ESP Biography



MICHAEL LIVSHITS, A non-affiliated Cambridge local




Major: Math

College/Employer: MIT

Year of Graduation: N/A

Picture of Michael Livshits

Brief Biographical Sketch:

Available on my home page at http://www.mathfoolery.com



Past Classes

  (Look at the class archive for more.)


Derivatives and integrals I: basic ideas and examples in Splash! 2009 (Nov. 21 - 22, 2009)
Let $$f(x)$$ be a polynomial and $$a$$ be a number. Then $$f(x)-f(a)$$ is also a polynomial in $$x$$, and $$a$$ is its root. Then, as you may remember from your algebra class, $$f(x)-f(a)=p(x,a)(x-a)$$, where $$p$$ is a polynomial in $$x$$ and $$a$$. The number $$p(a,a)$$ is called the derivative of $$f$$ at $$a$$ and written as $$f'(a)$$. This is differentiation, as understood by Descartes. You may notice that $$a$$ is a double root of $$f(x)-f(a)-f'(a)(x-a)$$, that is why the straight line $$y-f(a)=f'(a)(x-a)$$ is the tangent at the point $$(a,f(a))$$ to the curve $$y=f(x)$$, the graph of $$f$$, and why polynomials with positive derivarives are increasing functions. You can look at $$f$$' as a new function, the derivative of $$f$$. Now, for $$x > 0$$ take the triangle on the xy-plane with the vertices $$(0,0)$$, $$(x,x)$$ and $$(0,x)$$. Its area, $$A(x)=x^2/2$$ is "the area under the graph" of the function $$y=x$$ between $$0$$ and $$x$$. When you differentiate it, you get $$A'(x)=x$$, which is the function you had started with! This fact is called Newton-Leibniz theorem and it is useful for calculating areas, since for many functions $$f$$ it is easy to find an antiderivative, i.e., such a function $$F$$ that $$F'=f$$. Definite integrals are areas under the graphs, and indefinite integrals are antiderivatives. Looks rather simple so far? It is, but we have to deal not only with polynomials, and it gets a bit tricky. I will explain some of these tricks in a mathematically sound manner, but without obscuring the basic ideas with excessive generality and heavy technicalities, such as continuity, limits or real numbers.


Derivatives and integrals II: some theory and many variables in Splash! 2009 (Nov. 21 - 22, 2009)
Continuation of Derivatives and Integrals I. I will present a streamlined theory of differentiation and integration, based on some simple inequalities. Then I will show how the ideas that work for functions of one variable can be extended to two variables. I will discuss more than 2 variables briefly, depending on time and the audience.


Differentiation and Integration of Friendly Functions I: the Basics in Spark! Spring 2009 (Mar. 07, 2009)
The aim is to understand the basics of differentiation and integration, starting with simple examples. Concentrating on well-behaved, "friendly" functions, we will not have to wade through somewhat intimidating notions of continuity and limits to get going. This class is for people who are fluent in high school algebra and geometry, and are curious about differentiation and integration; some "precalculus" is a plus, familiarity with physics will help with motivation and appreciation. People who know some calculus may also find this unorthodox approach entertaining and/or thought-provoking. Most of the content of this class is summarized in the first 10 slides for the 15' talks that I gave at MathFest in 2004 and at the joint AMS-MAA meeting in 2006 (both fell on Friday the 13th), available at http://www.mathfoolery.org/talk-2004.pdf


Differentiation and Integration of Friendly Functions II: Some Theory in Spark! Spring 2009 (Mar. 07, 2009)
Continuation of Differentiation and Integration of Friendly Functions I. We will develop a streamlined theory of differentiation and integration based on some simple inequalities and prove the fundamental theorem of calculus. We may also a take a look at more sophisticated topics (power series, Taylor formula, integration and differentiation of multivariable functions, etc.) or discuss the relation of our approach to limits and continuity, if there is enough time and interest. The core content of this class is summarized in the last 3 slides for the talks that I gave at MathFest in 2004 and at the joint AMS-MAA meeting in 2006 (both fell on Friday the 13th), available at http://www.mathfoolery.org/talk-2004.pdf


Differentiation and Integration of Friendly Functions I: the Basics in Splash! 2008 (Nov. 22 - 23, 2008)
In this class we will try to understand the basics of differentiation and integration, starting with simple examples.We will concentrate on well-behaved, "friendly" functions, and will not have to wade through somewhat intimidating notions of continuity and limits to develop our theory. This class is for people who are fluent in high school algebra and geometry, and are curious about differentiation and integration; some "precalculus" is a plus, familiarity with physics will help with motivation and appreciation. People who know some Calculus may also find this unorthodox approach entertaining and/or thought-provoking. Most of the content of this class is summarized in the first 10 slides for the talks that I gave at MathFest in 2004 and at the joint AMS-MAA meeting in 2006 (both fell on Friday the 13th), available at http://www.mathfoolery.org/talk-2004.pdf


Differentiation and Integration of Friendly Functions II: Some Theory in Splash! 2008 (Nov. 22 - 23, 2008)
Continuation of Differentiation and Integration of Friendly Functions I. We will develop a streamlined theory of differentiation and integration based on some simple inequalities and prove the fundamental theorem of Calculus. We may akso a take a look at more sophisticated topics (power series, Taylor formula, integration and differentiation of multivariable functions, etc.) or discuss the relation of our approach to limits and continuity, if there is enough time and interest. The core content of this class is summarized in the last 3 slides for the talks that I gave at MathFest in 2004 and at the joint AMS-MAA meeting in 2006 (both fell on Friday the 13th), available at http://www.mathfoolery.org/talk-2004.pdf Prerequisite: Differentiation and Integration of Friendly Functions I.


Party Tricks, Geometry and Topology II: Rotations, Quaternions and Spinors in Splash! 2008 (Nov. 22 - 23, 2008)
Attach some loose strings to a chair and the other ends of these strings to some other furniture. Tumble the chair once. Can you untangle the strings without futher rotating the chair or moving the other furniture around? The answer is "no," and we will try to see why during this class. Now tumble the chair once more in the same direction. The strings become even more messed up, but amazingly, you can untagle them now. Visit http://gregegan.customer.netspace.net.au/APPLETS/21/21.html to get some idea how. Now grab a coffee cup by its bottom. Can you give the cup 2 full revolutions without spilling the coffee or twisting your arm and/or hand out of their joints? Hint: pass the cup under your forearm during the first revolution and keep it over your forearm during the second one. See page 1013 in section 23, chapter VII of the free physics book at http:/www.motionmountain.com for an illustration. Don't attempt this trick with hot coffee in the cup before you become good at it. In this class, besides practicing these and some other party tricks, you will learn several ways to mathematically describe rotations in 3 dimensions. One particularly elegant description uses quaternions and will be especially handy in seeing the connection with quantum mechanics of electrons and understanding why the table of chemical elements is periodic. Prerequisites: Fluency in high school algebra and 3-d geometry, familiarity with vectors, matrices, trigonometry and complex numbers.


Calculus Without Limits in Splash! 2008 (Nov. 22 - 23, 2008)
Sounds like an oxymoron, doesn't it? Come to this class and see that it is not. I'll show you how differentiation and integration can be built - rigorously - from the ground up by using only elementary tools (algebra, geometry, inequalities). We will start with some simple examples and end with a proof of the fundamental theorem. I'll follow pretty closely the slides for the talk I gave at 2004 Mathfest at http://www.mathfoolery.org/talk-2004.pdf This class is for those who already know calculus and want to take a fresh look at it, especially from a mathematical perspective. Those who don't know calculus yet may find it a bit too fast to understand everything, they may be better off taking Differentiation and Integration of Friendly Functions I and II.


Party Tricks, Geometry and Topology I: Mobius Band, Links, Knots and Projective Plane in Splash! 2008 (Nov. 22 - 23, 2008)
Come to this class to learn some tricks with rubber bands and paper ribbons, to learn about knots and links, to experiment with the Mobius band, to explore the projective plane and to understand why it doesn't live in our three-dimensional space.


From Algebra and Geometry to Differentiation and Integration in HSSP (2010)
Integration and differentiation developed from the attempts to extend the notions of area and tangency from the Euclidean geometry of ...


Derivatives and integrals I: basic ideas and examples in SPLASH (2010)
Let $$f(x)$$ be a polynomial and a be $$a$$ number. Then $$f(x)-f(a)$$ is also a polynomial in $$x$$, and $$a$$ ...


Derivatives and integrals II: some theory and many variables in SPLASH (2010)
Continuation of Derivatives and integrals I. I will present a streamlined theory of differentiation and integration, based on some simple ...


Party tricks, geometry and topology in SPLASH (2010)
We will do some party tricks with rubber bands, paper ribbons, coffee cups, belts and ropes and learn some topology ...


Calculus a la Euler II: Some Theory via Inequalities in SPLASHONWHEELS (2008)
Continuation of Calculus a la Euler I. I will present a streamlined theory of Calculus based on some simple inequalities ...


Calculus a la Euler I: The Basics via Algebra and Geometry in SPLASHONWHEELS (2008)
Leonard Euler, one of the most brilliant and prolific mathematicians ever, was born on April 15 1707. He lived and ...


Some Party Tricks, Geometry and Topology in SPLASHONWHEELS (2008)
Attach some loose strings to a chair and the other ends of these strings to some other furniture. Tumble the ...


Differentiation, Integration, Approximation. in HSSP (2008)
We will try to understand differentiation, integration, differential equations and approxmation, starting with a rather easy case of polynomials, and ...


Calculus Basics by Problem and Example in HSSP (2007)
We will try to understand Calculus by using some tools from algebra (polynomials, factoring, inequalities, the geometric series) and geometry ...


Calculus Basics by Problem and Example in HSSP (2007)
We will try to understand Calculus by using some tools from algebra (polynomials, factoring, inequalities, the geometric series) and geometry ...


Calculus a la Euler II: Some Theory via Inequalities in SPLASH (2007)
Continuation of Calculus a la Euler I. I will present a streamlined theory of Calculus based on some simple inequalities ...


Calculus a la Euler II: Some Theory via Inequalities in SPLASH (2007)
Continuation of Calculus a la Euler I. I will present a streamlined theory of Calculus based on some simple inequalities ...


Calculus a la Euler I: The Basics via Algebra and Geometry in SPLASH (2007)
Leonard Euler, one of the most brilliant and prolific mathematicians ever, was born on April 15 1707. He lived and ...


Calculus a la Euler I: The Basics via Algebra and Geometry in SPLASH (2007)
Leonard Euler, one of the most brilliant and prolific mathematicians ever, was born on April 15 1707. He lived and ...


Complex Numbers, Quaternions, Rotations and Spins in SPLASH (2007)
Many of you probably know that complex numbers are related to geometry in 2 dimensions. In this class we will ...


Some Party Tricks, Geometry and Topology in SPLASH (2007)
Attach some loose strings to a chair and the other ends of these strings to some other furniture. Tumble the ...


Some Party Tricks, Geometry and Topology in SPLASH (2007)
Attach some loose strings to a chair and the other ends of these strings to some other furniture. Tumble the ...


Caluclus Basics by Problem and Example in HSSP (2006)
We will try to understand Calculus by using some tools from algebra (polynomials, factoring, inequalities, the geometric series) and geometry ...


A Modern Approach to Calculus and Its Theory in HSSP (2006)
Have you ever wondered why your Calculus homework is mostly algebra (it's because differentiation is an algebraic operation related to ...


Elementary Calculus: Playing with Formulas in SPLASHONWHEELS (2006)
I will explain how to differentiate and how to integrate, and show some applications. My approach will be mostly heuristic, ...


Elementary Calculus: Playing with Inequalities in SPLASHONWHEELS (2006)
I will present a theory of Calculus based on some simple inequalities and prove the Fundamental Theorem. I will also ...


Magic with Paper Ribbons and Other Topological Tricks in SPLASHONWHEELS (2006)
Why did the chicken cross the Mobius band? To get to the same side! When do you have the ultimate ...


Elementary Calculus II: Some Theory via Inequalities in SPLASH (2006)
Continuation of Elementary Calculus I. I will present a streamlined theory of Calculus based on some simple inequalities and prove ...


Elementary Calculus II: Some Theory via Inequalities in SPLASH (2006)
This is the Sunday repetition of my Saturday class. I will present a streamlined theory of Calculus based on some ...


Elementary Calculus I: The Basics via Algebra and Geometry in SPLASH (2006)
This is the Sunday repetition of my Saturday class. I will explain the basic ideas of Calculus (differentiation, integration and ...


Elementary Calculus I: The Basics via Algebra and Geometry in SPLASH (2006)
I will explain the basic ideas of Calculus (differentiation, integration and the fundamental theorem), starting with some simple examples and ...


Some Party Tricks and Topology of Rotations in SPLASH (2006)
This is the repetition of my Saturday class. Attach some loose ropes to a chair and the other ends of ...


Some Party Tricks and Topology of Rotations in SPLASH (2006)
Attach some loose ropes to a chair and the other ends of these ropes to some other furniture. Tumble the ...


Elementary Calculus: Playing with Formulas in SPLASH (2005)
I will explain how to differentiate and how to integrate, and show some applications. My approach will be mostly heuristic, ...


Elementary Calculus: Playing with Formulas in SPLASH (2005)
I will explain how to differentiate and how to integrate, and show some applications. My approach will be mostly heuristic, ...


Elementary Calculus: Playing with Inequalities in SPLASH (2005)
I will present a theory of Calculus based on some simple inequalities and prove the Fundamental Theorem. I will also ...


Elementary Calculus: Playing with Inequalities in SPLASH (2005)
I will present a theory of Calculus based on some simple inequalities and prove the Fundamental Theorem. I will also ...


Magic with Paper Ribbons and Other Topological Tricks in SPLASH (2005)
Why did the chicken cross the Mobius band? To get to the same side! When do you have the ultimate ...


Magic with Paper Ribbons and Other Topological Tricks in SPLASH (2005)
Why did the chicken cross the Mobius band? To get to the same side! When do you have the ultimate ...


Elementary Calculus: Playing With Formulas in SPLASH (2004)
I will explain how to differentiate and how to integrate, and show some applications. My approach will be mostly heuristic, ...


Elementary Calculus: Playing With Formulas in SPLASH (2004)
I will explain how to differentiate and how to integrate, and show some applications. My approach will be mostly heuristic, ...


Elementary Calculus: Playing with Inequalities in SPLASH (2004)
I will present a theory of Calculus based on some simple inequalities and prove the Fundamental Theorem. I will also ...


Elementary Calculus: Playing with Inequalities in SPLASH (2004)
I will present a theory of Calculus based on some simple inequalities and prove the Fundamental Theorem. I will also ...


Magic with Paper Ribbons and Other Topological Tricks in SPLASH (2004)
Why did the chicken cross the Mobius band? To get to the same side! When do you have the ultimate ...


Magic with Paper Ribbons and Other Topological Tricks in SPLASH (2004)
Why did the chicken cross the Mobius band? To get to the same side! When do you have the ultimate ...


Elementary Calculus I in SPLASH (2003)
Prerequisites: Fluency in high school algebra and geometry, precalculus is a plus. The students in this class will learn how ...


Elementary Calculus I in SPLASH (2003)
Prerequisites: Fluency in high school algebra and geometry, precalculus is a plus. Repeat of M-081.


Elementary Calculus II in SPLASH (2003)
Prerequisites: Elementary Calculus I, interest in abstract mathematics is a plus. Continuation of Elementary Calculus I. Some more Calculus tricks, ...


Elementary Calculus II in SPLASH (2003)
Prerequisites: Elementary Calculus I, interest in abstract mathematics is a plus. Repeat of M-083.


Elementary Calculus I in SPLASH (2002)
Repetition of the above.


Elementary Calculus I in SPLASH (2002)
The students in this class will learn how to differentiate and integrate, starting with some simple examples. Our approach will ...


Elementary Calculus II in SPLASH (2002)
Repetition of the above.


Elementary Calculus II in SPLASH (2002)
Continuation of Elementary Calculus I. Some more calculus tricks, such as integration by substitution and by parts, Taylor's formula, power ...


Elementary Calculus in HSSP (2001)
After a quick review of the origins of Calculus, an informal description of its two basic operations (differentiation and integration) ...


An Algebraic Approach to Calculus in HSSP (2001)
After a brief review of the origins of Calculus, an informal description of its two basic operations (differentiation and integration) ...


An Algebraic Approach to Calculus in HSSP (2000)
After a brief review of the origins of Calculus, an informal description of its two basic operations (differentiation and integration) ...


An Algebraic Approach to Elementary Calculus, Part 1 in SPLASH (2000)
Repetition of M-035. See above for details.


An Algebraic Approach to Elementary Calculus, Part 1 in SPLASH (2000)
In this class, we will learn how to differentiate polynomials and other elementary functions, then we will figure out how ...


An Algebraic Approach to Elementary Calculus, Part 2 in SPLASH (2000)
Repetition of M-037. See above for details.


An Algebraic Approach to Elementary Calculus, Part 2 in SPLASH (2000)
In this class we will learn more calculus tricks, such as integration by parts and by substitution, will take a ...


Differentiation as Division: An Algebraic Approach to Calculus. in SPLASH (1999)
Repetition of M-01A. See above for details.


Differentiation as Division: An Algebraic Approach to Calculus. in SPLASH (1999)
An attempt to approach the basic notions of calculus algebraically, without using limits. Surprisingly, such an approach is more general ...


Real Numbers and Infinitesimals in SPLASH (1999)
In this class, we will try to understand what the real numbers are, how to construct them starting with the ...


Real Numbers and Infinitesimals in SPLASH (1999)
Repetition of M-02A. See above for details.