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Things to Do NOW

1. HubAnalytics
2. Look into Differential poset problem
3. Learn Discrete Morse Theory
4. Solve the P//G problem at least for rank 2 graph!! NO
5. Submit rainbow graph paper to a journal
6. Apply for Conferences
7. Apply for Internship: http://www.gs.com
8. Study Machine Learning: http://courses.csail.mit.edu/6.867/wiki/index.php?title=6.867_Machine_Learning_(2011_Fall)
9. Study Graphical Models: https://stellar.mit.edu/S/course/6/fa11/6.438/
10. CFA level 1
11. Coding theory and combinatorial optimization (linear programming)

Teaching

1. 18.02 Calculus

Housing

1. Housing

List of Projects

1. Affine Balanced Labellings
2. Affine Quasisymmetric Functions
3. Differential Posets
4. Permutation RSK distribution
5. Hopf Algebras
6. Groups Acting on Posets
7. Stanley's Cycle Problem: http://math.mit.edu/~ahmorales/mywiki/doku.php?id=cycles:stanley_s_cycle_problem
8. Eigenvalue Problem
9. Markov Chains and Schubert Polynomials
10. Toric Arrangements and Arithmetic Matroids
11. Restricted Partition Posets and Specht Modules

List of Big Problems

1. Finding the percolation threshold
2. Prove that 3-sum problem takes at least $n^2 \log n$ or find faster algorithm.
3. Shellsort increment sequence and time complexity http://en.wikipedia.org/wiki/Shellsort
4. Find efficient and "symmetric" algorithm for deletion in Binary Search Tree.

Theories I need to learn

1. Review algebraic geometry with cohomology theory
2. Coxeter groups, root systems, Hecke Algebras
3. Crystals
4. Promotion and Evacuation
5. Fulton's Young Tableaux
6. Tropical RSK Correspondence

Dictionary

1. Promotion and Evacuation
2. Hecke Algebras
3. Grothendieck Groups
4. Ito Calculus
5. Machine Learning

Random Ideas

• When proving a structure constant of some multiplication with conjectural structure constant C,
  define $\mu(X_a, X_B) := \sum C_{a,b}^c X_c$ and show that $\mu(1,X_a) = X_a$ and $\mu(X_r X_a,X_b) = \mu(X_r, X_a X_b)$ for $X_r, r\in S$ where $S$ generates R. Then $\mu$ should be the multiplication.

Android App Ideas

• Convert zip codes to city, State. (first need to build the database.)

Papers to read

• On the cd-index and gamma-vector of S*-shellable CW-spheres - Satoshi Murai, Eran Nevo

We show that the $\gamma$-vector of the order complex of any polytope is the f-vector of a balanced simplicial complex. This is done by proving this statement for a subclass of Stanley's S-shellable spheres which includes all polytopes. The proof shows that certain parts of the cd-index, when specializing $c=1$ and considering the resulted polynomial in $d$, are the f-polynomials of simplicial complexes that can be colored with "few" colors. We conjecture that the cd-index of a regular CW-sphere is itself the flag f-vector of a colored simplicial complex in a certain sense. http://arxiv.org/pdf/1102.0096v1

• KP solitons, total positivity, and cluster algebras - Yuji Kodamaa and Lauren K. Williams

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili [Kadomtsev BB, Petviashvili VI (1970) Sov Phys Dokl 15:539–541] proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. The regular soliton solutions that one obtains in this way come from points of the totally nonnegative part of the Grassmannian. In this paper we explain how the theory of total positivity and cluster algebras provides a framework for understanding these soliton solutions to the KP equation. We then use this framework to give an explicit construction of certain soliton contour graphs and solve the inverse problem for soliton solutions coming from the totally positive part of the Grassmannian. http://arxiv.org/abs/1106.0023

List of Problems

• Prove positivity for linear Laurent phenomenon algebras & polytopality of linear Laurent phenomenon complexes

http://arxiv.org/abs/1206.2611

• Bunkbed conjecture - Svante linusson

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=7930738 http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=7930750

• Bona's Conjecture on Product of Cycles

$u \cdot v = w$ where $u$ and $v$ are long cycles, probability that 1 and 2 are in the same cycle of $w$

• Conjecture of Thomas Lam on Affine NilCoxeter Algebras

http://arxiv.org/pdf/1007.2871v1

• Pieri rule for toric Schur functions and Affine Stanley symmetric functions

http://arxiv.org/pdf/math/0501335v1

• Let $n\geq 2$ and $t \geq 0$. Let $f(n,t)$ be the number of sequences with $n$ $x$'s and $2t$ $a_{ij}$'s, where $1\leq i < j \leq n$, such that each $a_{ij}$ occurs between the i-th $x$ and the j-th $x$ in the sequence. (Thus the total number of terms in each sequence is $n + 2t\binom{n}{2}$.) Then,

$f(n,t) = \frac{(n+tn(n-1))!}{n!t!^n(2t)!^{\binom{n}{2}}} \displaystyle\prod_{j=1}^{n} \frac{((j-1)t)!^2(jt)!}{(1+(n+j-2)t)!}$. (Problem 27 of http://math.mit.edu/~rstan/bij.pdf )

• The n-cube $C_n$ (as a graph) is the graph with vertex set $\{ 0,1\} ^n$ (i.e., all binary n-tuples), with an edge between u and v if they differ in exactly one coordinate. Thus $C_n$ has $2^n$ vertices and $n2^{n-1}$ edges. Then,

$c(C_n) = 2^{2^n-n-1}\displaystyle\prod_{k=1}^{n} k^{\binom{n}{k}}$,

where c(G) is the number of spanning trees of G.

• Find a bijection between a staircase tableaux with no $\delta$ of size $n$ and perfect matchings of $\{1,2,\ldots,2n+2\}$. http://math.mit.edu/~rstan/papers/staircase2.pdf

• Problem of finding volumes (Ehrhart polynomials) of valuation polytope (EC1 2nd ed. Exer. 4.62)

• Prove that $J(P)$ where $P = \{(a_0,a_1), (a_0,b_1), (a_1,a_2), (a_1,b_2), \ldots \}$ is the minimal distributive lattice with the property that it has antichains that has one element at each rank. ("An extremal problem for finite topologies and distributive lattices" - Richard Stanley http://math.mit.edu/~rstan/pubs/pubfiles/13.pdf)

Others

1. new desktop
2. Lenovo x220
3. math dept sage notebook server: http://localhost:8000/home/admin/ from runge.mit.edu
4. office sage server: https://mathstation051.mit.edu:8000/
5. home: https://tedyun.mit.edu:8000/
6. repn theory of quivers http://www-fourier.ujf-grenoble.fr/~mbrion/notes_quivers_rev.pdf
7. The Federal Budget Deficit: Causes, Consequences, and Potential Remedies
8. Computer Science Master's Program
9. Emacs
10. Networking 101
11. New England Tour