel (Ronnie) Nagloo, Bronx Community College (CUNY)</b><br>The Ax-Lindemann-Weierstrass with Derivatives and the Genus 0 Fuchsian Groups</FONT>
<p>The works of Pila and later Freitag and Scanlon, give the Ax-Lindemann-Weierstrass with derivatives for the Hauptmoduls of arithmetic subgroups of <I>PSL</I><sub>2</sub>(<I>Z</I>). A challenge has been to prove similar transcendence results for the Hauptmoduls of all Fuchsian groups of genus zero. In this talk I will explain recent progress towards the resolution of those problems. This is a report of a joint work with Guy Casale and James Freitag.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, September 21, 2018, 10:00–11:00 a.m.,<a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Thomas Dreyfus, University of Strasbourg</b><br>Differential Transcendence of Special Functions</FONT>
<p>One of the goals of difference Galois theory is to understand the algebraic relations between solutions of a linear functional equation. Recently, Hardouin and Singer developed a Galois theory that aims at understanding what are the algebraic and differential relations among solution of such equations. In this talk we are going to see recent results ensuring that in many situations, such solutions satisfy no algebraic differential relations.
<p>For a review of the lecture, please click <a href="https://youtu.be/g-Y1sBpjhTk">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, September 28, 2018, 10:00–11:00 a.m., <a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Doron Zeilberger, Rutgers University</b><br>The <I>C</I>-Finite Ansatz</FONT>
<p>A sequence belongs to the <I>C</I>-finite ansatz if it satisfies a linear recurrence equation with constant coefficients. For example, 2<sup><I>n</I></sup>, and the sequence of Fibonacci numbers, <I>F</I><sub><I>n</I></sub>. After describing some applications to enumerative combinatorics, I will describe yet another approach to the Ising model, different than the one Manuel Kauers talked about on September 6 (see <a href="http://qcpages.qc.cuny.edu/%7Eaovchinnikov/slides-Kauers.pdf">slides</a>) at the <a href="http://qcpages.qc.cuny.edu/~aovchinnikov/seminar.html">CUNY/NYU symbolic-numeric computing seminar</a>. This is also joint work with Manuel Kauers.
<p>For a review of the lecture, please click <a href="https://youtu.be/4_QAKERmZfQ">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, October 5, 2018, 10:00–11:00 a.m., <a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Peter Thompson, City University of New York</b><br>Input-Output Equations for Parameter Identifiability in Rational ODE Models</FONT>
<p>The problem of parameter identifiability is of great importance in modeling, for example in biological systems. One technique used in studying identifiability is the notion of input-output equations. Let S be a system of ordinary differential equations in several variables, some of which are observable and others of which are unobservable. The input-output equations are a subset of the consequences of S in which only observable variables appear, and from which information about the identifiability of certain parameters can be gained. We discuss input-output equation methods.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, October 12, 2018, <font color="red">14:00–15:00</font>, <a href="#into">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Peter Thompson, City University of New York</b><br>Input-Output Equations for Parameter Identifiability in Rational ODE Models, Part 2</FONT>
<p>We continue our discussion of input-output equation methods for the problem of parameter identifiability in ODE modeling. It is commonly assumed that the functions of parameters appearing as coefficients in input-output equations are identifiable. We discuss the validity of this in the single-output and multiple-output cases.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, October 19, 2018, 10:00–11:00 a.m., <a href="#into">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Françoise Point, University of Mons</b><br>Differential Expansions of Topological Larrge Fields and Transfer Results</FONT>
<p>Given a theory <I>T</I> of large topological fields of characteristic 0 admitting quantifier elimination (in some relational expansion<I> L</I> of the language of fields), we consider its (generic) expansion <I>T<sub>D</sub></I> to a theory of differential fields. Under some natural hypotheses, which we will detail, it is known that the class of existentially closed models of such expansions is axiomatizable and that its theory <I>T<SUP> *</SUP><span class="bksp5"><SUB>D</sub></span> </I> admits quantifier elimination in <I>L<sub>D</sub></I> (the language <I>L</I> to which we add the derivation <I>D</I>). For instance if one starts with the class of real-closed fields, M. Singer showed that one obtains the class of closed ordered differential fields (CODF). We will first review a number of known transfer results between <I>T</I> and <I>T<SUP> *</SUP><span class="bksp5"><SUB>D</sub></span> </I> and their consequences for the theory of dense pairs of models of <I>T</I>. Then we will concentrate on elimination of imaginaries, a property that allows one to associate with any definable set a code (for instance, the theory of differentially closed fields of characteristic zero has that property). Under the hypothesis that <I>T<SUP> *</SUP><span class="bksp5"><SUB>D</sub></span> </I> ; has open core, namely any open <I>L<sub>D</sub></I>-definable set is already <I>L</I>-definable, we will show transfer of elimination of imaginaries between <I>T</I> and <I>T<SUP> *</SUP><span class="bksp5"><SUB>D</sub></span> </I> using a topological argument due to M. Tressl in the case of CODF. This is a joint work with Pablo Cubidès Kovacsics (Caen).
<p>There will be no prerequisites in model theory.
<p>For a copy of the slides, please click <a href="https://cs.nyu.edu/~pogudin/ksda/Point.pdf">slides</a>.
<p>For a review of the talk, please click <a href="https://youtu.be/--pc13-PL7o">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, November 16, 2018, <font color="red">14:00–15:00</font>, <a href="#into">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Boris Kramer, MIT</b><br>Lifting Transformations in Dynamical Systems and Model Reduction </FONT>
<p>Model order reduction for large-scale nonlinear systems is a key enabler for design, uncertainty quantification, and control of complex systems. I will discuss a beneficial detour to deriving efficient reduced-order models for nonlinear systems. First, the nonlinear model is lifted to a model with more structure via variable transformations and the introduction of auxiliary variables. The lifted model is equivalent to the original model—it uses a change of variables, but introduces no approximations. When discretized, the lifted model yields a polynomial system of either ordinary differential equations or differential algebraic equations, depending on the problem and lifting transformation. In order to obtain computationally inexpensive models, we then proceed with reducing those lifted systems. Proper orthogonal decomposition (POD) is applied to the lifted models, yielding a reduced-order model for which all reduced-order operators can be pre-computed. We show several examples in form of a FitzHugh-Nagumo PDE and a tubular reactor PDE model, and show how this approach opens new pathways for rigorous analysis and input-independent model reduction via the introduction of the lifted problem structure.
<p>For a review of the talk, please click <a href="https://youtu.be/D0f2lfvgsHk">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, November 30, 2018, <font color="red">14:00–15:00</font>, <a href="#into">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>James Greene, Rutgers University</b><br>Mathematics Behind Induced Drug Resistance in Cancer Chemotherapy</FONT>
<p>Resistance to chemotherapy is a major impediment to successful cancer treatment that has been extensively studied over the past three decades. Classically, resistance is thought to arise primarily through random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests this evolution to resistance need not occur randomly, but instead may be induced by the application of the drug. Indeed, phenotype switching via epigenetic alterations is just recently beginning to be understood. In this work, we present a mathematical model to that describes both random and induced resistance. We discuss issues related to both structural and practical identifiability of model parameters. A time-optimal control problem is formulated and analyzed utilizing differential-geometric techniques. Specifically, the control structure is precisely characterized, and therapy outcome is analyzed for different levels of resistance induction through a combination of analytic and numerical results. Existence results are also discussed, as well as further extensions to combination therapies are also considered, and questions of combination vs. sequential therapy are studied.
<p>For a copy of the slides, please click <a href="PostedPapers/Greene113018.pdf">slides</a>.
<p>For a review of the lecture, please click <a href="https://youtu.be/beuKRZ-IKm4">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, December 7, 2018, 10:00–11:00, <a href="#into">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Li Guo, Rutgers University at Newark</b><br>Rota-Baxter Algebras and Quasi-Symmetric Functions</FONT>
<p>In the 1960s, Rota applied his first construction of free Rota-Baxter algebra and his algebraic formulation of Spitzer's identity to obtain the well-known Waring formula which relates elementary symmetric functions to power symmetric functions. He later suggested that there should be a close connection between Rota-Baxter algebras and generalizations of symmetric functions. He claimed, "In short, (Rota-)Baxter algebras represent the ultimate and most natural generalization of the algebra of symmetric functions." We present some results in support of Rota's claim. We show that a free commutative Rota-Baxter algebra can be interpreted as generalized quasi-symmetric functions from weak compositions. This result also equips the free commutative Rota-Baxter algebra with a natural Hopf algebra structure.<br>
This is joint work with Jean-Yves Thibon, Houyi Yu and Jianqiang Zhao.<br>
<p>For a copy of the slides, please click <a href="PostedPapers/LiGuo120718.pdf">slides</a>.
<p>For a review of the lecture, please click <a href="https://youtu.be/b0xxcYtS1l0">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, December 7, 2018, <font color="red">14:00–15:00</font>, <a href="#into">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Gleb Pogudin, New York University</b><br>Elimination of Unknowns in Delay-Differential Equations </FONT>
<p>Delay-differential equations are actively used in areas of applied mathematics ranging from mathematical biology to electrical engineering. Elimination of unknowns is a fundamental tool for studying solutions of equations (linear, polynomial, differential, etc.). In the talk, the first elimination algorithm for a system of delay-differential equations will be presented. For this algorithm, we develop new effective methods in differential algebraic geometry and combine them with parts of the approach taken in the first elimination algorithm for systems of difference equations designed recently by Ovchinnikov, Pogudin, and Scanlon.
This is a joint work with Wei Li, Alexey Ovchinnikov, and Thomas Scanlon.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, December 14, 2018, 10:00–11:00 a.m., <a href="#into">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Mirco Tribastone, School for Advanced Studies, Lucca</b><br>Maximal Aggregation of Polynomial Differential Equations</FONT>
<p><!--Institutions, Markets, Technologies-->Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science and engineering. Our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. In this talk I will present an aggregation technique which rests on two notions of equivalence that relate variables in polynomial differential equations whenever they have the same solution (backward criterion), or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of this approach is the encoding of a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This encoding enables the development of a partition-refinement algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimise probabilistic models of computation according to the notion of bisimulation. I will discuss the effectiveness as well as the physical interpretability of the aggregation in applications to biochemical reaction networks, gene regulatory networks, and evolutionary game theory.<br>
This is joint work with Luca Cardelli, Max Tschaikowski, and Andrea Vandin.<br>
For a copy of the slides, please click <a href="PostedPapers/Tribastone121418.pdf">slides</a>.<br>
For a review of the lecture, please click <a href="https://youtu.be/6cKJRNwjMVk">video</a>.
</BLOCKQUOTE>
<p><HR SIZE="6" width="100%" color="#003399"><a name="prelim"></a>
<p><p><FONT color="000000" size="4"> <b>Friday, February 15, 2019, 10:15–11:30 a.m.<a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Jason Bell, University of Waterloo</b><br>Invariant Hypersurfaces and Ideals Invariant Under an Endomorphism or a Derivation </FONT>
<p>We prove a general geometric theorem, which in the affine case can be phrased as follows: Suppose that <I>k</I> is a field of characteristic zero, <I>R</I> (an integral domain) and <I>S</I> are finitely generated commutative <I>k</I>-algebras, and <I>f</I>, <I>g</I> : <I>R</I> → <I>S</I> are injective <I>k</I>-algebra homomorphisms with the property that <I>f</I>(<I>R</I>) and <I>g</I>(<I>R</I>) do not contain zero divisors of <I>S</I> other than zero. Then if the set of (pure) height one radical ideals <I>I</I> of <I>R</I> such that the radical of <I>f</I>(<I>I</I>)<I>S</I> is equal to the radical of <I>g</I>(<I>I</I>)<I>S</I> is infinite, there is some <I>h</I> in the field of fractions of <I>R</I> that is not in <I>k</I> such that <I>f</I> (<I>h</I>)=<I>g</I>(<I>h</I>), where we have extended <I>f</I> and <I>g</I> to the fraction field of <I>R</I> in the natural way using the fact that <I>f</I>(<I>R</I>) and <I>g</I>(<I>R</I>) do not contain zero divisors other than zero. We show that this has numerous, somewhat unexpected applications, including recovering the work of Cantat on rational dynamics and the work of Jouanolou and Hrushovski on
<I>δ</I>-invariant ideals of a ring <I>A</I>, where <I>δ</I> is a derivation of <I>A</I>
<p>For a review of the talk, please click <a href="https://youtu.be/cNmqTNFp3sw">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, Feb 15, 2019, <font color="red">14:00–15:00 a.m. </font><a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Harm Derksen, University of Michigan </b><br>Singular Values of Tensors</FONT>
<p><I>This is a joint talk with the <a href="http://qcpages.qc.cuny.edu/~aovchinnikov/seminar.html">Courant/CUNY symbolic-numeric seminar</a>.</I></p>
Tensor decompositions have many applications, including chemometrics and algebraic complexity theory. Various notions, such as the rank and the nuclear norm of a matrix, have been generalized to tensors. In this talk I will present a new generalization of the singular value decomposition to tensors that shares many of the properties of the singular value decomposition of a matrix.
<p>For a review of the talk, please click <a href="https://www.youtube.com/watch?v=soXt00aIxmY">video</a>.
<p>For a copy of the slides, please click <a href="http://qcpages.qc.cuny.edu/~aovchinnikov/slides-Derksen.pdf">slides</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, Feb 22, 2019, <font color="red">14:00–15:00 </font><a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b> Malabika Pramanik, University of British Columbia
</b><br>Analysis and Geometry of Sparse Sets</FONT>
<p>Pattern identification in sets has long been a focal point of interest in analysis, geometry, combinatorics and number theory. No doubt the source of inspiration lies in the deceptively simple statements and the visual appeal of these problems. For example, when does a given set contain a copy of your favourite pattern (say specially arranged points on a line or a spiral, the vertices of a polyhedron or solutions of a functional equation)? Does the answer depend on how thin the set is in some quantifiable sense?
Here is another problem. Curves and surfaces form a class of thin sets in Euclidean space that is rich in analytic and geometric structure. They form the central core in many problems in harmonic and complex analysis (such as restriction phenomena and integral transforms) and play an important role in the study of partial differential equations with a geometric flavour. How well do properties of surfaces and submanifolds carry over to the setting of an arbitrary sparse set with no differential-geometric structure?
Problems of this flavour fall under the category of geometric measure theory. Under varying interpretations of size, they have been vigorously pursued both in the discrete and continuous setting, often with spectacular results that run contrary to intuition. Yet many deceptively simple questions remain open. I will survey the literature in this area, emphasizing some of the landmark results that focus on different aspects of the problem.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, March 1, 2019, 10:15–11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b> Carlos Arreche, University of Texas at Dallas
</b><br>Differential Transcendence of Elliptic Hypergeometric Functions Through Galois Theory </FONT>
<p>Elliptic hypergeometric functions arose roughly 10 years ago as a generalization of classical hypergeometric functions and <I>q</I>-hypergeometric functions. These special functions enjoy remarkable symmetry properties, like their more classical counterparts, and find applications in mathematical physics. After interpreting one of these symmetries as a linear difference equation over an elliptic curve, we apply the differential Galois theory of difference equations to show that these functions are always differentially transcendental for generic values of the parameters. This is joint work with Thomas Dreyfus and Julien Roques.
<p>For a review of the talk, please click <a href="https://youtu.be/SrmUzLTWXLI">video</a>.
<p>For a copy of the slides, please click <a href="PostedPapers/Arreche030119.pdf">slides</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, March 8, 2019, <font color="red">No Seminar</font></b> </p></FONT>
<p><p><FONT color="000000" size="5"> <b><font color="red">Monday, March 11, 2019 to Saturday, March 16, 2019 at CUNY/Courant</font> </b></FONT></p>
<BLOCKQUOTE>
<FONT color="000000" size="5"><b>Workshop on Model Theory, Differential/Difference Algebra, and Applications</b></FONT>
</BLOCKQUOTE>
<UL>
<li><p><p><FONT color="000000" size="4"> <b>Monday, March 11, 2019, 14:00–14:30; Room 3207 (Graduate Center, CUNY)</b> </p></FONT>
<BLOCKQUOTE>
<p>Discussions
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Monday, March 11, 2019, 14:30–15:30; Room 3207 (Graduate Center, CUNY)</b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Joel (Ronnie) Nagloo, City University of New York</b><br> The Generic Schwarz Triangle Equations</FONT>
<p>In this talk, I will focus on the ODEs satisfied by the Schwarz triangle functions. These are the conformal mappings from the circular triangles (in ℂ) onto the complex unit disk. I will explain how, building on my recent joint work with Casale and Freitag on the genus zero Fuchsian groups, one can give a full description of the structure of the set of solutions of a generic Schwarz triangle equation. More precisely, I will explain how one can show that the solution set is strongly minimal and also strictly disintegrated, that is, there are no algebraic relations between distinct solutions (including their derivatives).
<p>For a review of the talk, please click <a href="https://youtu.be/8bS5qgN_GVo">video</a>.
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Monday, March 11, 2019, 15:30–16:00; Room 3207 (Graduate Center, CUNY) </b> </p></FONT>
<BLOCKQUOTE>
<p>Discussions
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Monday, March 11, 2019, 16:00–17:00; Room 4214.03 (Graduate Center, CUNY)</b> </p></FONT>
<BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Tuesday, March 12, 11:45–12:45; Room 3207 (Graduate Center, CUNY) </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Michael Wibmer, University of Notre Dame</b><br> On the Dimension of Systems of Algebraic Difference Equations </FONT>
<p>We introduce and study a notion of dimension for the solution set of a system of algebraic difference equations. This dimension measures the degrees of freedom when determining a solution in the ring of sequences. This number need not be an integer, but as we show, it satisfies properties suitable for a notion of dimension. We also show that the dimension of a difference monomial is given by the covering density of its set of exponents.
<p>For a review of the talk, please click <a href="https://youtu.be/EzzGS1OjUv0">video</a>.
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Tuesday, March 12, 12:45–13:45; Room 3207 (Graduate Center, CUNY) </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Ivan Tomaai, Queen Mary University</b><br>A Topos-Theoretic View of Difference Algebra</FONT>
<p>Abstract difference algebra was founded by Ritt in the 1930s as the study of algebraic structures equipped with distinguished endomorphisms. This approach has a long and productive history, but attempts to develop methods of homological algebra within this context quickly reach insurmountable obstacles.
We will show how to use the methods of topos theory and categorical logic to resolve these issues and to elevate the study of difference algebraic geometry to the level of classical algebraic geometry.
<p>For a review of the talk, please click <a href="https://youtu.be/81bkd5zobiU">video</a>.
<p>For a copy of the slides, please click <a href="PostedPapers/Tomasic031219.pdf">slides</a>.
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Tuesday, March 12, 2019, 14:00–15:00; Room 3205 (Graduate Center, CUNY)</b> </p></FONT>
<BLOCKQUOTE>
<p>Discussions
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Wednesday, March 13, 2019, 15:10–18:00; Room 805 Warren Weaver Hall, Courant Institute (251 Mercer Street, Manhattan)</b> </p></FONT>
<BLOCKQUOTE>
<p>Discussions
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Thursday, March 14, 11:45–12:45; Room 3309 (Graduate Center, CUNY) </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Wei Li, Chinese Academy of Sciences</b><br>Sparse Resultants in Differential and Difference Algebra: An Overview </FONT>
<p>The (sparse) resultant, which gives conditions for an over-determined system of polynomial equations to have common solutions, is a basic concept in algebraic geometry, and emerges to be one of the most powerful computational tools in (sparse) elimination theory due to its ability to eliminate several variables simultaneously. In recent years, a theory has been developed for these analogous concepts in differential and difference algebra, and many new problems have arisen. In this talk, I will give an overview of the progress we have made in this area, and present several open problems.
<p>For a review of the talk, please click <a href="https://youtu.be/MOkzQwjjaRo">video</a>.
<p>For a copy of the slides, please click <a href="PostedPapers/WeiLi031419.pdf">slides</a>.
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Thursday, March 14, 12:45–13:45; Room 3309 (Graduate Center, CUNY) </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>James Freitag, University of Illinois at Chicago
</b><br>Algebraic Relations Between Solutions of Painlevé Equations</FONT>
<p>In this talk we will explain the origin and importance of Painlevé equations, before addressing the central question of the talk. What are the algebraic relations between solutions of Painlevé equations? The work of Pillay and Nagloo brought this question into focus, and following recent work of Nagloo on the sixth Painlevé equation, we can now give a complete answer when at least one coefficient in one of the equations we consider is transcendental. This is joint work with Ronnie Nagloo.
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Thursday, March 14, 2019, 14:00–15:00; Room 4214.03 (Graduate Center, CUNY)</b> </p></FONT>
<BLOCKQUOTE>
<p>Discussions
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Friday, March 15, 2019, 10:15–11:15 a.m. <a href="#info">Room 5382 (Graduate Center, CUNY)</a> </b> </p></FONT>
<BLOCKQUOTE><FONT color="000000" size="4"><b> Rémi Jaoui, University of Waterloo
</b><br>Disintegration and Planar Algebraic Vector Fields</FONT>
<p>A differential equation is disintegrated (or geometrically trivial) if any algebraic relation between an arbitrary number of its solutions can be decomposed into algebraic relations between couples of solutions. I will explain that disintegration is a typical property for complex planar algebraic vector fields of degree
<I>d</I> e" 3. This implies, for example, that the set of parameters for which this property holds has full Lebesgue measure in the parameter space of algebraic planar vector fields of degree
<I>d</I> e" 3.
<p>For a review of the talk, please click <a href="https://youtu.be/MGXjVP9T5Wc">video</a>.
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Friday, March 15, 2019, 14:00–15:00, <font color="red">Room 6417 </font> (Graduate Center, CUNY)</b> </p></FONT>
<BLOCKQUOTE><FONT color="000000" size="4"><b>Rahim Moosa, University of Waterloo
</b><br>Pullbacks Under the Logarithmic Derivative</FONT>
<p>Let <I>X</I>
be the Kolchin closed set defined by an algebraic differential equation of the form
<I>Dx</I> = <I>f</I>(<I>x</I>), where <I>f</I> is a rational function over constant parameters. Rosenlicht's theorem gives us a condition on <I>f</I>
that tells us when <I>X</I>
is (in model-theoretic terms) internal to the constants. In this talk I will describe a criterion in a similar spirit answering the question of when the pullback of <I>X</I>
under the logarithmic derivative is internal to the constants. The case of nonconstant parameters will also be discussed. These are results from my student Ruizhang Jin's recent thesis, as well as further joint work.
<p>For a review of the talk, please click <a href="https://youtu.be/6HMvYwnu0MI">video</a>.
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Friday, March 15, 2019, 15:00–16:00; Room 6417 (Graduate Center, CUNY)</b> </p></FONT>
<BLOCKQUOTE>
<p>Discussions
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Saturday, March 16, 2019, 9:00–10:00 a.m.; room 201, Warren Weaver Hall, Courant Institute (251 Mercer Street, Manhattan) </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Gleb Pogudin, New York University
</b><br>Primitive Element Theorem for Fields with Commuting Derivations and Automorphisms </FONT>
<p>The Primitive Element Theorem says that every finitely generated algebraic extension of fields of characteristic zero is generated by a single element. It is a classical tool in field theory and symbolic computation. It has been generalized to partial differential fields by Kolchin in 1942 and to difference fields (with a single automorphism) by Cohn in 1965. These theorems guarantee that if an extension <I>E</I> of <I>F</I>
is finitely generated and algebraic in an appropriate sense and the ground field <I>F</I>
is "nonconstant", then the extension can be generated by a single element. These generalizations played an important role in differential/difference algebra and its applications.
<p>However, both theorems by Kolchin and Cohn imposed an extra condition for the ground field <I>F</I>
to be "nonconstant" that made them not applicable to many important extensions coming from autonomous differential/difference equations or algebraic varieties equipped with a vector field or an automorphism. In 2015, I have partially resolved this issue by strengthening Kolchin's theorem in the case of one derivation so that the condition that <I>F</I>
contains a nonconstant was replaced by a natural condition that <I>E</I>
contains a nonconstant (otherwise, the derivation would be zero).
<p>In this talk, I will describe my recent result that generalizes the primitive element theorems by Kolchin, Cohn, and myself in two directions:
<p>• the existence of a primitive element is established for fields with any number of derivations and automorphisms commuting with each other (this includes, for example, partial difference and differential-difference fields);
<p>• no extra condition on the ground field is imposed.
<p>For a review of the talk, please click <a href="https://youtu.be/QVXG-n-WcWM">video</a>.
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Saturday, March 16, 2019, 10:00–11:00 a.m.; room 201, Warren Weaver Hall, Courant Institute (251 Mercer Street, Manhattan)</b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b> Henry Towsner, University of Pennsylvania
</b><br>Ultraproducts: What Are They Good For?</FONT>
<p>The use of ultraproducts as a technique for proving results in algebra and differential algebra is well established. We will discuss how ultraproduct arguments can be transformed into explicit, constructive arguments. Along the way, we will be able to identify what features of a proof can make them suitable for simplifying using an ultraproduct.
<p>For a review of the talk, please click <a href="https://youtu.be/CE2dDPkq1EI">video</a>.
</BLOCKQUOTE>
<li><p><p><FONT color="000000" size="4"> <b>Saturday, March 16, 2019, 11:00–12:00; Room 605 Warren Weaver Hall, Courant Institute (251 Mercer Street, Manhattan)</b> </p></FONT>
<BLOCKQUOTE>
<p>Discussions
</UL>
<p><p><FONT color="000000" size="4"> <b>Friday, March 22, 2019, 10:15–11:30, <a href="#into">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Joseph Scott, Clemson University</b><br>Rapid and Accurate Reachability Analysis for Nonlinear Systems by Exploiting Model Redundancy</FONT>
<p>The presentation will cover recent advances in techniques for rapidly and accurately propagating rigorous uncertainty bounds through complex dynamic models. In applications from autonomous aircraft to biochemical networks, the ability to quantify the effects of uncertainty is essential for designing systems that are passively robust to uncertainty, as well as for making optimal, real-time control decisions under uncertainty. Moreover, methods that can provide rigorous bounds on the system states achievable under uncertainty are uniquely useful in their ability to guarantee that a particular course of action will satisfy all relevant constraints (e.g., in aircraft collision avoidance). Although it has long been possible to compute such bounds efficiently using interval methods, the results are often too conservative to be of any practical use (i.e., the upper and lower bounds tend to ± ∞ over short time-scales). In contrast, modern bounding strategies can achieve remarkably sharp bounds, even for highly nonlinear systems with large uncertainties, but are far too costly for real-time decision making when the number of states and uncertain parameters exceeds ≈ 5. Thus, there is a critical need for an alternative approach to uncertainty propagation in nonlinear dynamic systems that is simultaneously rigorous, accurate, fast enough for real-time applications, and scalable to much larger systems.
<p>Toward this end, our key insight is that the conservatism of fast interval methods can be dramatically reduced through the use of model redundancy. Indeed, our recent work shows that bounds produced by these methods often enclose large regions of state-space that violate redundant relations implied by the dynamics, such as conservation laws, and that these can be exploited to obtain much sharper bounds for a limited class of systems. Motivated by these observations, we have developed an innovative new approach for arbitrary systems based on the deliberate introduction of model redundancy to reduce conservatism. This technique lies at interface of numerical and symbolic computing and has been shown to lead to remarkably sharp bounds at low cost in a variety of challenging applications. We will discuss the mechanisms by which redundancy leads to improved bounds, strategies for introducing redundant equations that are effective in this context, and preliminary results on automating the construction of these equations. Finally, our methods will be demonstrated on uncertain dynamic system arising in the chemical and aerospace domains.
<p>For a copy of the slides, please click <a href="PostedPapers/Scott032219.pdf">slides</a>.
<p>For a review of the lecture, please click <a href="https://youtu.be/cOalHP9HTNk">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, March 22, 2019, <font color="red">14:00–15:30 a.m. </font><a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Varadharaj Ravi Srinivasan, IISERM (Indian Institute of Science Education and Research Mohali</b><br>Integration in Finite Terms: Error Functions, Logarithmic Integrals and Polylogarithmic Integrals</FONT>
<p>The talk concerns the problem of integration in finite terms with special functions. Our main result extends the classical theorem of Liouville in the context of elementary functions to various classes of special functions: error functions, logarithmic integrals, dilogarithmic and trilogarithmic integrals. The results are important since they provide a necessary and sufficient condition for an element of the base field to have an antiderivative in a field extension generated by transcendental elementary functions and special functions. A special case of our main result simplifies and generalizes a theorem of Baddoura on integration in finite terms with dilogarithmic integrals. Our results can be naturally generalized to include polylogarithmic integrals and to this end, a conjecture will be stated for integration in finite terms with transcendental elementary functions and polylogarithmic integrals.
<p>This is a joint with Yashpreet Kaur.
<p>For a copy of the slides, please click <a href="PostedPapers/Srinivasan032219.pdf">slides</a>.
<p>For a review of the lecture, please click <a href="https://youtu.be/9_gYa0nC2Tw">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, March 29, 2019, 10:15–11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Martin Hils, University of Münster
</b><br>Imaginaries in Separably Closed Valued Fields</FONT>
<p>Let <I>p</I> be a fixed prime number and let <I>SCV F<sub>p</sub></I>
be the first order theory of separably closed non-trivially valued fields of characteristic
<I>p</I>. In the talk, we will see that, in many ways, from a model-theoretic point of view, the step from algebraically closed VALUED fields in characteristic be the first order theory of separably closed non-trivially valued fields of characteristic
<I>p</I> to
to <I>SCV F<sub>p</sub></I>
is not more complicated than the one from algebraically closed fields to separably closed fields in characteristic <I>p</I>.
<p>At a basic level, this is true for quantifier elimination (Delon), for which it suffices to add parametrized <I>p</I>-coordinate functions to any of the usual languages for valued fields. At a more sophisticated level, in finite degree of imperfection, when a
<I>p</I>-basis is named by constants or when one just works with Hasse derivations, the imaginaries (i.e. definable quotients) are classified by the so-called geometric sorts of Haskell-Hrushovski-Macpherson, certain higher-dimensional analogs of the residue field and the value group. This classification is proved by a reduction to the algebraically closed case, using prolongations.
<p>This is joint work with Moshe Kamensky and Silvain Rideau.
<p>For a review of the talk, please click <a href="https://youtu.be/C3UY-vpGzF4">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, March 29, 2019, <font color="red">11:45–13:45, Room 6114</font> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Peter Thompson, City University of New York</b><br> A Differential Algebra Approach to Commuting Polynomial Vector Fields and to Parameter Identifiability in ODE Models (Ph.D. defence)</FONT>
<p>In the first part, we study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field. One motivating factor is that we can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. We first show that a linear vector field admits a full complement of commuting vector fields. Then we study a type of planar vector field for which there exists an upper bound on the degree of a commuting polynomial vector field. Finally, we turn our attention to conservative Newton systems and show the following result. Let
<I>f</I> ∈ <I>K</I>[<I>x</I>], where <I>K</I> is a field of characteristic zero, and <I>d</I> is the derivation that corresponds to the differential equation
<I>x''</I> = <I>f</I>(<I>x</I>)
in a standard way. We show that if deg <I>f</I> e" 2, then any
<I>K</I>-derivation commuting with <I>d</I> is equal to <I>d</I>
multiplied by a conserved quantity. For example, the classical elliptic equation
<I>x</I>'' = 6<I>x</I><sup>2</sup> + <I>a</I>, where
<I>a</I> ∈ ℂ, falls into this category.
<p>In the second part, we study structural identifiability of parameterized ordinary differential equation models of physical systems, for example, systems arising in biology and medicine. A parameter is said to be structurally identifiable if its numerical value can be determined from perfect observation of the observable variables in the model. Structural identifiability is necessary for practical identifiability. We study structural identifiability via differential algebra. In particular, we use characteristic decompositions. A system of ODEs can be viewed as a set of differential polynomials in a differential ring, and the consequences of this system form a differential ideal. This differential ideal can be described by a finite set of differential equations called a characteristic decomposition. The technique of studying identifiability via a set of special equations, sometimes called "input-output" equations, has been in use for the past thirty years. However it is still a challenge to provide rigorous justification for some conclusions that have been drawn in published studies. Our work provides justification for some cases, and provides a computable condition that can be used to justify the others. We present a computable condition on the elements of the characteristic decomposition such that if this condition is satisfied, then the conclusions about identifiability drawn from this decomposition are correct. We proceed to show that all linear systems of ODEs with one observable variable satisfy this condition.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, Apr 5, 2019, 10:15–11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b> Nigel Pynn-Coates, University of Illinois at Urbana-Champaign
</b><br>Asymptotic Valued Differential Fields </FONT>
<p>The general goal is to do valuation theory for differential fields given an appropriate condition on the interaction between the valuation and the derivation. In this talk, I will consider asymptotic valued differential fields, introduced by Aschenbrenner, van den Dries, and van der Hoeven during their work on transseries, extending work of Rosenlicht. I will present analogues of three fundamental results from valuation theory that go through in this setting, concerning (differential-algebraically) maximal immediate extensions and their connection with differential-henselianity.
<p>For a review of the talk, please click <a href="https://youtu.be/8r9XaD4KRXk">video</a>.
<p>For a copy of the slides, please click <a href="PostedPapers/PynnCoates040519.pdf">slides</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, Apr 12, 2019, 10:15–11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Richard Gustavson, Manhattan College
</b><br>Algebraic Structures from Integral Equations </FONT>
<p>Integral calculus is in general more complicated than differential calculus. For functions of one variable, integral equations involving integrals of the form
<B>∫</B><I><sup><sup>x</sup></sup><span class="bksp5"><sub><sub>a</sub></sub></span></I> <I>f</I>(<I>t</I>)<I>dt</I>,
for some unknown function <I>f</I> have been studied algebraically using the theory of Rota-Baxter algebras. In this talk we will discuss the algebraic structures of Volterra integral equations, which are equations involving integrals of the form <B>∫</B><I><sup><sup>x</sup></sup><span class="bksp5"><sub><sub>a</sub></sub></I></span> <I>K</I>(<I>x,t</I>) <I>f</I>(<I>t</I>)<I>dt</I>, for some unknown function <I>f</I>
and given kernel <I>K</I>. While there are methods for finding solutions to Volterra equations, the presence of the kernel <I>K</I>(<I>x,t</I>)
as a function of <I>x</I> and <I>t</I>
makes these equations much more difficult to study from an algebraic perspective. This talk is based on joint work with Li Guo and Yunnan Li.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, Apr 12, 2019, <font color="red">12:15–13:45 Room 6114 </font></b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Mengxiao Sun, Graduate Center, CUNY
</b><br>On the Complexity of Computing Galois Groups of Differential Equations (Ph.D. defense) </FONT>
<p>The differential Galois group for a linear differential equation is an analogue of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions. Hrushovski first proposed an algorithm for computing the differential Galois group of a general linear differential equation. The complexity, in the worst case, of computing a Gröbner basis is doubly exponential in the number of variables. Hence, a double-exponential degree bound for computing Gröbner bases would be involved if one chooses to represent an algebraic subgroup by the generating set of its defining ideal. Triangular decomposition provides an alternative and relatively efficient method. In order to give a better bound, we represent an algebraic subgroup by the triangular sets instead of the generating sets. We apply Szántó's modified Wu-Ritt type decomposition algorithm and make use of the numerical bound for Szántó's algorithm to adapt to the complexity analysis of Hrushovski s algorithm. We present a triple exponential bound which is the degree bound of the polynomials used in the first step of Hrushovski's algorithm for finding a proto-Galois group.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, Apr 19–28, 2019, Spring Recess. No seminar.</b>
<p><p><FONT color="000000" size="4"> <b>Friday, May 3, 2019, 10:15–11:30 a.m. <a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b> Jorge Vitório Pereira, Instituto Nacional de Matemática Pura e Aplicada
</b><br>Effective Integration of Polynomial Differential Equations</FONT>
<p>The plan is to discuss the following question: "Can one (algorithmically) decide if a polynomial vector field on the plane admits a rational first integral/Liouvillian first integral?" After briefly recalling the history of the analogue problem for linear differential equations, I will review some recent results on the subject obtained in collaborations with Gael
and <a href="https://arxiv.org/abs/1604.05276">Alcides Lins Neto</a> and <a href="https://arxiv.org/abs/1612.06932">Roberto Svaldi</a>.
<p>For a copy of the slides, please click <a href="PostedPapers/Pereira050319.pdf">slides</a>.
<p>For a review of the lecture, please click <a href="https://youtu.be/ZEj7WVSgE_Y">video</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b>Friday, May 10, 2019, <font color="red">14:00–15:00 </font><a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b> Yunnan Li, Guangzhou University
</b><br>Extension of Gröbner-Shirshov Basis of an Algebra to Its Generating Free Differential Algebra</FONT>
<p>The free differential algebra on a set is well-understood as the polynomial algebra on the differential variables. More generally, a free differential algebra on an algebra can be defined, giving the left adjoint functor of the forgetful functor from differential algebras to algebras, instead of sets. Little is known for such free objects. In this talk, we show that generator-relation properties of a base algebra can be extended to the free differential algebra on this algebra. More precisely, Gröbner-Shirshov basis property of the base algebra can be extended, allowing construction of these more general free differential algebras in concrete terms. Examples are given as illustrations. Finally, the free differential Lie algebra on a Lie algebra \(\mathfrak{g}\)
are introduced and are shown to be generated within the free differential associative algebra generated by the enveloping algebra
\(U(\mathfrak{g})\) of \(\mathfrak{g}\), similar to the classical case of generating free Lie algebras in a free associative algebra.
<p>This is joint work with Li Guo.
<p>For a copy of the slides, please click <a href="PostedPapers/YunnanLi051019.pdf">slides</a>.
</BLOCKQUOTE>
<p><p><FONT color="000000" size="4"> <b><font color="red">Thursday, May 30, 2019, 10:15–11:30 a.m. </font><a href="#info">Room 5382</a> </b> </p></FONT>
<BLOCKQUOTE>
<FONT color="000000" size="4"><b>Maria Pia Saccomani, University of Padova</b><br>Structural Identifiability of Rational ODE Models in Biological Systems:<br> A Real-World Application of Differential Algebra</FONT>
<p>ODE models used to describe biological systems often depend on many unknown parameters. Structural identifiability concerns the uniqueness of the model parameters as determined from input-output data, under ideal conditions of noise-free observations. It is thus a prerequisite for parameter estimation to provide reliable and accurate results from experimental data. Often these ODE models consist of rational or even polynomial differential equations. In this context, the aims of this talk are
<ol> <li> to present a differential algebra method to test structural identifiability based on the structure of the characteristic set of the differential ideal generated by the polynomials defining the model,
<li> to explain the important role played by the model initial conditions in the characteristic set and the role of a system-theoretic property called accessibility, crucial to correctly check identifiability,
<li>to illustrate how one can combine our structural identifiability test with practical identifiability approaches in order to calculate either all the multiple parameter solutions of a locally identifiable model or, the analytic relations between the infinite number of solutions of a nonidentifiable model. These different solutions are equivalent to describe the observable input-output behaviour but they generally yield different dynamic behaviours of unmeasurable variables, whose prediction is often the main goal of mathematical modeling.
</ol>
The relevance of structural identifiability analysis in biological modeling is shown by some recent examples including HIV and oncological models. Structural identifiability is tested with our freely available software DAISY (Differential Algebra for Identifiability of Systems).
<p>For a copy of the slides, please click <a href="PostedPapers/Saccomani053019.pdf">slides</a>.
<p>For a review of the talk, please click <a href="https://youtu.be/7PRdrbLvgHw">video</a>.
</BLOCKQUOTE>
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