The undergraduate Mathematics Colloquium (aka Math Table) sponsored by the
Mathematics Department, meets most Tuesday nights at 5:30 in Mather House
Private Dining Rooms A & B. Its purpose is to expose students to diverse
topics of pure and applied mathematics, to suggest potential Senior Thesis
topics. Speakers include faculty, guest speakers, graduate students and
undergraduate students. All students, irrespective of their mathematical
backgrounds, are strongly encouraged to attend and/or to give talks.
The two best undergraduate talks at the colloquium each year will win the
Rogers Prize.
For more information on the Mathtable, please contact
Noam Elkies (elkies@math.harvard.edu).
Next Mathtable
The undergraduate Mathematics Colloquium
meets most Tuesday nights at 5:30 PM in Mather
House PrivateDining Rooms A & B.
12/05/2017
Putnam Postmortem, by Noam Elkies, Harvard University
Click on the large icon to see all mini posters.
Solutions, outline, and/or mathematical context will be given for
some of the problems from last Saturday's Putnam exam,
likely starting with problems A2, A6, and B4.
We are signing up students to give talks for next year. If you are interested, or have questions about the process, please reach out to Rosalie
(rbr@math.harvard.edu).
Fall 2017
12/05/2017
Putnam Postmortem, by Noam Elkies, Harvard University
11/28/2017
Moonshine for All, by Neekon Vafa and Xavier Gonzalez Harvard Undergraduates, Harvard Undergraduates
11/14/2017
Knots and the Spirit World, by Sander Kupers, Harvard University
11/07/2017
Introduction to Morse Theory, by Rohil Prasad, Harvard Undergraduate
10/31/2017
The Monster group and scary character tables, by Brian Warner, Harvard Undergraduate
10/24/2017
Cayley Graphs and the Grothendieck Construction, by David Spivak , MIT
10/17/2017
6561101970383!, by Noam Elkies, Harvard University
10/10/2017
How to Compute a Cobordism Group, by Carlos Albors-Riera, Harvard Undergraduate
10/03/2017
Poincare-Birkhoff: an island of stability within chaos, by Davis Lazowski, Harvard Undergraduate
09/26/2017
Graduate School Panel, by Dennis Tseng, Robert Cass and Isabel Vogt, Harvard University and MIT
09/19/2017
Vector Bundles and K-Theory, by Cameron Krulewski, Harvard Undergraduate
09/12/2017
Adapting to a Chaotic World, by Rosalie Belanger-Rioux, Harvard University
09/05/2017
The Chairman's Welcoming Address: Solving the Quintic by Iteration, by Curtis T. McMullen, Harvard University
08/29/2017
Fall 2017 Tutorial in Mathematics: Arithmetic of elliptic curves, by Zijian Yao, Harvard University
Spring 2017
05/01/2017
Representation Theory and Quantum mechanics, Binary Quadratic Forms and Class Field Theory , by Justin Campbell and Zijian Yao, Harvard Mathematics Department
04/25/2017
Two (En)-Lightening Talks, by Adam Atanas and Justin Cavitt, Harvard Undergraduates
04/18/2017
Say What?!: Proving the Non-Context-Freeness of Natural Language, by Jennifer Hu, Harvard Undergraduate
04/11/2017
The Ramanujan-Nagell Equation, by Xavier Gonzalez, Harvard Undergraduate
04/04/2017
Shift Registers in Cryptography, by George Torres, Harvard Undergraduate
03/28/2017
Algebraic Functions and the Representability Zoo, by Julian Salazar, Harvard Undergraduate
03/21/2017
Who wants to be a math concentrator?, by Shanelle Eavis, Michael Fountaine, Sophie Garrett, and Kevin Yang, Harvard Undergraduates
03/07/2017
Partial Differential Equations and Universality in Probability Theory, by Kevin Yang, Harvard Undergraduate
02/28/2017
Infinitely Many Proofs of Infinitely Many Primes, by Andrew Gordon, Harvard Undergraduate
02/21/2017
Auction Theory, by Brabeeba Wang , Harvard Undergraduate
02/07/2017
Conical Algebraic Geometry, by Rohil Prasad, Harvard Undergraduate
01/31/2017
Division by Three, by Reuben Stern, Harvard Undergraduate
01/24/2017
Spring 2017 Tutorials:Category theory, Representations and cohomology of groups, by Justin Campbell and Meng Guo, Harvard University,
Please congratulate the recipients of the Department's undergraduate
prizes and awards:
The Mumford Prize in 2017 will be shared by Hannah Larson and Ashvin Swaminathan
The Herb Alexander Award is given this year to Kevin Yang
The Robert Fletcher Rogers Prizes are given to Jennifer Hu and
Julian Salazar. Honorable mentions to: Johann (Hans) Gaebler
and George Torres for "Shift Registers in Cryptography".
The Friends prizes are given this year to
Hannah Larson and Johann Gaebler.
Congratulate the recipients of this year's Hoopes Prizes.
Johann Gaebler,
Benjamin Lee, Gregory Parker, and Greg Yang.
Congratulations again to all of our award recipients for their excellent work.
Fall 2016
12/06/2016
Putnam Postmortem, by Noam Elkies, Harvard University
11/29/2016
Understanding Inequality: A Kinetic Model of Wealth, by Marek Johnson, Undergraduate at Tufts University
11/15/2016
A look at the Mercator Projection, by Carlos Albors-Riera, Harvard Undergraduate
11/08/2016
An Introduction to Stochastic Modeling, by Adam Al-Natsheh, Harvard Math Alumnus
11/01/2016
Graduate School Panel, by Eva Belmont, Chi-Yun Hsu, Ben Landon, Morgan Opie and Krishanu Sankar, Harvard University and MIT
10/25/16
The Ergodic Theorem and Benford's Law, by Hans Gaebler, Harvard Undergraduate
10/18/16
The Bowen-Lanford Zeta function of a graph, by Oliver Knill, Harvard University
10/11/16
Counting Stacks of Blocks, by Ben Wetherfield, Harvard Undergraduate
10/04/16
Dynamics of Toppling for the Abelian Sandpile model, by Drisana Mosaphir, Harvard Undergraduate
09/27/16
Topological Data Analysis, by Julian Salazar, Harvard Undergraduate
9/20/16
Oriented Matroids and Gale diagrams, by Greg Yang, Harvard Undergraduate
9/6/16
Combinatorial Game Theory, by Jacob Lurie, Harvard University
08/30/2016
Fall 2016 Tutorial in Mathematics: Arithmetic of elliptic curves, by Zijian Yao, Harvard University
Spring 2016
05/03/2016
Summer 2016 Tutorials in Mathematics, by Geoffrey Smith: Coding Theory, Jake Marcinek and Zijian Yao: Complex and arithmetic dynamics, Meng Guo: Category theory, Harvard University
04/26/2016
Partition Party, by Hannah Larson, Harvard Undergraduate
04/19/2016
Geometric Complexity Theory, by Sitan Chen, Harvard Undergraduate
04/12/2016
The parity problem in sieve theory, by Xiaoyu He, Harvard Undergraduate
04/05/2016
Adam the Manifold Doctor, by Adam Al-Natsheh, Harvard Undergraduate
03/29/2016
Who wants to be a math concentrator?, by Students interested in a variety of areas in and near mathematics, Harvard Undergraduates
03/22/2016
Destroying Arithmetic Progressions in Abelian Groups, by Ashvin Swaminathan, Harvard Undergraduate
03/08/2016
Wu characteristic, by Oliver Knill, Harvard University
03/01/2016
Bisection Envelopes, by Noah Fechtor-Pradines, Harvard Undergraduate
02/24/2016
Parameter spaces for polygons , by Isabel Vogt , Harvard Undergraduate
02/16/2016
Mathematics of Cell Membrane Topology and Transport, by Siddhartha Jena, Harvard Undergraduate
02/09/2016
Dr. Skolem, or: How I Learned to Stop Worrying and Love the Countable, by Johann Gaebler , Harvard Undergraduate
02/02/2016
Interpolation in Algebraic Geometry, by Aaron Landesman, Harvard Undergraduate
01/26/2016
Spring 2016 Tutorials:Morse Theory and Theory of Partitions, by Boyu Zhang and Konstantin Matveev, Harvard University
Please congratulate the recipients of the Department's undergraduate
prizes and awards this year.
The Mumford Prize is awarded in 2016 to Aaron Landesman.
The Wister Prize is awarded in 2016 to Sitan Chen.
The Herb Alexander Award is given this year to Kate Donahue.
The Robert Fletcher Rogers Prizes are given this year to Hannah Larson, Nat Mayer and Ashvin Swaminathan.
The Friends prizes are given this year to Aaron Landesman and Xiaoyu He.
Congratulations again to all of our award recipients for their excellent work.
Fall 2015
12/08/2015
Putnam Postmortem, by Noam Elkies, Harvard University
12/1/2015
Applications of Topology to a Problem in Algebra, by Adam Al-Natsheh , Harvard Undergraduate
11/17/2015
The Surreal Numbers and Combinatorial Game Theory, by Roman Berens , Harvard Undergraduate
11/10/2015
The Heilbronn Triangle Problem, by Xiaoyu He, Harvard Undergraduate
11/03/2015
Graduate School Panel, by Jun Hou Fung, Akhil Mathew, Alexander Smith, Isabel Vogt, Harvard University and MIT
10/27/2015
Elliptic Curve Cryptography , by Julian Salazar, Harvard Undergraduate
10/20/2015
The Gromov Non-Squeezing Theorem, by Greg Parker, Harvard Undergraduate
10/13/2015
How small is that prime?, by Ashvin Swaminathan, Harvard Undergraduate
10/06/2015
Volume Densities of Hyperbolic Knots, by Nat Mayer , Harvard Undergraduate
9/29/2015
The Banach-Tarski Paradox, by George Torres, Harvard Undergraduate
9/22/2015
Why exp(pi 163^{1/2}) is almost an integer, by Hannah Larson, Harvard Undergraduate
9/15/2015
Riemann-Roching out on the Projective Line, by Aaron Landesmann, Harvard Undergraduate
9/08/2015
Rotations and the four-color theorem, by Peter Kronheimer, Harvard University
Please congratulate the recipients of the Department's undergraduate
prizes and awards this year.
The Mumford Prize is awarded in 2015 to Evan O'Dorney.
The Wister Prize is awarded in 2015 to Arpon Raksit
The Herb Alexander Award is given this year to Lily Hu.
The Robert Fletcher Rogers Prizes are given this year to Xiaoyu He and Barry Tng.
The Friends prizes are given this year to Arpon Raksit and Barry Tng.
Congratulations again to all of our award recipients for their excellent
work.
5/05/2015
Summertutorials 2015, by Justin Campbell, Erick Knight, Yi Xie and Boyu Zhang, Harvard University
4/21/2015
Conspiracies among quadratic residues, by Kevin Yang, Harvard Undergraduate
4/14/2015
The Rubik's Group, by George Torres, Harvard Undergraduate
4/07/2015
Who wants to be a math concentrator?, by Students interested in a variety of areas in and near mathematics, Harvard Undergraduates
3/31/2015
Integral geometry and the size of a cookie, by Barry Tng , Harvard undergraduate
3/24/2015
The Erdos-Ginzburg-Ziv Problem, by Xiaoyu He , Harvard undergraduate
3/09/2015
Topological Field Theories, by Adam Al-Natsheh, Harvard undergraduate
3/02/2015
How large is n! = n(n-1)(n-2) ... 3.2.1?, by Benedict Gross, Harvard University
2/24/2015
Universal limits in Probability Theory, by Ben Adlam, Harvard University
2/17/2015
Accidental Algorithms, by Sitan Chen, Harvard Undergraduate
2/3/2015
Counting lattice points in triangles and the Fibonacci numbers, by Dan Cristofaro-Gardiner, Harvard University
1/28/2015
Summer Tutorial: Cyclotomic Fields and Fermat's Last Theorem, Math 157 info, by Tom Lovering and Atanas Atanasov, Harvard Math Department, 4-5:30 SC30
Fall 2014
12/09/14
Putnam Postmortem, by Noam Elkies, Harvard University
12/02/14
Peck Posets , by Aaron Landesman , Harvard Undergraduate
11/18/14
Pseudorandomness and Fourier Analysis, by Sitan Chen, Harvard Undergraduate
11/11/14
Graduate School Panel, by Sherry Gong, Igor Rapinchuk, Isabel Vogt and Fan Wei, Harvard University and MIT
11/04/14
The Riemann zeta function, prime numbers, and the explicit formula, by Carl Wang Erickson, Brandeis University
10/21/14
Large Prime Gaps , by Xiaoyu He, Harvard Undergraduate
10/14/14
Algebraic Geometry and Automated Proofs, by Julian Salazar , Harvard Undergraduate
10/7/14
Lattice theory for the working Matchmaker, by Scott Kominers, Harvard University
9/30/14
Morse Theory, by Adam Al-natsheh, Harvard Undergraduate
9/23/14
Reverse Mathematics, by Stephen Mackereth , Harvard Undergraduate
9/16/14
Arboreal Galois Representations, by Ashvin Swaminathan, Harvard Undergraduate
9/9/14
Combinatorial Game Theory, by Jacob Lurie, Harvard University
9/2/14
Fall 2014 Tutorial in Mathematics: Complex multiplication, by Rong Zhou and Yihang Zhu, Harvard University
Spring 2014
5/06/2014
Summer Tutorial: Mathematical Origami , by Erick Knight , Harvard Math Department
4/29/2014
Why it takes 7 shuffles to mix a 52 card deck, by Seth Neel , Harvard Undergraduate
4/22/2014
Galois groups, by Akhil Mathew, Harvard Undergraduate
4/15/2014
What are formal groups and what are they good for?, by Isabel Vogt, Harvard Undergraduate
4/08/2014
Who wants to be a Math concentrator?, by Levent Alpoge, Sam Horan, Scott Johnson, Seth Neel, Caitlin Stanton, Isabel Vogt, Allen Yuan, Harvard Undergraduates
04/01/2014
Ignorance is Strength, by Eric Wepsic, Harvard Undergraduate
03/23/2014
How many integral points does the average elliptic curve have?, by Levent Alpoge, Harvard Undergraduate
03/11/2014
The de Rham theorem, by Evan O'Dorney, Harvard Undergraduate
03/04/2014
Categories and Kan extensions, by Marina Lehner, Harvard Undergraduate
02/25/2014
Borsuk-Ulam and combinatorics, by Sukhada Fadnavis, Harvard University
02/18/2014
Puzzles that don't exist: Hilbert's third problem, by Omar Camarena, Harvard University
02/11/2014
Metric graphs and their Jacobians , by Nathan Pflueger , Harvard University
02/04/2014
Elliptic Curves and Hypergeometric Functions, by Keenan Monks, Harvard Undergraduate
Fall 2013
12/10/13
Putnam Postmortem, by Noam Elkies, Harvard University
12/03/13
Ideas behind an amazing recent proof of bounded gaps, by Levent Alpoge, Harvard Undergraduate
11/19/13
3472073^{7} + 4627011^{7} ≈ 4710868^{7} and other LLL tricks., by Noam Elkies, Harvard Math Department
11/12/13
Perfect Powers in Recurrence Sequences, by Isabel Vogt, Harvard Undergraduate
11/05/13
Tournament Dynamics , by Daniel Conney , Harvard Undergraduate
10/29/13
Graduate School Panel, by Sukhada Fadnavis, Ben Landon, Max Menzies, Emily Riehl, Danny Shi, Hiro Tanaka, and Fan Wei, Harvard University
10/22/13
Rational distance problems , by Anand Patel , Boston College
10/15/13
Surreal Analysis: An Analogue of Real Analysis for Surreal Numbers, by Ashvin Swaminathan, Harvard Undergraduate
10/8/13
Cool theorems proved by undergrads, by Ken Ono, Emory University
10/1/13
The Ramanujan-Nagell equation x^{2} + 7 = 2^{n}, by Noam Elkies, Harvard University
9/23/13
Schubert Calculus, by John Sheridan, Harvard undergraduate
9/16/13
The isoperimetric problem on surfaces, by Adam Jacob, Harvard University
9/9/13
Area-preserving maps, by Peter Kronheimer , Harvard University
9/3/13
Fall 2013 Tutorial in Mathematics: Structure, Randomness and Pseudorandomness, by Konstantin Matveev, Harvard University
Please congratulate the recipients of the Department's undergraduate
prizes and awards this year.
The Mumford Prize is awarded in 2013 to Eric Larson
The Wister Prize is awarded in 2013 to Tony Feng
The Herb Alexander Award is given this year to Ben Adlam
Two of our graduating seniors won 'Hoopes' Prizes for their theses this
year: please congratulate
Christian Anderson and Ashok Cutkosky.
The Rogers prize is award to Lucia Mocz, Akhil Mathew and
Dmitri Gekhtman. The Friends prize want to
Eric Larson. Tony Feng was the runner up.
Spring 2013
4/23/13
Modular Forms in String Theory, by Murad Alim , Harvard University
4/16/13
What is the Birch and Swinnerton-Dyer Conjecture?, by Lucia Mocz, Harvard Undergraduates
4/09/13
Who wants to be a Math concentrator?, by Zachary Abel,Tony Liu,Lucia Mocz,Robert Nishihara,John Sheridan,Caitlin Stanton, Harvard Alumni or Harvard Undergraduates
4/02/13
The Unique Games Conjecture and Hardness of Approximation , by Elena-Madalina Persu, Harvard Undergraduate
3/26/13
The sine integral Int(sin(x)/x,x,0,INFTY)=Pi/2 by elementary calculus, by Noam Elkies, Harvard University
3/12/13
An Elementary Proof of Ostrowski's Theorem, by Seth Neel, Harvard Undergraduate
3/05/13
Topological Field Theories, by Gijs Heuts, Harvard University
2/26/13
The Poincaré Dodecahedral Space, by Andrew Cotton-Clay, Harvard University
2/19/13
Solving Equations in the Integers is Hard, by Levent Alpoge, Harvard Undergraduate
2/12/13
Finite fields and topology, by Akhil Mathew , Harvard Undergraduate
2/05/13
The controversy around the Church-Turing Thesis, by Rachel Epstein , Harvard University
1/29/13
Spring 2013 Tutorials in Mathematics, by Konstantin Matveev and Alex Perry, Harvard University
Fall 2012
12/04/12
Putnam Postmortem , by Noam Elkies, Harvard University
11/27/12
Mathematics and Finance: How Math Can Make You Money, by Ben Conlee, '07, and Peter Green, Ph.D. '05, Ellington Management Group
11/13/12
A purely analytic proof of Quadratic Reciprocity, by Levent Alpoge, Harvard Undergraduate
11/6/12
Graduate School Panel, by Zach Abel, Sukhada Fadnavis, Adam Jacob, Eric Knight, Emily Riehl, and Krishnanu Sankar, Harvard and MIT Mathematics Departments
10/30/12
Heads or Tails? Coin tossing, fair and biased, by Emily Riel , Harvard University
10/23/12
Heads or Tails? Coin tossing, fair and biased, by Alex Bloemendal , Harvard University
10/16/12
Continued Fractions and Hyperbolic Geometry, by Krishna Dasaratha, Harvard Undergraduate
10/9/12
Braid Groups , by Allen Yuan , Harvard Undergraduate
10/2/12
Bug Dynamics on Manifolds, by Dimitry Gekhtman, Harvard Undergraduate
9/25/12
Covariance Matrices, by Ben Adlam , Harvard Undergraduate
9/18/12
Pellynomials, by Lucia Mocz, Harvard Undergraduate
9/11/12
Chairman's Welcoming Address: The ABC Conjecture and why we care, by Noam Elkies , Harvard University
Simplifying Complicated Simplicial Complexes: Discrete Morse Theory and its Applications, by Rachel Zax, Harvard Undergraduate
4/17/12
The Julia and Fatou sets of a polynomial map, by Benjamin Dozier , Harvard Undergraduate
4/10/12
Special K. What is K-theory and why do we care?, by Eva Belmont, Harvard Undergraduate
4/03/12
M,1,21,23,25,55...? Who wants to be a Math Concentrator?, by Panelists: Zachary Abel '10, Eva Belmont '12, Halimeda Glickman-Hoch '12, Katrina Evtimova '13, Tony Feng '13, and Yacoub Kureh '13., Harvard Graduates and Undergraduates
3/19/12
The Ubiquity of the Bernoulli Numbers, by Levent Alpoge , Harvard Undergraduate
2/28/12
The Grothendieck group , by Akhil Mathew , Harvard Undergraduate
2/28/12
The Euler Characteristic , by Caitlin Stanton , Harvard Undergraduate
2/21/12
What can be computed in mathematics?, by Anand Deopurkar, Harvard University
2/14/12
The space (locale) of random sequences, by Emily Riehl, Harvard University
2/7/12
Euler Integrals, and Gauss and Jacobi Sums, by Noam Elkies, Harvard University
1/31/12
Zolotarev's combinatorial proof of quadratic reciprocity, by Joseph Rabinoff , Harvard University
1/24/12
Spring 2012 Tutorials in Mathematics: Groupoids in Topology and The Figure Eight Knot: Introduction to Hyperbolid Geometry, by Omar Antolin Camarena and Stergios Antonakoudis, Harvard University
Fall 2011
12/6/11
Putnam Postmortem, by Noam Elkies, Harvard University
11/29/11
Fun with Carl and Carl, Messing Around with Gauss and Jacobi Sums, by Levent Alpoge, Harvard Undergraduate
11/15/11
Exotic spheres, by Akhil Mathew , Harvard Undergraduate
11/8/11
Orthogonal polynomials, Gaussian quadrature, and Rolle's theorem, by Noam Elkies, Harvard University
11/1/11
Graduate School Panel, by Zach Abel,George Boxer,Nathan Kaplan,Eric Riedl,Peter Smillie,Ethan Street,Meghan Anderson,Jennifer Balakrishnan, Harvard and MIT Mathematics Departments
10/11/25
How to count like Schubert, by Jack Huizenga, Harvard University
10/11/18
Why Many Published Experimental Findings Are Probably False, by Daniel Benjamin '99, Cornell and National Bureau of Economic Research
10/11/11
Bruhat-Tits Buildings of GL_{n}(Q_{p}) and SL_{n}(Q_{p}), by Lucia Mocz, Harvard Undergraduate
10/4/11
Interval exchange maps, by Vaibhav Gadre, Harvard University
9/20/11
Unfolding Polyhedra is NP-Hard, by Zach Abel `10, MIT
9/13/11
6561101970383!, by Noam Elkies, Harvard University
9/6/11
How I Learned the Fundamental Theorem of Algebra, by Peter Kronheimer, Harvard University
8/30/11
Fall 2011 Tutorial:Category Theory, by Atanas Atanasov, Harvard University
Spring 2011
5/3/11
Summer Tutorials 2011, by Aliakbar Daemi, Nathan Kaplan, Sam Raskin: Symplectic Geometry, Coding theory, Representation Theory, Harvard University
4/26/11
Irreducibility of M_{g,}, by Francois Greer, Harvard Undergraduate
4/19/11
Scaling Dimensions with Fractal Curves, by Adrian Sanborn, Harvard Undergraduate
4/12/11
Interactive Proofs, by Akhil Mathew , Harvard Undergraduate
4/05/11
M,1,21,23,25,55...? Who wants to be a Math Concentrator?, by Panelists: Zachary Abel '10, Tiffany Cai '12, Eva Belmont '12, Francois Greer '11, Dmitri Vaintrob '11, Harvard
3/29/11
Quaternions in everyday life, by Joe Rabinoff, Harvard University
3/22/11
Three faces of the Rado graph: What other multi-visaged structures are out there?, by Rehana Patel,
3/8/11
Crossing numbers of graphs: origins and recent progress, by John Mackey, Carnegie Mellon
3/1/11
pi proofs that pi^{2} smaller 10, by Noam Elkies, Harvard University
2/15/11
Solving an MC Escher Mystery With Mathematics, by Juliana Belding, Harvard University
2/8/11
Continued fractions and interval exchange maps, by Vaibhav Gadre, Harvard University
1/25/11
Spring Tutorial 2011, by Stergios Antonakoudis. "Operators on Hilbert spaces: analysis and application", Harvard University
Please congratulate the recipients of the Department's undergraduate
prizes and awards this year.
David Mumford Undergraduate Mathematics Prize
From the income of the gift in 1997 of Peter L. Falb, A.B. 1957, A.M.
1957, Ph.D. 1961 to establish the David B. Mumford Undergraduate
Mathematics Prize, to be given annually to the most promising senior
concentrator in mathematics, provided such concentrator is outstanding.
The David Mumford Undergraduate Mathematics Prize is awarded in 2011 to Arnav Tripathy.
Winster Prize
From the gift of Susan Alexander, the Herb Alexander Award recognizes
each year one of our outstanding undergraduates, and provides money to
be used for research and travel
The Wister Prize is awarded in 2011 to Sherry Gong.
Herb Alexander Award
From the gift of Susan Alexander, the Herb Alexander Award recognizes
each year one of our outstanding undergraduates, and provides money to
be used for research and travel.
The Herb Alexander Award is given this year to Jonathan Wang.
Robert Fletcher Rogers Prizes
From the income of the gift in 1921 of the late Robert Fletcher
Rogers, A.B. 1886, there have been established a first prize and second
prize to be awarded annually to the College students who present the
best talks before the Mathematics Table during an academic year.
The Rogers Prizes are awarded this year as follows:
First prize: Adrian Sanborn, for his talk Scaling Dimensions with Fractal Curves
Second prize: to be divided equally between Eric Larson for his talk
"An Inequality Involving the Perimeter and Curvature of Plane Curve", and
Akhil Mathew for his talk "Translation-Invariant Binary Representations".
Hoopes Prize
Michael Viscardi has been awarded the Hoopes' Prize this year, for his
senior thesis "Alternative compactifications of the moduli space of genus on maps."
Hoopes' Prizes were also won by two of our joint concentrators: Andrei
Cristea (Economics and Math), and Chrisopher Lim (Music and Math).
Congratulations to all!
Peter Kronheimer, Director of Undergraduate Studies, Department of Mathematics
Fall 2010
12/7/10
Putnam Postmortem, by Noam Elkies, Harvard University
11/30/10
Determinacy, by Joel Specter, Visiting Harvard Undergraduate
11/23/10
Stably Surrounding Sphere with Smallest String Size, by Zach Abel ('10), MIT graduate student
11/16/10
Large Games IV: A New Hope, by Scott Kominers ('09), Economics graduate student at HBS
11/9/10
Graduate School Panel, by Zach Abel, Nathan Kaplan, Erich Riedl, Ethan Street, Charmaine Sia and Sarah Koch, Harvard and MIT Mathematics Departments
11/2/10
Inversive Geometry, by Noam Elkies, Harvard Mathematics Department
10/26/10
Translation-Invariant Binary Representations, by Akhil Mathews , Harvard Undergraduate
10/19/10
On a conjecture of Lubin, by Joel Specter, Visiting Harvard Undergraduate
10/12/10
Tannakian Duality for Finite Groups, by Eric Larson , Harvard Undergraduate
10/5/10
Block Monoids and Factorizations of Algebraic Integers, by Scott Chapman, Sam Houston State University (Huntsville, Texas)
9/28/10
Fourier Series and Elliptic Regularity, by Akhil Mathew , Harvard Undergraduate
9/21/10
Limits, measures and democracy , by Gabor Lippner, Harvard University
9/14/10
An Inequality Involving the Perimeter and Curvature of Plane Curves, by Eric Larson , Harvard Undergraduate
9/7/10
Chairman's Welcoming Address: The maximum principle, by Shing-Tung Yau , Harvard University
Photos of the mathtable of September 7, 2010. Shing-Tung Yau gives a talk on the maximum principle.
[See larger panorama photo].
Please congratulate the recipients of the Department's undergraduate
prizes and awards this year.
David Mumford Undergraduate Mathematics Prize
From the income of the gift in 1997 of Peter L. Falb, A.B. 1957, A.M.
1957, Ph.D. 1961 to establish the David B. Mumford Undergraduate
Mathematics Prize, to be given annually to the most promising senior
concentrator in mathematics, provided such concentrator is outstanding.
The Mumford Prize is awarded to Michael Viscardi.
Herb Alexander Award
From the gift of Susan Alexander, the Herb Alexander Award recognizes
each year one of our outstanding undergraduates, and provides money to
be used for research and travel.
The Herb Alexander Award is given this year to Zachary Abel.
Robert Fletcher Rogers Prizes
From the income of the gift in 1921 of the late Robert Fletcher
Rogers, A.B. 1886, there have been established a first prize and second
prize to be awarded annually to the College students who present the
best talks before the Mathematics Table during an academic year.
The two prizes are awarded equally this year to:
Zhou Fan, for his talk "Random Matrices, Catalan Numbers, and Wigner's Semicircle Law;"
Dmitri Vaintrob, for his talk "1+2+3+ ... = -1/12: Values of the Zeta Function at Negative Integers."
Hoopes Prize
Michael Viscardi has been awarded the Hoopes' Prize this year, for his
senior thesis "Alternative compactifications of the moduli space of genus on maps."
Hoopes' Prizes were also won by two of our joint concentrators: Andrei
Cristea (Economics and Math), and Chrisopher Lim (Music and Math).
Congratulations to all!
Peter Kronheimer, Director of Undergraduate Studies, Department of Mathematics
Spring 2010
4/27/10
Summertutorials 2010, by Ronen Mukamel: Teichmüller Theory and Thurston's Classification of Mapping Classes, Ryan Reich: Linear Algebraic Groups, Jack Thorne: p-adic Arithmetic and p-adic Analysis., Harvard University
4/20/10
Empirical Knot Theory, by Zachary Abel, Harvard undergraduate
4/13/10
M,1,21,23,25,55...? Who wants to be a Math Concentrator?, by Junior and senior math concentrators, Harvard undergraduate
4/6/10
Newton Polygons, by Elena-Madalina Persu, Harvard undergraduate
3/30/10
The Spirit of Moonshine, by Jeremy Booher, Harvard undergraduate
3/23/10
Representation by quadratic forms, by Christian Zamora Jaen, Harvard undergraduate
3/9/10
A Slice of Pi, by Luke Anderson, Harvard Administration and founder of TeachPi.org
3/2/10
The outer automorphism of S_{6}, by Noam Elkies, Harvard University
2/23/10
Spaces that Don't Quite Fit in R^{3}, by Laura Starkston, Harvard Undergraduate
2/16/10
On Oranges and the Sun, by Nir Avni , Harvard University
2/9/10
Supersymmetric nonlinear sigma model, by Youngsub Yoon, Harvard Unvergraduate
2/8/10
Low dimensional topology for fun and profit (Monday, SciC 232, 4 PM), by Clifford Stoll, Teacher and Entrepreneur
2/2/10
1 + 2 + 3 + ... = -1/12. Values of the zeta function at negative integers., by Dmitry Vaintrob, Harvard Undergraduate
1/26/10
Spring Tutorial: Combinatorial and geometric group theory and symmetric functions in algebraic combinatorics, by Thomas Koberda and Greta Panova, Harvard Math Department
Fall 2009
12/8/09
Putnam Postmortem, by Noam Elkies, Harvard University
12/1/09
Random Matrices, Catalan Numbers, and Wigner's Semicircle Law, by Zhou Fan, Harvard Undergraduate
11/24/09
The McKay Correspondence, by Yi Sun, Harvard Undergraduate
11/17/09
Extremal functions of forbidden double permutation matrices, by Jesse Geneson, Harvard Undergraduate
11/10/09
The Partial Assignment Problem, by Olga Zverovich, Harvard Undergraduate
11/3/09
Graduate School Panel, by Ronen Mukamel, Ethan Street, Wei Ho and Joe Rabinoff, Harvard University
10/27/09
Calculus on Non-Standard Domains, by Harrison Pugh, Harvard Undergraduate
10/20/09
Matings of Polynomials, by Sarah Koch , Harvard University
10/13/09
What Is the LLL Algorithm?, by Zachary Abel, Harvard Undergraduate
10/6/09
The real and p-adic lives of n!, by Benedict Gross, Harvard Math Department
9/29/09
Thin Lehman Matrices and Their Graphs, by Johnathan Wang, Harvard Undergraduate
9/15/09
3472073^{7} + 4627011^{7} ~=~ 4710868^{7} and other LLL tricks., by Noam Elkies, Harvard Math Department
9/8/09
Welcoming Address: Knots and Complexity, by Peter Kronheimer, Head Tutor in Mathematics, Harvard University
9/1/09
Fall 2009 Tutorial: Dynamics of analytic maps: small divisor problems Abstract., by Giulio Tiozzo, Harvard University
Please congratulate the recipients of the Department's undergraduate
prizes and awards this year.
David Mumford Undergraduate Mathematics Prize
From the income of the gift in 1997 of Peter L. Falb, A.B. 1957, A.M.
1957, Ph.D. 1961 to establish the David B. Mumford Undergraduate
Mathematics Prize, to be given annually to the most promising senior
concentrator in mathematics, provided such concentrator is outstanding.
The Mumford Prize is awarded in 2009 jointly to John Lesieutre
and Shrenik Shah.
Wister Prize
From a bequest of Charles J. Wister, a prize is awarded in alternate
years to the Senior concentrating in Mathematics or Music who has the
highest record in his/her field of concentration. In 2008-09 the prize
is open to concentrators in Mathematics, in 2009-10 to concentrators in
Music.
The Wister Prize is awarded to Samuel Lichtenstein.
Herb Alexander Award
From the gift of Susan Alexander, the Herb Alexander Award recognizes
each year one of our outstanding undergraduates, and provides money to
be used for research and travel.
The Herb Alexander Award is given this year to Nike Sun.
Robert Fletcher Rogers Prizes
From the income of the gift in 1921 of the late Robert Fletcher
Rogers, A.B. 1886, there have been established a first prize and second
prize to be awarded annually to the College students who present the
best talks before the Mathematics Table during an academic year.
First Prize: Zachary Abel for his talk "Hinged dissections exist".
Second Prize: Laura Starkston, for her talk "Khovanov homology".
Hoopes Prize
Scott Kominers for his submission "Weighted Generating Functions
and Configuration Results for Type II Lattices and Codes".
Shrenik Shah for his submission "Framed Deformation and
Modularity".
Nils Wernerfelt for his submission "The Evolution of Cooperation
on Dynamic Graphs".
Congratulations again to all
Peter Kronheimer, Director of Undergraduate Studies, Department of Mathematics
Spring 2009
5/12/09
Summertutorials 2009, by Ivana Bozic: Stochastic Models in Biology, Tanya Kobylyatskaya and Ethan Street: Knot Theory Le Anh Vinh: Additive Combinatorics, Harvard University
5/5/09
The Rap Battle of the Millennium, by Zachary Abel and Scott Kominers , Harvard undergraduates
4/28/09
Phylogenetic tropical geometry, by Kelley Harris , Harvard undergraduate
4/21/09
New Application of the Inclusion-Exclusion Principle to Graph Coloring, by Olga Zverovich , Harvard undergraduate
4/14/09
Koch's Condition on Type II Codes of Length 24, by Scott Kominers, Harvard undergraduate
4/7/09
The Curry-Howard Correspondence, by Grant Dasher , Harvard undergraduate
3/31/09
Khovanov Homology , by Laura Starkston , Harvard undergraduate
3/17/09
Who wants to be a Math Concentrator?, by Junior and senior math concentrators, Harvard undergraduates
3/10/09
A Slice of Pi , by Luke Anderson, Harvard Administration and founder of TeachPi.org
3/3/09
Computation at a Distance, by Lawren Smithline, Center for Communications Research at Princeton
2/24/09
Treasure Hunting Perfect Euler Bricks, by Oliver Knill, Harvard University
2/17/09
Pairings on Elliptic Curves in Cryptography: Friend or Foe, by Juliana Belding, Harvard University
2/10/09
The secret lives of class numbers, by David Hansen, Brown University Undergraduate
2/03/09
Why stop at bilinear forms?, by Noam Elkies, Harvard Math Department
1/27/09
Spring Tutorial: Morse Theory , by Chen-Yu Chi , Harvard Math Department
Fall 2008
12/9/08
Putnam Postmortem, by Noam D. Elkies, Harvard Math Department
12/02/08
Analyzing Complex Semisimple Lie Algebras, by Christian Zamora Jaen , Harvard Undergraduate
11/25/08
Many Proofs of Quadratic Reciprocity, by Jeremy Booher, Harvard Undergraduate
11/18/08
The Math Graduate School Discussion Panel, by recent Ph.D.'s and graduate students , from Berkeley, Cambridge, MIT, Harvard, Princeton etc.
11/11/08
Matchmaker, matchmaker, clear out my house., by Scott Kominers, Harvard Undergraduate
11/4/08
Hinged dissections exist, by Zachary Abel, Harvard Undergraduate
10/28/08
Dynkin diagrams, or what to tell the aliens, by John Duncan, Harvard University
10/21/08
Knots, Spirals, and Superinvariants, by Laura Starkston , Harvard Undergraduate
10/14/08
Mathtable joins Clay Math public lecture. "A Tribute to Euler", by William Dunham, Harvard University
10/7/08
The gap between the first two eigenvalues of the Schroedinger operator and methods from geometry, by Sing-Tung Yau, Harvard University
9/23/08
The Weil representation, by Noam Elkies, Harvard University
9/16/08
Fall 2008 Tutorials: Tropical Geometry and Algebraic Graph Theory, by Ryan Reich and LeAnh Vinh, Harvard University
Please join me in congratulating the recipients of the Department's
undergraduate prizes and awards this year.
David Mumford Undergraduate Mathematics Prize
The Mumford prize is awarded to Dustin Clausen.
From the income of the gift in 1997 of Peter L. Falb, A.B. 1957, A.M.
1957, Ph.D. 1961 to establish the David B. Mumford Undergraduate
Mathematics Prize, to be given annually to the most promising senior
concentrator in mathematics, provided such concentrator is outstanding.
The Herb Alexander Award
The Herb Alexander Award is given this year to Elena Yudovina.
From the gift of Susan Alexander, the Herb Alexander Award recognizes
each year one of our outstanding undergraduates, and provides money to
be used for research and travel to mathematics conferences.
The Robert Fletcher Rogers Prizes:
First Prize: Scott Kominers, for his talk
"C=15".
Second Prize: Christian Zamora Jaen, for his talk
"Fermat's Last Theorem".
From the income of the gift in 1921 of the late Robert Fletcher
Rogers, A.B. 1886, there have been established a first prize and second
prize to be awarded annually to the College students who present the
best talks before the Mathematics Table during an academic year.
Hoopes Prizes
Dustin Clausen and Alison Miller were recipients of the Hoopes
Prizes this year, for their outstanding senior theses.
Congratulations again to all!
Peter Kronheimer, Director of Undergraduate Studies, Department of Mathematics
Spring 2008
5/06/08
Division Algebras over R, by Jeremy Booher, Harvard Undergraduate
4/29/08
Self-Reference and Provability, by Jesse Geneson , Harvard Undergraduate
4/22/08
General Relativity and the Mathematics of the Universe, by Lydia Bieri, Harvard University
4/15/08
Mono-monostatic bodies: the story of the Gomboc, by Gabor Domokos and Peter Varkonyi, Budapest University of Technology and Economics.
4/8/08
Beyond Batting Averages, by Rohit Acharya and Alex D'Amour, Harvard Undergraduates
4/1/08
Fermat's Last Theorem, by Christian Zamora Jaen, Harvard Undergraduate
3/18/08
The Fraisse Construction and its Descendants, by Rehana Patel, Harvard University
3/14/08
Pi-day 2008, by Pi, Mathematical constant
3/4/08
Color by Numbers, by Daniel Litt, Harvard Undergraduate
2/26/08
Probabilistic Group Theory, by Bret Benesh, Harvard University
2/5/08
Curves, Cryptography and Primes of the Form x^{2} + D y^{2}, by Juliana Belding , University of Maryland
Gabor Domokos visits Harvard on April 15'th 2008 and demonstrates the gomboc.
His mathtable talk was a big success.
The Gomboc was created
by Gabor Domokos and Peter Varkonyi, Budapest University of Technology and Economics.
The 'Gomboc' became very popular.
The website attracts hundreds of thousand of visitors (mostly
non-mathematicians) from over hundred countries.
References:
Domokos, G., Ruina, A., Papadopoulos, J.: Static equilibria of rigid
bodies: is there anything new? J. Elasticity, 36 (1994), 59-66.
Varkonyi, P., Domokos G.: Static equilibria of rigid bodies:
dice, pebbles and the Poincare-Hopf Theorem. J. Nonlinear Science 16
(2006), 255-281.
Abstract: The weeble (also called the 'Comeback Kid') is the favorite
of many children: whenever knocked over, it always returns to the same
(stable) equilibrium position. This toy is, of course, not homogenous,
spontaneous self-righting is guaranteed by the weight at the bottom. We
may also observe that most weebles have only one unstable balance point,
at the top.
When we look at homogeneous objects, the problem becomes less trivial. In
two dimensions, it is relatively easy to prove that each homogeneous,
convex slab has at least two stable (S=2) and two unstable (U=2)
equilibria when rolling under gravity on a horizontal surface, i.e.
homogeneous 2D weebles do not exist.
One can imagine several three dimensional generalizations to this
statement: a) S>1 b) U>1 c) S+U>2 While a) and b) can be shown to be
false, c) posed a considerable puzzle for many years. Finally, in 1995,
V.I. Arnold conjectured that c) is NOT true either, in other words,
convex, homogeneous solids with just one stable and one unstable point
of equilibrium (also called mono- monostatic) may exist. These are
'special weebles' which share the number and type of equilibria of the
toy, however, no weight is added.
Not only did his conjecture turn out to be true, the newly discovered
objects show various interesting features. Our goal is to give an
overview of these findings with emphasis on a qualitative approach based
on dynamical systems theory, as well as to present some new results. We
will point out that mono-monostatic bodies are neither flat, nor thin,
they are not similar to typical objects with more equilibria and they
are hard to approximate by polyhedra. Despite these "negative" traits,
there seems to be strong indication that these forms appear in Nature
due to their special mechanical properties.
These new forms, called 'Gomboc' became very popular on the internet,
the website attracting hundreds of thousand of visitors (mostly
non-mathematicians) from over hundred countries.
Fall 2007
12/11/07
Integers and Sequences, by Tanya Khovanova, Lead Analyst at BAE Systems
12/4/07
Putnam Postmortem, by Noam D. Elkies, Harvard Math Department
11/27/07
"Binary Quadratic Forms and Ideal Class Groups", by Christian Zamora Jaen, Harvard Undergraduate
11/20/07
"Playing Modulor: Combination and the Golden Ratio in the Work of Le Corbusier", by Julia Shlozman, Harvard Undergraduate
11/06/07
"Relativistic Electromagnetism", by Arnav Tripathy, Harvard Undergraduate
10/30/07
"C=15", by Scott Kominers, Harvard Undergraduate
10/23/07
"Firefly Self-Synchronization models and their applications", by Pablo Azar , Harvard Undergraduate
10/16/07
"The Mathematics of fixed income Markets", by Andrew Lesniewski, Managing Director Ellington Management Group
10/09/07
"Mathematics used at Google", by A Representative from the Company, Google
10/02/07
"The Icosahedral Group", by Bret Benesh, Harvard University
9/25/07
Knots, and how to unknot them , by Peter Kronheimer , Head Tutor in Mathematics Harvard Math Department.
9/18/07
Introductions, Announcements, and Problem Solving, by Bret Benesh, Harvard University
Photos of the mathtable of September 25, 2007. Peter Kronheimer gives a talk on knots.
[See larger panorama photo].
Spring 2007
4/24/07
A Queuing Model for Use of Laundry Machines in Student Dormitories, by Serena Rezny, Harvard Undergraduate
4/17/07
Presentations for 3 Summer Tutorials: Ramsey Theory, Enumerative Geometry, Elliptic Curves, by Thomas Barnet-Lamb, Dawei Chen, David Geraghty and Jeechul Woo, Harvard University
4/3/07
Quantum Computation and Shor's Algorithm, by Shrenik Shah, Harvard Undergraduate
3/13/07
Abelian extensions and analytic functions, by Kaloyan Slavov, Harvard Undergraduate
2/27/07
How many irreducibles?, by Noam Elkies, Harvard University
2/20/07
How many primes?, by Zachary Abel, Harvard Undergraduate
2/13/07
Young tableaux and permutations, by Thomas Lam, Harvard University
1/30/07
Spring 2007 Tutorials: Elliptic functions, Geometry in Real and Complex Projective Space, by Chung Pang Mok, Rina Anno, Harvard University
Please join me in congratulating the recipients of the Department's
undergraduate prizes and awards this year.
David Mumford Undergraduate Mathematics Prize
The Mumford prize is awarded to Kaloyan Slavov.
From the income of the gift in 1997 of Peter L. Falb, A.B. 1957, A.M.
1957, Ph.D. 1961 to establish the David B. Mumford Undergraduate
Mathematics Prize, to be given annually to the most promising senior
concentrator in mathematics, provided such concentrator is outstanding.
The Wister Prize
The Wister Prize is awarded to Anatoly Preygel.
From a bequest of Charles J. Wister, a prize is awarded in alternate
years to the Senior concentrating in Mathematics or Music who has the
highest record in his/her field of concentration. In 2006-07 the prize
is open to concentrators in Mathematics, in 2007-08 to concentrators in
Music.
The Herb Alexander Award
The Herb Alexander Award is given this year to Igor Rapinchuk.
From the gift of Susan Alexander, the Herb Alexander Award recognizes
each year one of our outstanding undergraduates, and provides money to
be used for research and travel to mathematics conferences.
The Robert Fletcher Rogers Prizes:
First Prize: Robin Walters, for his talk
"Generalizations of the Hexagonal Honeycomb Theorem".
Second Prize: Gerardo Con Diaz, for his talk
"Cutting and Pasting Doughnuts: Three Ways of Constructing Lens Spaces".
Honorable Mention: Alison Miller, for her talk "Exp(pi*Sqrt(163)) and Friends".
From the income of the gift in 1921 of the late Robert Fletcher
Rogers, A.B. 1886, there have been established a first prize and second
prize to be awarded annually to the College students who present the
best talks before the Mathematics Table during an academic year.
Hoopes Prizes
Mark Lipson and Kaloyan Slavov were recipients of the Hoopes
Prizes this year, for their outstanding senior theses. The Hoopes Prize
committee awarded Goutham Seshadri an honorable mention.
Congratulations again to all!
Peter Kronheimer, Director of Undergraduate Studies, Department of Mathematics
Mathtable Icontable
As of January 1, 2007, there were 184 mathtable talks in our
database. Click on the icon mosaic, to see a table of all icons.
Exp(pi*Sqrt(163)) and Friends, by Alison Miller, Harvard Undergraduate
12/5/06
Putnam Postmortem , by Noam D. Elkies, Harvard Math Department
11/28/06
Dunking Donuts: Morse theory and the Poincare-Hopf theorem, by Alexander Ellis, Harvard Undergraduate
11/21/06
Opetopia, by Scott Kominers, Harvard Undergraduate
11/14/06
SO(4)/{1,-1} = SO(3) x SO(3) , by Noam Elkies, Harvard Math Department
10/31/06
Inversive Geometry , by Noam Elkies, Harvard Mathematics Department
10/24/06
Generalizations of the Hexagonal Honeycomb Theorem , by Robin Walters, Harvard Undergraduate
10/17/06
Cutting and Pasting Doughnuts: Three Ways of Constructing Lens Spaces, by Gerardo Con Diaz, Harvard Undergraduate
10/11/06
Mathematics and financial modeling , by Andrew Lesniewski, Managing Director, Ellington Management Group
10/3/06
Rubik's Hyperhexahedra , by Stewart Wilcox, Harvard Mathematics Department
9/26/06
Spectral flow , by Clifford Taubes, William Petschek Professor of Mathematics and Chair of the Harvard Mathematics Department
9/19/06
Tutorials "Enumerative Geometry", "Morse Theory" "Geometry and Physics" , by Dawei Chen, Chen-Yu Chi, Aleksandar Subotic, Harvard Math Department
This year, the Mumford Prize will be split between
Stefan Patrikis, and Nikita Rozenblyum.
"From the income of the gift in 1997 of Peter L. Falb, A.B. 1957, A.M.
1957, Ph.D. 1961 to establish the David B. Mumford Undergraduate
Mathematics Prize, to be given annually to the most promising senior
concentrator in mathematics, provided such concentrator is
outstanding."
The Herb Alexander Award is given this year to
Sheel Ganatra.
"From the gift of Susan Alexander, the Herb Alexander Award recognizes
each year one of our outstanding undergraduates, and provides money to
be used for research and travel to mathematics conferences."
The Robert Fletcher Rogers Prizes
are split equally between:
Gerardo Con Diaz, Igor Rapinchuk and
Inna Zakharevich.
"From the income of the gift in 1921 of the late Robert Fletcher
Rogers, A.B. 1886, there have been established a first prize and
second prize to be awarded annually to the College students who
present the best talks before the Mathematics Table during an academic
year."
In addition Stefan Patrikis and Gregory Valliant were
recipients of Hoopes Prizes this year,
for their outstanding senior theses.
Congratulations to all!
Peter Kronheimer, Director of Undergraduate Studies, Department of Mathematics
(Winners of the Department's prizes and awards should contact Ruby
in the main office.)
Spring 2006
5/2/06
The 30 Minute Introduction to Algebraic Geometry, by Corina Tarnita Patrascu, Harvard Undergraduate
4/25/06
Homotopy and Simplicial Sets, by Inna Zakharevich, Harvard Undergraduate
4/18/06
The Summer tutorials: Category theory", "Group cohomology", "Modular forms", "Morse Theory, by Thomas Barnet-Lamb, John Francis, Mike Hill, Andrew Lobb, Harvard Math Department
4/11/06
The maximal abelian extension of Q_{p}, by Emily Riehl, Harvard Undergraduate
4/4/06
The inefficiency of selfishness, by Greg Valiant, Harvard Undergraduate
3/21/06
Dirichlet's Prime Number Theorem: Algebraic and Analytic Aspects , by Igor Rapinchuk, Harvard Math Department
3/14/06
Why Pi = 2.2222 ..., by Paul Bamberg, Harvard Math Department
2/28/06
The outer automorphism of S_{6}, by Noam Elkies, Harvard Math Department
2/21/06
Metric-Preserving Functions, by Scott Kominers, Harvard Undergraduate
2/14/06
Zero-Knowledge Proofs, by Joshua Kroll, Harvard Undergraduate
2/1/06
Spring 2006 Tutorials 'Height Functions in Diophantine Geometry', 'Sphere Packing', 'Geometry and Physics', by Sonal Jain, Abhinav Kumar, Aleksandar Subotic, Harvard Math Department
Photos of the mathtable of November 22, 2005. Also present is
John Tate, who had attended some Math table talks here at Harvard
as an undergraduate 60 years ago.
Fall 2005
12/13/05
The Mathematics Contest in Modeling: A Better QuickPass System for Amusement Parks, by Ivan Corwin, Sheel Ganatra, and Nik Rozenblyum, Harvard Undergraduates
12/6/05
Putnam Post Mortem, by Noam Elkies, Harvard Math Department
11/29/05
Measurable dynamics of the p-adic shift and its relatives, by Alex Levin and Anatoly Preygel, Harvard Undergraduates
11/22/05
3472073^{7} + 4627011^{7} ~=~ 4710868^{7} and other LLL tricks., by Noam Elkies, Harvard Math Department
11/15/05
Bugs, blobs, and rotor-routers: an introduction to quasirandomness, by Jim Propp, University of Wisconsin at Madison
11/8/05
Quadratic extensions of Q_{p}, by Emily Riehl, Harvard Undergraduate
11/1/05
Trefoils on your Shoes,Knots, Surfaces and Matrices, by Gerardo Con Diaz, Harvard Undergraduate
10/25/05
Elliptic Curves and Mordell's Theorem, by Igor Rapinchuk, Harvard Undergraduate
10/18/05
Modeling Financial Risk, by Michael Vranos and Andrew Lesniewski, CEO, Ellington Management Group, rsp. Director of Quantitative Research
10/11/05
What do the integers from 1 to 10 know about the Cantor set?, by Benedict Gross, Leverett Professor of Mathematics and Dean of Harvard College.
9/27/05
Knaster's conjecture, and level landing sites for lunar modules., by Peter Kronheimer, Head Tutor in Mathematics Harvard Math Department.
9/20/05
Tutorials 'Binary quadratic forms', 'Holomorphic vector bundles', by Chung Pang Mok, Sabin Cautis, Harvard Math Department
Image credit: Doug Dunham, based on pictures of
the Dutch artist M. C. Escher.
Congratulations to the winners of the prizes for undergraduates 2004-2005:
1. DAVID MUMFORD PRIZE will go to Gabriel Carroll and Ian Le
DAVID B. MUMFORD UNDEGRADUATE MATHEMATICS PRIZE
From the income of the gift in 1997 of Peter L. Falb, A.B. 1957, A.M.
1957, Ph.D. 1961 to establish the David B. Mumford Undergraduate
Mathematics Prize, to be given annually to the most promising senior
concentrator in mathematics, provided such concentrator is
outstanding.
2. ROBERT FLETCHER ROGERS PRIZES for 2004-2005:
First prize: Ronen Mukamel for his talk on
"Four Things You Can Do in the Hyperbolic Plane that Your Parents Couldn't in the Euclidean"
Two Second Prizes will go to:
Mike Hamburg for his talk on
"Compass and Straightedge Constructions for Projectile and Orbital Motion"Adam Levine for his talk on "Word-Hyperbolic Coxeter Groups"
ROBERT FLETCHER ROGERS PRIZE.
From the income of the gift in 1921 of the late Robert Fletcher Rogers,
A.B. 1886, there have been established a first prize and second prize
to be awarded annually to the College students who present the best
talks before the Mathematics Table during an academic year. In making
the awards, emphasis will be laid primarily on the excellence of the
oral presentation. It is not essential that the material presented be
original, but mastery of the subject and adaptation of the presentation
will be regarded as important. The prizes will be awarded by the
Department of Mathematics on the recommendation of the Mathematics
Table. For further information, please contact the Mathematics
Department.
3. The WISTER PRIZE will go to Andrei Jorza
WISTER PRIZE. From a bequest of Charles J. Wister, a prize is awarded
in alternate years to the senior concentrating in Mathematics or Music who has the
highest record in his/her field of concentration. In 2004-05 the prize
is open to concentrators in Mathematics, in 2005-06 to concentrators in
Music.
Spring 2005
5/10/05
Four things you can do in the Hyperbolic plane that your parents couldn't in the Euclidean, by Ronen Mukamel, Harvard Undergraduate
5/3/05
A Modular Proof of the Class Number One Problem, by Tseno Tselkov, Harvard Undergraduate
4/26/05
The Riemann-Weierstrass Function, by Ian Le, Harvard Undergraduate
4/19/05
World-Hyperbolic Coxeter Groups, by Adam Levine, Harvard Undergraduate
4/12/05
Tropical Geometry, by Ilia Zharkov, Harvard Math Department
4/5/05
Summertutorials 'Bilinear forms in algebra', 'topology and arithmetic', 'Knot theory', 'Spectral sequences, How to talk to physicists, Groups, symmetry and topology', by John Francis, Andrew Lobb, Mike Hill, Daniel Larson, Harvard Math Department
3/22/05
Math and Logic Puzzles: Base 3 Representations, Modular Arithmetic and other fun techniques, by Daniel Gardiner, Harvard Undergraduate
3/15/05
Your Answer May Vary: Probability Enigmas Beyond Monty Hall, by Eric Wepsic, D.E. Shaw \& Co
3/8/05
Are we living in 10 dimensional space-time?, by René Reinbacher, Harvard Math Department
3/1/05
The curious power series, by Noam Elkies, Harvard Math Department
2/22/05
The busy Laplacian, by Mark Reeder, Boston College
2/15/05
Solving equations using unique factorization, by Robert Pollack, Boston University
2/8/05
An introduction to Geometric Knot Theory, by Elizabeth Denne, Harvard Math Department
2/1/05
Tutorials: Complex Multiplication and Sheaves in Logic and Geometry, by Sug Woo Shin and Teru Yoshida, Jay Pottharst and Cameron Freer, Harvard Math Department
Fall 2004
12/7/04
Putnam Post Mortem, by Noam Elkies, Harvard Math Department
11/30/04
A Sharp Bound for the Degree of Proper Monomial Mappings Between Balls: a combinatorial approach, by Emily Riehl, Harvard Undergraduate
11/23/04
Compass and straightedge constructions for projectile and orbital motion, by Mike Hamburg, Harvard Undergraduate
The Math Graduate School Discussion Panel, by recent Ph.D.'s and current graduate students, from Harvard, MIT, Princeton, Berkeley, Chicago, etc
11/2/04
Cryptography and Electronic Elections" , by Alex Levin, Harvard Undergraduate
10/19/04
cos(2 Pi/17), by Matt Leingang, Mathematics Department
10/12/04
Mathematical Topics in Quantum Physics, by Robbie Martinez, Division of Engeneering and Applied Sciences
10/5/04
Wavelets, Uniform and Otherwise, by Derek Bruff, Mathematics Department
9/28/04
Welcoming Address: "The Evolution of Plane Curves , by Joe Harris , Chair of the Mathematics Department
9/21/04
Tutorials, by Additive Number Theory and Algebraic Surfaces and Complex Manifolds of Higher Dimension, Michael Schein and Sabin Cautis
Congratulations to the winners of the Math Prizes 2003-2004:
1. DAVID MUMFORD PRIZE goes to Dimitar Jetchev
2. ROBERT FLETCHER ROGERS PRIZES for 2003-2004:
Corina Patrascu - the First Prize for a talk on
"Computing order statistics in the Farey sequence"
Jonathan Bloom - the Second Prize for a talk on
"The Local Structure of Smooth Maps of Manifolds"
Spring 2004
4/27/04
The Local Structure of Smooth Maps of Manifolds, by Jonathan Bloom, Harvard Undergraduate
4/20/04
Number Theory in Function Fields: Dirichlet's Theorem about Primes in Arithmetic Progressions, by Athanasios Papaioannou, Harvard Undergraduate
4/13/04
Summer Tutorials Riemann Surfaces, Spinors, Differential Geometry with a view toward Physics and Ramsey Theory, by Deepee Khosla, John Francis, Aaron Tievsky and Alexander Pekker, Harvard University
4/6/04
Computing order statistics in the Farey sequence, by Corina Patrascu, Harvard Undergraduate
3/23/04
Automatic Groups, by Kazem Mahdavi, SUNY Potsdam and Harvard Math Department
3/16/04
On Quaternions and Octonions, by Andrew Ostergaard, Harvard Undergraduate
3/2/04
The Incredible Shrinking Data, by Derek Bruff, Harvard Math Department
2/24/04
Applications of Subharmonic Functions, by Oliver Knill, Harvard Math Department
2/17/04
The Ising model in Statistical Mechanics, by Alberto DeSole, Harvard Math Department
Fall 2003
12/9/03
This years Putnam Exam, by Noam Elkies, Harvard Math Department
11/25/03
Infinite-dimensional Lie algebras and the Virasoro algebra, by Alberto DeSole, Harvard Math Department
11/18/03
A proof of a theorem of Euclid, by Richard Rivero, Harvard Undergraduate
11/11/03
Lost in 3-Space: Electric Circuits and Polya's Theorem on Random Walks, by Phil Zeyliger, Harvard Undergraduate
11/4/03
The Graduate School Discussion Panel , by recent Ph.D.'s and current graduate students, from Harvard, MIT, Princeton, Berkeley, Chicago, Stanford
10/28/03
Perfect Matchings and the Octahedron Recurrence, by Gabriel Carroll, Harvard Undergraduate
10/21/03
Three Dimensional Manifolds and Surgery, by Tom Coates, Harvard Math Department
10/14/03
Testing for Mersenne Primes" by, by Benedict H. Gross, Dean of Harvard College
10/6/03
The Cayley-Menger Determinant and You, by Matt Leingang, Harvard Math Department
9/30/03
Mathematics as a Cultural System, by Colin Kelly, Harvard Undergraduate
9/23/03
Quantum Gravity, by Cliff Taubes, Harvard Math Department
9/16/03
Tutorials: 'Geometric Topology' 'Morse Theory' , by Curt McMullen and Erick Matsen and Ciprian Manolescu, Harvard Math Department
Math Prizes 2002-2003:
1. DAVID MUMFORD PRIZE goes to Andrew Cotton
2. WISTER PRIZE goes to James Fowler
3. ROBERT FLETCHER ROGERS PRIZES:
Andrew Cotton and James Fowler
Spring 2003
5/13/03
An Introduction to the Word Problem in Finitely Presented Groups, by Jordanna Schutz , Harvard Undergraduate
5/6/03
Holomorphic curves and minimal surfaces, by Andy Cotton , Harvard Undergraduate
4/29/03
SL_{2}(Q_{2}) and the Homogeneous Tree of Degree 3, by Joe Rabinoff , Harvard Undergraduate
4/22/03
Singular Moduli, by Tony Varilly , Harvard Undergraduate
4/15/03
Julia Sets, by Dustin Cartwright , Harvard Undergraduate
4/8/03
Summer Tutorials 'Symplectic and Contact Geometry', Real and Complex Manifolds' Topics in Group Theory Geometry and General Relativity, by Ciprian Manolescu, Deepee Khosla, Abhinav Kumar, Vivek Mohta, Mathematics Department
3/18/03
Primes in Arithmetic Sequences" , by Gordon Savin, University of Utah
3/11/03
The Banach-Tarski Paradox" , by Matt Bainbridge, Harvard Graduate Student
3/4/03
How to Triangulate a Polytope" , by Laura Matusevich , Harvard Math Department
2/25/03
Multiple Zeta Values" , by Richard Cudney , Harvard Undergraduate
2/18/03
The m-step, same-step and any-step competition graphs" , by Wei Ho , Harvard Undergraduate
2/11/03
Approximation by Polynomials with Integral Coefficients" , by Laurent Berger, Harvard Math Department
2/4/03
Surreal games" , by Jim Fouler, Harvard Undergraduate
1/28/03
Tutorials: K-Theory and its Applications ,Quantum Mechanics for the Masses and the Mass-less , by Nick Ramsey, Greg Grigorov, Spiro Karigiannis , Harvard Math Department
Fall 2002
12/10/02
The 2002 Putnam exam , by Noam Elkies , Harvard Math Department
12/3/02
Graduate Discussion Panel , by Recent Ph.D's and current Grad students , Harvard/MIT/Berkeley,Chicago,Columbia
11/26/02
Domains of Holomorphy and the Levi Problem, by Andreea Nicoara , Harvard Math Department
11/19/02
Mortgages Mathematics, by Michael Vranos and John Geanakoplos , CEO and Managing Partner Ellington Management Group rsp. Professor of Economics, Yale University
11/12/02
Geometric Group Theory, by Kathy Paur, Harvard Math Department
11/5/02
Reachability of Points in Phase Space, by Michael Brian McElroy, Graduate Student at DEAS, Harvard Math Department
10/29/02
Foundations of Decision Theory, by Howard Raiffa, Frank P. Ramsey Professor of Managerial Economics Emeritus, Harvard Math Department
10/22/02
Kronecker's Theorem on the Density of Cyclic Subgroups of (R/Z)^{k}, by Peter Green , Harvard Math Department
10/15/02
Loves Me, Loves Me Not: what you shouldn't do after you dissect a Square into Identical Triangles, by Jim Fowler , Harvard Undergraduate
10/8/02
The Singular Value Decomposition and Image Storage, by Thomas Judson , Harvard Math Department
10/1/02
Computer Generation of the Maskit Slice, by Zak Stone , Harvard Undergraduate
9/24/02
Are Cubics Rational? (Welcoming Address), by Joe Harris, Chair of the Mathematics Department
9/17/02
Math Tutorials: A Plethora of Polynomials, Bundles of Bundles, by Matt Bainbridge, Alina Marian , Harvard Math Department
Math Prizes: 2001-2002:
Lionel Levine, for his talk on "Clifford Algebras and Vector Fields on
Spheres" and Alex Healy, for his talk on "Zero Knowledge Proofs"
Spring 2002
5/14/02
Soliton Cellular Automata, by Richard Cudney , Harvard undergraduate
5/7/02
Steenrod Powers, the Serre Spectral Sequence and the Computation of Homotopy Groups of Spheres Using Cohomology, by Mike Hill, Harvard undergraduate
4/30/02
The generalized Poincaré Conjecture, Exotic Spheres and the Surgery Theoretic Classification of High-Dimensional Manifolds, by Jonathan Kelner, Harvard undergraduate
4/23/02
Three Proofs of Quadratic Reciprocity in 33 Minutes, by Noah Snyder, Harvard undergraduate
4/16/02
Do Math, Have Fun, Get Paid: R.E.A.C.H.-ing new results in Combinatorics, by Jim Propp , Harvard Math Department
4/9/02
Cleverly Painted Mules: Epistemic Closure, Logic, and Evidence, by Haiwen Chu , Harvard Undergraduate
4/2/02
Summer Tutorials!, by Ciprian Manolescu, David Dumas, Noah Snyder and Peter Green , Harvard Math Department
3/19/02
The 7-Color Theorem, by Stephen DeBacker , Harvard Math Department
3/12/02
Is there a cube-tiling of dimension 7 in which no pair of cubes share a complete 6-dimensional face?, by John Mackey , Harvard Math Department
3/5/02
How Many Equilibria Are There? An Introduction to Morse Theory, by Robin Forman, Rice University (Harvard Ph.D. '85)
2/26/02
Clifford Algebras and Vector Fields on Spheres, by Lionel Levine , Harvard Undergraduate
2/19/02
Finite Projective Planes: A Mathematical Definition of 'Nice', by Gregg Musiker , Harvard Undergraduate
2/12/02
Zero Knowledge Proofs, by Alex Healy, Harvard Undergraduate
Math Tutorials: Statistics and Applications to Biology, Milnor's Seven Spheres - Algebraic and Differential Topology, by Stan Sawyer, Jake Rasmussen, Harvard Math Department
Photos from the event on
Wednesday, September 19. 2001, Cliff Taubes gave the welcoming talk on
the impact of path integrals in Mathematics.
Fall 2001
12/12/01
The COMAP Mathematical Modelling Competition, The Putnam Exam for Applied Mathematics, by Michael Brenner, Harvard Math Department (DEAS)
12/5/01
Noam Elkies, by Problems from this years Putnam exam, Harvard Math Department
11/27/01
1+1 \<1, by Richard Cudney, Harvard Undergraduate
11/20/01
q-series Identities and a New Result on the Representation of Integers as Sums of Triangular Numbers, by Jayce Getz, Harvard Undergraduate
11/13/01
Graduate Discussion Panel, by Recent Ph.D's and current grad students, Harvard Math Department/MIT/Berkeley,Chicago,Columbia
11/6/01
Geometric Evolution, by Albert Chau, Harvard Math Department
10/30/01
Well-covered graphs and the roller coaster conjecture, by Phil Matchett, Harvard Undergraduate
10/23/01
Groups of Loops, Bounding the Volume of Hyperbolic 3-manifolds, by Jim Fowler, Harvard Undergraduate
10/16/01
Tony Varilly, by Harvard Undergraduate , Primes mod 4 and Dirichlets Theorem on Arithmetic Progressions
10/09/01
The Volume and Area of the n-Sphere, by Youngsub Yoon, Harvard Undergraduate
10/02/01
Cause we living in an Euclidean world, 'Proofs' of the Parallel Postulate, by Ari Schwayder, Harvard Undergraduate
9/25/01
Kant and the Axiom of Abstraction, by Sarah Moss , Harvard Undergraduate
9/16/01
The impact of path integrals in Mathematics (Photos), by Cliff Taubes , Harvard Math Department
9/11/01
The REACH Program and Exciting Math Tutorials, by Jim Propp, Eddy Lee and Marty Weissman, Harvard Math Department
Food and chat before the talk.
Spring 2001
5/8/01
Rearrangeable Oompaloompas: Recent Open Questions in Graph Theory, by Sarah Moss, Harvard Undergraduate
5/1/01
Lifting the p-th Power Map to Characteristic 0, by Anna Medvedovsky, Harvard Undergraduate
4/24/01
Fun with p-Orderings, by Davesh Maulik, Harvard Undegraduate
4/17/01
Mandelbrot's Conjecture and Critical Exponents for Brownian Motion, by Nick Roussanov, Harvard Undergraduate
4/3/01
All Functions Are Continuous, by Richard Cudney, Harvard Undergraduate
3/20/01
Doing Gauss one better: Construction of the regular 257-gon using Mathematica , by Qian (Pinky) Zhang, Harvard Undergraduate
2/20/01
Accounting for Variability: Algebra, Statistics, and Measure Theory, by Rick Cleary, Associate Dean at Cornell
2/13/01
Finite Geometries, by Lionel Levine, Harvard Undergraduate
2/6/01
Richard Cudney, by y x = q x y, Harvard Undergraduate
1/30/01
Tutorials: Codes and Algebraic Curves, Symmetric Groups, Fourier Analysis , by Greg Warrington, Russ Mann, Andy Engelward and John Boller , Harvard Math Department
The speaker Cliff Taubes on "4 dimensions" 26. September 2000
Fall 2000
12/12/00
Really Big Numbers, by Jim Fowler, Harvard Undergraduate
12/5/00
Ramsey Theory, or Why Are There So Many People Who Don't Know Each Other at my Party, by Andrew Ostergaard, Harvard Undergraduate
11/28/00
The Symplectic Camel and other animals, by Dusa McDuff, SUNY Stony Brook
11/21/00
The Graduate School Discussion Panel, by recent Ph.D.'s and current graduate students , Harvard, MIT, etc
11/14/00
Voter Distances, by Ken Fan, Harvard University
11/7/00
Wild Topology in 2 and 3 Dimensions, by Danny Calegari, Harvard Math Department
10/31/00
Why exp(pi 163^{1/2}) is almost an integer... or April Fools, Elliptic Curves and Complex Multiplication, by Matthew Baker, Harvard Math Department
10/24/00
Dennis Clark, by Harvard Undergraduate , Lattices on Parabolic Trees
10/17/00
An Intriguing Open Problem in Chaos Theory, by Oliver Knill, Harvard Math Department
10/10/00
Permutations, reduced decompositions, and Young tableaux, by Richard Stanley, MIT and Harvard Math Departments
10/3/00
Keller's Cube Tiling Conjecture, by John Mackey, Harvard Math Department
9/26/00
Four Dimensions, by Cliff Taubes, Harvard Math Department
9/19/00
Tutorials: Making Waves: Fourier Analysis and Applications Computational Algebraic Geometryg , by Russ Mann and Hal Schenck, Harvard Math Department
Spring 2000
5/14/00
Seniors' Thesis Reflections , by Various speakers, Harvard Undergraduates
5/9/00:
Efficient Methods for Almost Proving Things , by Lauren Williams, Harvard Undergraduate
5/2/00
Breakfast on the Beach: Tiling Beach Balls and Bagels , by Lisa Powell, Harvard Undergraduate
4/25/00
Spinors in Mathematics , by Dan Lee, Harvard Undergraduate
4/18/00
A "new" proof that the square root of 2 is irrational , by Max Lieblich, Harvard Undergraduate
4/11/00
Spacetime Geometry, Graph Theory , by Spiro Karigiannis, Ian Dowker, Harvard Math Department
4/4/00:
Coloring Links in the Plane , by Travis Schedler, Harvard Undergraduate
3/21/00
Quantum Factorization , by Richard Cudney, Harvard Undergraduate
3/14/00
Contrary to Popular Belief, R^{n} is not the same as R^{m}, by Mike Hill, Harvard Undergraduate
3/7/00:
Zeta(3) is irrational , by Stephen Miller, Yale University
2/29/00
Iterates of a Spiky Function , by Lionel Levine, Harvard Undergraduate
2/22/00
Lie Groups and Geometry , by Megan Kerr, Wellesley College
2/15/00
Knots and Knotted Graphs , by Daniel Allcock, Harvard Math Department
2/8/00:
The Ham Sandwich Theorem , by Tara Holm, MIT Grad student
DAVID B. MUMFORD UNDERGRADUATE MATHEMATICS PRIZE
From the income of the gift in 1997 of Peter L. Falb, A.B. 1957, A.M.
1957, Ph.D. 1961 to establish the David B. Mumford Undergraduate
Mathematics Prize, to be given annually to the most promising senior
concentrator in mathematics, provided such concentrator is outstanding.)
WISTER PRIZE
From a bequest of Charles J. Wister, a prize is awarded in alternate years
to the Senior concentrating in Mathematics or Music who has the highest
record in his/her field of concentration. In 2001-2002 the prize is open to
concentrators in Music, in 2002-03 to concentrators in Mathematics. Further
information may be obtained from the Departments of Mathematics and Music.)
ROBERT FLETCHER ROGERS PRIZE
From the income of the gift in 1921 of the late Robert Fletcher Rogers,
A.B. 1886, there have been established a first prize and second prize to be
awarded annually to the College students who present the best talks before
the Mathematics Table during an academic year. In making the awards,
emphasis will be laid primarily on the excellence of the oral presentation.
It is not essential that the material presented be original, but mastery of
the subject and adaptation of the presentation will be regarded as
important. The prizes will be awarded by the Department of Mathematics on
the recommendation of the Mathematics Table.
Fall 1999
12/7/99
Pondering Putnam Problems , by Noam Elkies, Harvard Math Department
11/30/99
Making Money with Math , by Eric Wepsic, Harvard Alumnus
11/23/99
Hyperspheres, symmetries and Diophantine equations , by Damien Wisniewski, Harvard Undergrduate
11/16/99
Panel Discussion on Graduate Schools , by Recent Ph.D.'s and current graduate students, from Harvard, MIT, Princeton, Berkeley, Chicago, etc
11/2/99
All triangles are isosceles and other strange p-adic facts , by John Boller, Harvard Math Department
10/26/99
Polytopes in four dimensions , by Eric Sommers, Harvard Math Department
10/19/99
Polytopes and their fans , by Kalle Karu, Harvard Math Department
10/12/99
Inverse Galois theory , by Tammy Lefcourt, Harvard Math Department
10/5/99
The geometric structure of a 1-point space , by Brian Conrad, Harvard Math Department
9/28/99
Lattices in Euclidean Space , by Benedict Gross, Harvard Math Department