Vivek Dhand

firstname dot lastname at gmail dot com
Here is a pdf copy of my CV.

Research

My interests lie in the interactions between representation theory, algebraic geometry, and combinatorics. For example, one can apply various algebraic tools (e.g. Stanley-Reisner ideals, the representation theory of quantum SL_2, etc.) to the problem of symmetric chain decomposition of posets. I am also studying certain arithmetic aspects of the Geometric Langlands program: forms of group schemes and generalizations of the geometric Satake correspondence.

Here is my page on the tropical decomposition of Young's partition lattice.

Here are the slides from a recent talk.

Preprints and papers

  1. Geometric Langlands duality and forms of reductive groups. (Thesis)
  2. Symmetric chain decomposition of necklace posets. Elec. J. Combin. 19 (2012) P26.
  3. Tropical decomposition of Young's partition lattice.
  4. Quasi-split group schemes and Langlands duality. (Joint work in progress w/ K. Vilonen).
  5. Symmetric chain decompositions and quantum SL_2. (Joint work in progress w/ P. Magyar).

Teaching

Spring 2011:

Math 310: Abstract Algebra and Number Theory

Fall 2010:

Math 309: Linear Algebra

Math 481: Discrete Math I


Spring 2010:

Math 310: Abstract Algebra and Number Theory

Math 482: Discrete Math II


Fall 2009:

Math 481: Discrete Math I


Summer 2009:

Math 310: Abstract Algebra and Number Theory


Spring 2009:

Math 481: Discrete Math I

Math 482: Discrete Math II


Fall 2008:

Math 481: Discrete Math I

Notes

Applets

The following applets visualize some basic mathematical structures. Each applet is controlled by scrollbars whose respective parameter values are displayed in the status bar. Where applicable, the size of the smallest drawable unit is given in pixels, and the refresh delay time is given in milliseconds.

Stars: draws n points in a circle with edges drawn between points that are k steps apart. Each connected component gets a different color. The number of components is equal to gcd(n,k). For a fixed n, the stars can also be generated by extending the sides of a regular n-gon.

Multiplication on the circle: draws n points in a circle and draws a line from the point with angle x to the point with angle kx. Each edge gets a different color, and the applet automatically animates by increasing the given value of k in increments of 0.001. The curved geometric shapes which emerge are called epicycloids.

The integer lattice points in the plane that are visible to an observer at the origin, i.e. those (a,b) such that gcd(a,b) = 1. Here's a picture of Euclid's orchard.

Iterations of the dragon curve with a specified number of colors. One can also tile the plane with four such dragons.

Cobweb plot for the doubling map on R/Z, starting at a given rational point k/n.

Orbits of the Fibonacci matrix: acting on R/Z x R/Z, starting at a given rational point (a/n,b/n). Below the applet are music files corresponding to the orbits from n = 2 to 8. One can also visualize the action of this map on all rational points with denominator n.

Iterates of the Collatz map which sends a postive odd integer n to the odd part of 3n+1. The maximum value reached during the orbit is displayed in the status bar.

Pascal's triangle mod n.

Not an applet, but a comparison of the just-tempered and well-tempered scales in Music Theory. Here's a picture of Ford circles.

Origami

Here is my page of Origami creations.
Department of Mathematics, Michigan State University.
Last modified: Tuesday, Jan. 24, 2012.