Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mccme.ru/dfc/2012/reports/2013-Vyugin_Report.pdf Äàòà èçìåíåíèÿ: Sun Dec 15 22:02:58 2013 Äàòà èíäåêñèðîâàíèÿ: Sat Mar 1 21:20:10 2014 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï

REPORT ON THE SIMONS­IUM FELLOWSHIP 2013 AND DYNASTY FOUNDATION
ILYA VYUGIN

Results of the 2013 year On solvability of linear differential systems. Consider the system dy = B (z )y , dz y ( z ) Cn , B (z ) Matn
ân

( C)

with rational coefficient matrix B (z ), and irregular singular points a1 , . . . , am . Suppose that the formal exponents j are pairwise distinct, satisfy the condition i Re j > - i 1 , m(n - 1)

and for every pair j , l one of the following two conditions i i 1)Re j - Re l Q i/ i Im j = Im i
l i

holds. Then this system is solvable by quadratures, if and only if there exists a constant matrix C GL(n, C) such that the matrix C B (z )C -1 is upper-triangular (see [1]). On the linear indep endence of the system of functions. Consider the scalar Fuchsian linear diferential equation u(n) + b1 (z )u(
n-1)

+ . . . + bn (z ) = 0

with Fuchsian singularities a1 , . . . , am , u1 (z ), . . . , un (z ) -- are fundamental soluj j tions. Let suppose that all exponents i satisfy Rei > D. Consider the polynomial form (z ) = C u1 . . . un (z ). n 1
1 +...+n
If the form (z ) is not identity to zero then the order of zero in any nonsingular point z0 {a1 , . . . , am } satisfy / ordz0 (z ) (C
n-1 2 M +n-1 )

(m - 2)

2 C z
t0 u1
1

- M mnD.

Let us consider the linear combination ~ (z ) =
1 +...+n n-1 M +n-1

. . . un (z ). n

- M mD and u1 (z ), . . . , un (z ) are linearly indpenSuppose that t > 2 ~ dent. Then the linear form (z ) is not identity to zero. Pap ers
1

)(m-2)


2

ILYA VYUGIN

[1] (With R.R. Gontsov) Solvability of linear differential systems in the Liouvillian sense arXiv:1312.2518 , 2013, (submited to journal). The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular singular points as well as those with (non-resonant) irregular ones and propose some criteria of solvability for systems whose (formal) exponents are sufficiently small. [2] On the linear independence of some system of functions Proceedings of the young mathematician conference, 2013, p. 14-18 (an extended version will be submited to journal soon). The question of linear independece of some system of functions has been studied. The system consists of products of the functions xt and powers of fundumental solutions yi i (x) of some linear differential equation. Such system is linearly independent if t is sufficiently large. The estimate of t has been obtained. Scientific conferences and seminar talks [1] Workshop "Formal and Analytic Solutions of Differential, Difference and Discrete Equations", Warsaw, 25.08.2013-31.08.2013, Talk "On solvability of linear differential systems by quadratures" (joined with R.R. Gontsov). [2] International conference dedicated to the centenary of Israel Gelfand, Moscow, 22.07.2013-15.07.2013, Talk "Linear independence of functions and intersections of subsets of Zp ". [3] Conference "Geometric Days in Novosibirsk-2013", Novosibirsk, 28.08.10131.09.1013, Talk "Local form of solutions of some Painlev´ equations". e [4] Seminar "Analitic theory of differential equations" (D.V Anosov, V.P. Lexin), MIAN RAS Talk "Linear independence of functions and the orders of zeros". Teaching [1] Calculus (lectures and seminars). Independent University of Moscow, I year students, Feburary-May 2013, 2+2 hours per week. Program 1. Functions of several variables. 2. Implicit function theorem and its corollaries. Morse lemma. 3. Jordan measure and Lebesgue measure. 4. Measurable functions. 5. Lebesgue integral. 6. Fubini's theorem and the theorem of Radon-Nikodym. 7. Space L2 . 8. Orthogonal system of functions. Fourier series. [2] Analitic theory of differential equations (lectures) Special course in HSE and IUM. Program


REPORT ON THE SIMONS­IUM FELLOWSHIP 2013 AND DYNASTY FOUNDATION

3

Linear differential equations 1. Linear differential equations: monodromy, singular points. Levelt decomposition. 2. Elements of global theory. 3. Hypergeometric equation and hypergeometric functions. Painlev´ prop erty for first order nonlinear equations. e 4. Fuchs conditions. 5. Riccati equation. 6. Equations of the genius zero are equivalent to Riccati equation. 7. Some of second order nonlinear differential equations. Painlev´ equations. e Theory of the lo cal normal forms. 8. Formal normal forms. Poincar´ e-Dulac theorem. 9. Poincar´ and Siegel domains. Theorem of Poincar´ e e. [3] Calculus (seminars-exercises) Higher School Economics, II year students, January-June 2013, 2 hours per week. Program 1. Jordan measure and Lebesgue measure. 4. Measurable functions. 5. Lebesgue integral. 6. Absolutely continuous functions. 7. Space L2 . 8. Orthogonal system of functions. Fourier series. 9. Spatial integrals. [4] June 1. 2. 3. 4. 5. 6. 7. PDE (seminars-exercises). Higher School Economics, II year students, January2013, 2 hours per week. Canonical form of a linear second order PDE. The wave equation. The Dalamber formula. The heat equation. The Poisson equation. Fourier method. The wave equation in the second and third dimensional spaces. Harmonic functions.