Your course grade is based on a 20-30 minute presentation on a topic relevant to the course.
Thursday 15 December (341 Altgeld Hall)
- 2:00pm – Faruk Temur “Birman-Schwinger inequalities“
- 2:30pm – Kunwoo Kim “Stochastic PDEs“
P.-L. Chow. Stochastic Partial Differential Equations. Chapter 3. Chapman and Hall/CRC, 2007.
J. B. Walsh. An Introduction to Stochastic Partial Differential Equations. Lecture Notes in Math, 1180, 1986. Chapter 3.
Da Prato and Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge University Press, 1992. - 3:00pm – Sarah Son “Bounds for eigenvalues and eigenvectors of symmetric operators“
R. S. Laugesen and B. A. Siudeja. Dirichlet eigenvalue sums on triangles are minimal for equilaterals. Commun. Anal. Geom., to appear. arXiv:1008.1316.
L. Fox, P. Henrici and C. Moler. Approximations and bounds for eigenvalues of elliptic operators. SIAM J. Numer. Anal. 4 (1967), 89-102. - 3:30pm – Tom Carty “Crystal lattice problem and Floquet theory“
Lecture notes by John K. Hunter at UC-Davis, in particular Section 4.5 – Periodic Potentials (pp. 113-117)
M. S. P. Eastham. Spectral Theory of Periodic Differential Equations. Chatto & Windus, London, 1973.
Friday 16 December (341 Altgeld Hall)
- 2:00pm – Qiang Zeng “Hot spots problem“
R. Atar and K. Burdzy. On Neumann eigenfunctions in lip domains. J. Amer. Math. Soc. 17 (2004), 243¿265.
R. Banuelos and K. Burdzy. On the ¿hot spots¿ conjecture of J. Rauch. J. Funct. Anal. 164 (1999), 1¿33.
K. Burdzy and W. Werner. A counterexample to the ¿hot spots¿ conjecture. Ann. of Math. (2) 149 (1999), 309¿317.
K. Burdzy. The hot spots problem in planar domains with one hole. Duke Math. J. 129 (2005), 481¿502.
Background (synchronous coupling, reflected BM and Skorokhod problem):
P. L. Lions and A. S. Sznitman. Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37 (1984), 511-537.
D. Revuz and M. Yor. Continuous Martingales and Brownian motion. Corrected 3rd edition, Springer (2005), chapter VI.
K. Burdzy and Z.-Q. Chen. Coalescence of synchronous couplings. Probab. Theory Related Fields 123 (2002), 553-578. - 2:30pm – Lisa Hickok “Turing instability“
L. C. Evans. Partial Differential Equations, Second Edition. Pages 172-175. American Mathematical Society.
P. Markowich. Applied Partial Differential Equations: A visual Approach. Chapter 4. Springer, 2007. - 3:00pm – Brian Benson “Eigenvalues of the p-Laplacian and Cheeger’s constant“
B. Kawohl and V. Fridman. Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carolin., 44 (2003), 659-667.
A.-M. Matei. First eigenvalue for the p-Laplace operator. Nonlinear Anal. Ser. A Theory Methods 39 (2000), 1051-1068. - 3:30pm – Han Wang “Polya’s Conjecture for the Dirichlet eigenvalues“
G. Polya. On the eigenvalue of vibrating membranes. Proc. London Math. Soc., 11 (1961), 419-433.
R. Kellner. On a theorem of Polya. American Mathematical Monthly, 73, No. 8 (Oct., 1966), 856-858.
R. Courant and D. Hilbert. Methods of Mathematical Physics, Vol I. Interscience New York, 1953.