An Introduction to the Combinatorics and Geometry of Schubert Varieties

Transcription

1 An Introduction to the Combinatorics and Geometry of Schubert Varieties Sara Billey University of Washington billey Nebraska Conference for Undergraduate Women in Mathematics January 28, 22

2 Famous Quotations Arnold Ross (and PROMYS). Think deeply of simple things. Angela Gibney. Why do algebraic geometers love moduli spaces? It is just like with people, if you want to get to know someone, go to their family reunion. Goal. Focus our attention on a particular family of varieties which are indexed by combinatorial data where lots is known about their structure and yet lots is still open.

3 Schubert Varieties A Schubert variety is a member of a family of projective varieties whose points are indexed by matrices and whose defining equations are determinantal minors. Typical properties: This family of varieties is indexed by combinatorial objects; e.g. partitions, permutations, or group elements. Some are smooth and some are singular. Their topological structure is often encoded by subsets and quotients of polynomial rings including the symmetric functions.

4 Enumerative Geometry Approximately 50 years ago... Grassmann, Schubert, Pieri, Giambelli, Severi, and others began the study of enumerative geometry. Early questions: What is the dimension of the intersection between two general lines in R 2? How many lines intersect two given lines and a given point in R 3? How many lines intersect four given lines in R 3? Modern questions: How many points are in the intersection of 2,3,4,...Schubert varieties in general position?

5 Why Study Schubert Varieties?. It can be useful to see points, lines, planes etc as families with certain properties. 2. Schubert varieties provide interesting examples for test cases and future research in algebraic geometry. 3. Applications in discrete geometry, computer graphics, economics, chemistry and computer vision.

6 Outline. Review of vector spaces and projective spaces 2. Introduction to Grassmannians, Flag Manifolds and Schubert Varieties 3. Five Fun Facts on Schubert Varieties 4. References for more information 5. Advice to offer

7 Vector Spaces V is a vector space over a field F if it is closed under addition and multiplication by scalars in F. B = {b,...,b k } is a basis for V if for every a V there exist unique scalars c,...,c k F such that a = c b + c 2 b c k b k = (c,c 2,...,c k ) F k. The dimension of V equals the size of a basis. A subspace U of a vector space V is any subset of the vectors in V that is closed under addition and scalar multiplication. Fact. Any basis for U can be extended to a basis for V.

8 Projective Spaces Defn. P(V ) = {-dim subspaces of V } = V d a=a. -dim subspaces = lines in V through 0 points in P(V ) 2-dim subspaces = planes in V through 0 lines in P(V ) Given a basis B = {b,b 2,...,b k } for V, the line spanned by a = c b + c 2 b c k b k P(V ) has homogeneous coordinates [c : c 2 : : c k ] = [dc : dc 2 : : dc k ] for any d F.

9 The Grassmannian Varieties Definition. Fix a vector space V over C (or R, Z 2,...) with basis B = {e,...,e n }. The Grassmannian variety G(k,n) = {k-dimensional subspaces of V }. Question. How can we impose the structure of a variety or a manifold on this set?

10 The Grassmannian Varieties Answer. Relate G(k,n) to the vector space of k n matrices. U =span 6e + 3e 2, 4e + 2e 3, 9e + e 3 + e 4 G(3,4) M U = U G(k,n) rows of M U are independent vectors in V some k k minor of M U is NOT zero.

11 Plücker Coordinates Define f j,j 2,...,j k (M) to be the polynomial given by the determinant of the matrix x,j x,j2... x,jk x 2,j x 2,j2... x 2,jk.... x kj x kj2... x kjk G(k,n) is an open set in the Zariski topology on k n matrices defined as the complement of the intersection over all k-subsets of {,2,...,n} of the varieties V (f j,j 2,...,j k ) where f j,j 2,...,j k (M) = 0. All the determinants f j,j 2,...,j k are homogeneous polynomials of degree k so G(k,n) can be thought of as an open set in projective space.

12 The Grassmannian Varieties Canonical Form. Every subspace in G(k, n) can be represented by a unique matrix in row echelon form. Example. U =span 6e + 3e 2, 4e + 2e 3, 9e + e 3 + e 4 G(3,4) e + e 2, 2e + e 3, 7e + e 4

13 Subspaces and Subsets Example. U = RowSpan Definition G(3,0) position(u) = {3,7,9} If U G(k,n) and M U is the corresponding matrix in canonical form then the columns of the leading s of the rows of M U determine a subset of size k in {,2,...,n} := [n]. There are 0 s to the right of each leading and 0 s above and below each leading. This k-subset determines the position of U with respect to the fixed basis.

14 The Schubert Cell C j in G(k,n) Defn. Let j = {j < j 2 < < j k } [n]. A Schubert cell is C j = {U G(k,n) position(u) = {j,...,j k }} Fact. G(k,n) = C j over all k-subsets of [n] Example. In G(3, 0), C {3,7,9} = Observe dim(c {3,7,9} ) = 3. In general, dim(c j ) = j i i.

15 Schubert Varieties in G(k, n) Defn. Let j = {j < j 2 < < j k } [n]. A Schubert variety is X j = Closure of C j under Zariski topology. Question. In G(3,0), what polynomials vanish on C {3,7,9}? C {3,7,9} = Answer. All minors f j,j 2,j 3 with 4 j 8 or j = 3 and 8 j 2 9 or j = 3,j 2 = 7 and j 3 = 0 In other words, the canonical form for any subspace in X j = C j has 0 s to the right of column j i in each row i.

16 k-subsets and Partitions Defn. A partition of a number n is a weakly increasing sequence of nonnegative integers λ = (λ λ 2 λ k ) such that n = λ i = λ. Partitions can be visualized by their Ferrers diagram (2,5,6) Fact. There is a bijection between k-subsets of {,2,...,n} and partitions whose Ferrers diagram is contained in the k (n k) rectangle given by shape : {j <... < j k } (j,j 2 2,...,j k k).

17 A Poset on Partitions Defn. A partial order or a poset is a reflexive, anti-symmetric, and transitive relation on a set. Defn. Young s Lattice If λ = (λ λ 2 λ k ) and µ = (µ µ 2 µ k ) then λ µ if the Ferrers diagram for λ fits inside the Ferrers diagram for µ. Facts.. X j = shape(i) shape(j) C i. 2. The dimension of X j is shape(j). 3. TheGrassmannianG(k,n) = X {n k+,...,n,n} isaschubertvariety!

18 Enumerative Geometry Revisited Question. How many lines intersect four given lines in R 3? Translation. Given a line in R 3, the family of lines intersecting it can be interpreted in G(2, 4) as the Schubert variety X {2,4} = ( ) with respect to a suitably chosen basis determined by the line. Reformulated Question. How many subspaces U G(2,4) are in the intersection of 4 copies of the Schubert variety X {2,4} each with respect to a different basis? Modern Solution. Use Intersection Theory!

19 Intersection Theory Schubert varieties induce canonical elements of the cohomology ring for G(k,n) called Schubert classes: [X j ]. Multiplication of Schubert classes corresponds with intersecting Schubert varieties with respect to different bases. [X i ][X j ] = [ X i (B ) X j (B 2 ) ] Schubert classes add and multiply just like Schur functions. Schur functions are a fascinating family of symmetric functions indexed by partitions which appear in many areas of math, physics, theoretical computer science. Expanding the product of two Schur functions into the basis of Schur functions can be done via linear algebra: S λ S µ = c ν λ,µ S ν. The coefficients c ν λ,µ are non-negative integers called the Littlewood- Richardson coefficients. In a 0-dimensional intersection, the coefficient of [X {,2,...,k} ] is the number of subspaces in X i (B ) X j (B 2 ).

20 Enumerative Solution Reformulated Question. How many subspaces U G(2,4) are in the intersection of 4 copies of the Schubert variety X {2,4} each with respect to a different basis?

21 Enumerative Solution Reformulated Question. How many subspaces U G(2,4) are in the intersection of 4 copies of the Schubert variety X {2,4} each with respect to a different basis? Solution. [ X{2,4} ] = S() = x + x 2 + x 3 + x 4 By the recipe, compute [ X{2,4} (B ) X {2,4} (B 2 ) X {2,4} (B 3 ) X {2,4} (B 4 ) ] = S 4 () = 2S (2,2) + S (3,) + S (2,,). Answer. The coefficient of S 2,2 = [X,2 ] is 2 representing the two lines meeting 4 given lines in general position.

22 Littlewood-Richardson Rules There are many combinatorial algorithms to compute the Littlewood-Richardson coefficients c ν λ,µ. One of my favorites is based on Knutson-Tao-Woodward puzzles. Example. (Warning: picture is not accurate without description.)

23 Recap. G(k,n) is the Grassmannian variety of k-dim subspaces in R n. 2. The Schubert varieties in G(k, n) are nice projective varieties indexed by k-subsets of [n] or equivalently by partitions in the k (n k) rectangle. 3. Geometrical information about a Schubert variety can be determined by the combinatorics of partitions. 4. Intersection theory applied to Schubert varieties can be used to solve problems in enumerative geometry.

24 Generalizations Nothing is more disagreeable to the hacker than duplication of effort. The first and most important mental habit that people develop when they learn how to write computer programs is to generalize, generalize, generalize.: Neil Stephenson In the Beginning was the Command Line Same goes for mathematicians!

25 The Flag Manifold Defn. A complete flag F = (F,...,F n ) in C n is a nested sequence of vector spaces such that dim(f i ) = i for i n. F is determined by an ordered basis f,f 2,...f n where F i = span f,...,f i. Example. F = 6e + 3e 2, 4e + 2e 3, 9e + e 3 + e 4, e

26 The Flag Manifold Canonical Form. F = 6e + 3e 2, 4e + 2e 3, 9e + e 3 + e 4, e = e + e 2, 2e + e 3, 7e + e 4, e Fl n (C) := flag manifold over C n n k= G(n,k) ={complete flags F } = B \ GL n (C), B = lower triangular mats.

27 Flags and Permutations Example. F = 2e +e 2, 2e +e 3, 7e +e 4, e Note. If a flag is written in canonical form, the positions of the leading s form a permutation matrix. There are 0 s to the right and below each leading. This permutation determines the position of the flag F with respect to the reference flag E = e, e 2, e 3, e

28 Many ways to represent a permutation = [ ] = 234 = matrix notation two-line notation one-line notation rank table = = (,2,3) 234 diagram of a permutation string diagram reduced word

29 The Schubert Cell C w (E ) in Fl n (C) Defn. C w (E ) = All flags F with position(e,f ) = w = {F Fl n dim(e i F j ) = rk(w[i,j])} Example. F = C 234 = : C Easy Observations. dim C (C w ) = l(w) = # inversions of w. C w = w B is a B-orbit using the right B action, e.g b, b 2, b 2, b 2, b 2,2 0 0 b 3, b 3,2 b 3,3 0 = b 3, b 3,2 b 3,3 0 b 4, b 4,2 b 4,3 b 4, b 4, b 4,2 b 4,3 b 4,4 b, 0 0 0

30 The Schubert Variety X w (E ) in Fl n (C) Defn. X w (E ) = Closure of C w (E ) under the Zariski topology = {F Fl n dim(e i F j ) rk(w[i,j])} where E = e, e 2, e 3, e 4. Example X 234(E ) = Why?

31 Five Fun Facts Fact. The closure relation on Schubert varieties defines a nice partial order. X w = C v = v w v w Bruhat order (Ehresmann 934, Chevalley 958) is the transitive closure of w < wt ij w(i) < w(j). X v Example. Bruhat order on permutations in S Observations. Self dual, rank symmetric, rank unimodal.

32 Bruhat order on S

33 Bruhat order on S 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

34 0 Fantastic Facts on Bruhat Order. Bruhat Order Characterizes Inclusions of Schubert Varieties 2. Contains Young s Lattice in S 3. Nicest Possible Möbius Function 4. Beautiful Rank Generating Functions 5. [x, y] Determines the Composition Series for Verma Modules 6. Symmetric Interval [ˆ0, w] X(w) rationally smooth 7. Order Complex of (u, v) is shellable 8. Rank Symmetric, Rank Unimodal and k-sperner 9. Efficient Methods for Comparison 0. Amenable to Pattern Avoidance

35 Singularities in Schubert Varieties Defn. X w is singular at a point p dimx w = l(w) < dimension of the tangent space to X w at p. Observation. Every point on a Schubert cell C v in X w looks locally the same. Therefore, p C v is a singular point the permutation matrix v is a singular point of X w. Observation 2. The singular set of a varieties is a closed set in the Zariski topology. Therefore, if v is a singular point in X w then every point in X v is singular. The irreducible components of the singular locus of X w is a union of Schubert varieties: Sing(X w ) = v maxsing(w) X v.

36 Singularities in Schubert Varieties Fact 2. (Lakshmibai-Seshadri) A basis for the tangent space to X w at v is indexed by the transpositions t ij such that vt ij w. Definitions. Let T = invertible diagonal matrices. The T-fixed points in X w are the permutation matrices indexed by v w. If v, vt ij are permutationsin X w theyareconnectedby at-stable curve. The set of all T-stable curves in X w are represented by the Bruhat graph on [id,w].

37 Bruhat Graph in S 4 (4 3 2 ) (4 2 3 ) (4 3 2) (3 4 2 ) (4 3 2) (4 2 3) (2 4 3 ) (3 4 2) (3 2 4 ) (4 2 3) ( 4 3 2) (2 4 3) (3 4 2) (2 3 4 ) (3 2 4) ( 4 2 3) (2 4 3) ( 3 4 2) (3 2 4) (2 3 4) ( 2 4 3) (2 3 4) ( 3 2 4) ( 2 3 4)

38 Tangent space of a Schubert Variety Example. T 234 (X 423 ) = span{x i,j t ij w}. (4 2 3 ) (3 2 4 ) (4 2 3) (4 3 2) (2 4 3 ) (3 2 4) (3 4 2) (2 3 4 ) (4 2 3) (2 4 3) ( 4 3 2) (3 2 4) (2 3 4) (2 4 3) ( 3 4 2) ( 4 2 3) (2 3 4) ( 3 2 4) ( 2 4 3) ( 2 3 4) dimx(423)=5 dimt id (423) = 6 = X(423) is singular!

39 Five Fun Facts Fact 3. There exists a simple criterion for characterizing singular Schubert varieties using pattern avoidance. Theorem: Lakshmibai-Sandhya 990 (see also Haiman, Ryan, Wolper) X w is non-singular w has no subsequence with the same relative order as 342 and 423. Example: w = contains = X is singular w = avoids 423 = X is non-singular &342

40 Five Fun Facts Consequences of Fact 3. (Billey-Warrington, Kassel-Lascoux-Reutenauer, Manivel 2000) The bad patterns in w can also be used to efficiently find the singular locus of X w. (Bona 998, Haiman) Let v n be the number of w S n for which X(w) is non-singular. Then the generating function V (t) = n v nt n is given by V (t) = 5t + 3t2 + t 2 4t 6t + 8t 2 4t 3. (Billey-Postnikov 20) Generalized pattern avoidance to all semisimple simply-connected Lie groups G and characterized smooth Schubert varieties X w by avoiding these generalized patterns. Only requires checking patterns of types A 3,B 2,B 3,C 2,C 3,D 4,G 2.

41 Five Fun Facts Fact 4. There exists a simple criterion for characterizing Gorenstein Schubert varieties using modified pattern avoidance. Theorem: Woo-Yong (Sept. 2004) X w is Gorenstein w avoids 3542 and 2453 with Bruhat restrictions {t 5,t 23 } and {t 5,t 34 } for each descent d in w, the associated partition λ d (w) has all of its inner corners on the same antidiagonal.

42 Five Fun Facts Fact 5. Schubert varieties are useful for studying the cohomology ring of the flag manifold. Theorem (Borel): H (Fl n ) = Z[x,...,x n ] e,...e n. The symmetric function e i = k < <k i n x k x k2...x ki. {[X w ] w S n } form a basis for H (Fl n ) over Z. Question. What is the product of two basis elements? [X u ] [X v ] = [X w ]c w uv.

43 Cup Product in H (Fl n ) Answer. Use Schubert polynomials! Due to Lascoux-Schützenberger, Bernstein- Gelfand-Gelfand, Demazure. BGG: If S w [X w ]mod e,...e n then S w s i S w x i x i+ [X wsi ] if l(w) < l(ws i ) [X id ] x n x n 2 2 x n i>j(x i x j )... Here deg[x w ] = codim(x w ). LS: Choosing [X id ] x n x n 2 2 x n works best because product expansion can be done without regard to the ideal!

44 Schubert polynomials for S 4 S w0 (234) = S w0 (234) = x S w0 (324) = x 2 + x S w0 (324) = x 2 S w0 (234) = x x 2 S w0 (324) = x 2 x 2 S w0 (243) = x 3 + x 2 + x S w0 (243) = x x 3 + x x 2 + x 2 S w0 (423) = x x x 2 + x 2 S w0 (423) = x 3 S w0 (243) = x x x2 x 2 S w0 (423) = x 3 x 2 S w0 (342) = x 2 x 3 + x x 3 + x x 2 S w0 (342) = x 2 x 3 + x 2 x 2 S w0 (432) = x 2 2 x 3 + x x 2 x 3 + x 2 x 3 + x x x2 x 2 S w0 (432) = x 3 x 3 + x 3 x 2 S w0 (342) = x 2 x2 2 S w0 (432) = x 3 x2 2 S w0 (234) = x x 2 x 3 S w0 (324) = x 2 x 2x 3

45 Cup Product in H (Fl n ) Key Feature. Schubert polynomials have distinct leading terms, therefore expanding any polynomial in the basis of Schubert polynomials can be done by linear algebra just like Schur functions. Buch: Fastest approach to multiplying Schubert polynomials uses Lascoux and Schützenberger s transition equations. Works up to about n = 5. Draw Back. Schubert polynomials don t prove c w uv s are nonnegative (except in special cases).

46 Cup Product in H (Fl n ) Another Answer. By intersection theory: [X u ] [X v ] = [X u (E ) X v (F )] Perfect pairing: [X u (E )] [X v (F )] [X w0 w(g )] = c w uv [X id] [X u (E ) X v (F ) X w0 w(g )] The Schubert variety X id is a single point in Fl n. Intersection Numbers: c w uv = #X u(e ) X v (F ) X w0 w(g ) Assuming all flags E, F, G are in sufficiently general position.

47 Intersecting Schubert Varieties Example. Fix three flags R, G, and B : Find X u (R ) X v (G ) X w (B ) where u,v,w are the following permutations: R R 2 R 3 G G 2 G 3 B B 2 B 3 P P 2 P 3

48 Intersecting Schubert Varieties Example. Fix three flags R, G, and B : Find X u (R ) X v (G ) X w (B ) where u,v,w are the following permutations: R R 2 R 3 G G 2 G 3 B B 2 B 3 P P 2 P 3

49 Intersecting Schubert Varieties Example. Fix three flags R, G, and B : Find X u (R ) X v (G ) X w (B ) where u,v,w are the following permutations: R R 2 R 3 G G 2 G 3 B B 2 B 3 P P 2 P 3

50 Intersecting Schubert Varieties Schubert s Problem. How many points are there usually in the intersection of d Schubert varieties if the intersection is 0-dimensional? ( ) n Solving approx. n d equations with variables is challenging! 2 Observation. We need more information on spans and intersections of flag components, e.g. dim(e x E 2 x 2 E d x d ).

51 Permutation Arrays Theorem. (Eriksson-Linusson,2000)ForeverysetofdflagsE,E2,...,Ed, there exists a unique permutation array P [n] d such that dim(e x E 2 x 2 E d x d ) = rkp[x]. B B 2 B 3 R R 2 R 3 R R 2 R 3 R R 2 R G G 2 G 3

52 Totally Rankable Arrays Defn. For P [n] d, rk j P = #{k x P s.t. x j = k}. P is rankable of rank r if rk j (P) = r for all j d. y = (y,...,y d ) x = (x,...,x d ) if y i x i for each i. P[x] = {y P y x} P is totally rankable if P[x] is rankable for all x [n] d. X Union of dots is totally rankable. Including X it is not.

53 Permutation Arrays 2 O O Points labeled O are redundant, i.e. including them gives another totally rankable array with same rank table. Defn. P [n] d is a permutation array if it is totally rankable and has no redundant dots. [4] 2. Open. Count the number of permutation arrays in [n] k.

54 Permutation Arrays Theorem. (Eriksson-Linusson) Every permutation array in [n] d+ can be obtained from a unique permutation array in [n] d by identifying a sequence of antichains. This produces the 3-dimensional array P = {(4,4,),(2,4,2),(4,2,2),(3,,3),(,4,4),(2,3,4)}

55 Unique Permutation Array Theorem Theorem.(Billey-Vakil, 2005) If X = X w (E ) X w d(ed ) is nonempty 0-dimensional intersection of d Schubert varieties with respect to flags E,E2,...,Ed in general position, then there exists a unique permutation array P [n] d+ such that X = {F dim(e x E 2 x 2 E d x d F xd+ ) = rkp[x].} () Furthermore, we can recursively solve a family of equations for X using P. Open Problem. Can one find a finite set of rules for moving dots in a 3-d permutation array which determines the c w uv s analogous to one of the many Littlewood-Richardson rules? Recent Progress. Izzet Coskun s Mondrian tableaux.

56 Generalizations of Schubert Calculus for G/B : A Highly Productive Score. A: GL n B: SO 2n+ C: SP 2n D: SO 2n Semisimple Lie Groups Kac-Moody Groups GKM Spaces cohomology quantum equivariant K-theory eq. K-theory Recent Contributions from: Bergeron, Berenstein, Billey, Brion, Buch, Carrell, Ciocan-Fontainine, Coskun, Duan, Fomin, Fulton, Gelfand, Goldin, Graham, Griffeth, Guillemin, Haibao, Haiman, Holm, Huber, Ikeda, Kirillov, Knutson, Kogan, Kostant, Kresh, S. Kumar, A. Kumar, Lam, Lapointe, Lascoux, Lenart, Miller, Morse, Naruse, Peterson, Pitti, Postnikov, Purhboo, Ram, Richmond, Robinson, Shimozono, Sottile, Sturmfels, Tamvakis, Thomas, Vakil, Winkle, Yong, Zara...

57 Some Recommended Further Reading. Schubert Calculus by Steve Kleiman and Dan Laksov. The American Mathematical Monthly, Vol. 79, No. 0. (Dec., 972), pp The Symmetric Group by Bruce Sagan, Wadsworth, Inc., Young Tableaux by William Fulton, London Math. Soc. Stud. Texts, Vol. 35, Cambridge Univ. Press, Cambridge, UK, Determining the Lines Through Four Lines by Michael Hohmeyer and Seth Teller, Journal of Graphics Tools, 4(3):-22, Honeycombs and sums of Hermitian matrices by Allen Knutson and Terry Tao. Notices of the AMS, February 20; awarded the Conant prize for exposition.

58 Some Recommended Further Reading 6. A geometric Littlewood-Richardson rule by Ravi Vakil, Annals of Math. 64 (2006), Flag arrangements and triangulations of products of simplices by Sara Billey and Federico Ardila, Adv. in Math, volume 24 (2007), no. 2, A Littlewood-Richardson rule for two-step flag varieties by Izzet Coskun. Inventiones Mathematicae, volume 76, no 2 (2009) p Sage:Creating a Viable Free Open Source Alternative to Magma, Maple, Mathematica, and Matlab by William Stein. sagebook/sagebook.pdf, Jan. 22. Generally, these published papers can be found on the web. The books are well worth the money.

59 Advice to Offer (STRONGLY) Work hard. Ask questions. Prove lemmas beyond what is written in the textbook. Memorize all of the mathematical vocabulary words and named theorems asap, preferably by the next lecture. Trust nothing in a math lecture or textbook, check ever single statement to practice your skills. Learn to program, and write programs to learn math. You are worth more on the job market as a mathematician if you can program. Do an REU. Take a class in mathematical modeling.

60 Advice to Offer (STRONGLY) Apply for every fellowship for grad school you can find. Check out the National Physical Science Consortium, the NSF, DOE, the Navy, the Ford Foundation, AWM, MAA, and the AMS.

61 Advice to Offer (STRONGLY) Find a way to enjoy the process. Be proactive about making your environment work for you. Look for people who can be your mentors at all stages. Be a mentor to others. Teach what you know. Be generous.

62 Advice to Offer (STRONGLY) Read Cathy O Neil s blog: mathbabe.org. More of my advice is linked on my website; search on sara math.

63 Advice to Offer (STRONGLY) Work hard. Ask questions. Prove lemmas beyond what is written in the textbook. Memorize all of the mathematical vocabulary words and named theorems asap, preferably by the next lecture. Trust nothing in a math lecture or textbook, check ever single statement to practice your skills. Learn to program, and write programs to learn math. You are worth more on the job market as a mathematician if you can program. Do an REU. Take a class in mathematical modeling. Take feedback on your work with an enthusiastic attitude toward learning. Every question or comment is a gift. If you find yourself feeling defensive, ponder the statement Don t take it personally. Apply for every fellowship for grad school you can find. Check out the National Physical Science Consortium, the NSF, DOE, the Navy, the Ford Foundation, AWM, MAA, and the AMS. Find a way to enjoy the process. Be proactive about making your environment work for you. Look for people who can be your mentors at all stages. Be a mentor to others. Teach what you know. Be generous. Read Cathy O Neil s blog: mathbabe.org. More of my advise is linked on my website; search on sara math.