# Rodolfo H. Torres

College of Liberal Arts and Sciences - Mathematics**Primary office**:

Room 201

**Second office**:

Room 546

Rodolfo Torres received his PhD in mathematics in 1989 from Washington University in St. Louis. He held postdoctoral positions at the Courant Institute of Mathematical Sciences of New York University and the University of Michigan, Ann Arbor. In 1996 he moved to the University of Kansas (KU), where he advanced through the ranks becoming a Full Professor in 2003 and a University Distinguished Professor in 2016. His research interests include Fourier analysis and applications in partial differential equations, signal analysis, and biology. He specializes in the study of singular integrals, function spaces, and decomposition techniques. Since 1993 his research has been supported by several grants form the National Science Foundation. Through his research collaborations in mathematics he has given numerous lectures around the world and his interdisciplinary work in biology has received considerably media attention, including articles in The New York Times, Science Magazine, and Discovery Channel on-line. He regularly mentors graduate and undergraduate students and has received various teaching awards for his efforts with students, including a Teaching Excellent Award from the T. Kemper Foundation. He was elected to the inaugural class of Fellows of the American Mathematical Society. He is a former Faculty Senate President at KU and in 2012 was appointed Associate Vice Chancellor for Research. He currently serves as Interim Vice Chancellor and President of the University of Kansas Center for Research Inc.

### Education

Ph.D., Mathematics, Washington University

M.A., Mathematics, Washington University

Licenciatura, Mathematics, Universidad Nacional de Rosario

### Research

Prof. Torres works in Fourier analysis. He is interested in singular integrals, Calderón-Zygmund theory, multilinear operators, function spaces, and discrete decompositions such as wavelets. Fourier analysis is a mathematical tool that permits the decomposition of a signal into a combination of oscillating waves of different frequencies and amplitudes, very much in the same way that a prism separates a beam of light into a rainbow of colors of different wavelengths. This analysis decodes information present in the signal and provides a precise mechanism to quantify oscillations, sudden changes, patterns, and symmetries in the data. Applications of Torres' theoretical research include partial differential equations and signal analysis. He has also done contributions in the spectral analysis of coloration in nanostructured biological tissues.

### Research Interests

- Fourier analysis
- Applications in partial differential equations
- Signal analysis
- And biology

### Selected Publications

Chaffee, L. Chen, P. Han, Y. Torres, R. H., & Ward, L. (in press). Characterization of compactness of commutators of bilinear singular integral operators. *Proc. Amer. Math. Soc.*

Torres, R. H., & Ward, E. L. (in press). Several results in classical and modern harmonic analysis in mixed Lebesgue spaces. *RIMS Kokyuroku Bessatsu *.

Hart, J. Torres, R. H., & Wu, X. (in press). Smoothing Properties of Bilinear Operators and Leibniz-Type Rules in Lebesgue and Mixed Lebesgue Spaces. *Trans. Amer. Math. Soc.*

Chaffee, L. Torres, R. H., & Wu, X. (2017). Multilinear weighted norm inequalities under integral type regularity conditions . In S. Chanillo, B. Franchi Pereyra, G. Lu, C. Perez, & E. T. Sawyer (Eds.), *Harmonic Analysis, Partial Differential Equations and Applications in Honor of Richard L. Wheeden*. Springer.

Benyi, A. & Torres, R. H. (2017). The discrete Calderón reproducing formula of Frazier and Jawerth. In M. Cwikel & M. Millman (Eds.), *Functional Analysis, Harmonic Analysis, and Image Processing: A Collection of Papers in Honor of Björn Jawerth *. AMS.

Torres, R. H. (2016). Cora Sadosky: Her mathematics, mentorship, and professional contributions. In M. C. Pereyra, M. Marcantognini, A. M. Stokolos, & W. Urbina (Eds.), *Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory. Celebrating Cora Sadosky’s Life. Volume 1,*. Springer.

Chaffee, L. & Torres, R. H. (2015). Characterization of compactness of the commutators of bilinear fractional integral operators. *Potential Analysis*, *43*(3), 481–494.

Bényi, A. Damian, W. Moen, K. & Torres, R. H. (2015). Compact bilinear commutators: The weighted case. *Michigan Math. J.*, *64*(1), 39-51.

Bényi, A. Damian, W. Moen, K. & Torres, R. H. (2015). Compactness properties of commutators of bilinear fractional integrals. *Math. Z.*, *280*(1-2), 569-582.

Torres, R. H., & Ward, E. (2015). Leibniz’s rule, sampling and wavelets on mixed Lebesgue spaces. *J. of Fourier Anal. and Appl.*, *21*(5), 1053-1076.

Pérez, C. Pradolini, G. Torres, R. H., & Trujillo-González, R. (2014). End-point estimates for iterated commutators of multilinear singular integrals. *Bull. London Math. Soc.*, *46*(1), 26-42.

Pérez, C. & Torres, R. H. (2014). Minimal regulatory conditions for the end-point estimate of bilinear Calderón-Zygmund operators. *Proc. Amer. Math. Soc. Ser. B *, *1*, 1-13.

Prum, R. O., & Torres, R. H. (2013). Fourier blues: structural coloration of biological tissues. In T. D, Andrews, R. Balan, J. Benedetto, W. Czaja, & K. A Okoudjou (Eds.), *Excursions in harmonic analysis. Volumen 2* (pp. 401–421). New York: Birkhauser/Springer.

Bényi, A. & Torres, R. H. (2013). Compact bilinear operators and commutators. *Proc. Amer. Math. Soc.*, *141*(10), 3609-3621.

Torres, R. H. (2013). *Merrill Advanced Studies Center Report No. 117 *. *Research Analytics: Facilitating the Use of Metrics to Improve the Research Profile of Academic Programs, *., 96-105.

Carro, M. J., Soria, J. & Torres, R. H. (2012). Rubio de Francia's extrapolation Theory: estimates for the distribution function. *J. London Math. Soc.*, *85*, 430-454.

Bernicot, F. & Torres, R. H. (2011). Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus. *Anal. PDE*, *4*, 551-571.

Grafakos, L. Liu, L. Pérez, C. & Torres, R. H. (2011). The multilinear strong maximal function. *J.Geom. Anal.*, *21*(1), 118-149.

Bényi, &. Maldonado, D. Nahmod, A. & Torres, R. H. (2010). Bilinear paraproducts revisited. *Math. Nach.*, *283*, 1257-1276.

Bényi, &. Maldonado, D. Naibo, V. & Torres, R. H. (2010). On the Hörmander classes of bilinear pseudodifferential operators. *J. Integral Eq. Oper. Theory*, *67*, 341-364.

Lacey, M. Moen, K. Pérez, C. & Torres, R. H. (2010). Sharp weighted bounds for fractional integral operators. *J. Func. Anal.*, *259*(5), 1073-1097.

Bényi, &. Demeter, C. Nahmod, A. Thiele, C. Torres, R. H., & Villarroya, P. (2009). Modulation invariant bilinear T(1) Theorem. *J. Anal. Math.*, *109*, 279-352.

Torres, R. H. (2009). Multilinear singular integrals with variable coefficients. *Rev. Un. Mat. Argentina*, *50*, 153-170.

Lerner, A. Ombrosi, S. Pérez, C. Torres, R. H., & Trujillo-González, R. (2009). New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. *Adv. in Math.*, *220*, 1222-1264.

Gavosto, E., Korten, M., Moore, C., & Torres, R. H. (Eds.). (2007). *AMS Contemporary Mathematics Series, #428*. *Harmonic Analysis, Partial Differential Equations, and Related Topics* (E. Gavosto, M. Korten, C. Moore, & R. H. Torres, Eds.).

Prum, R. Quinn, T. & Torres, R. H. (2006). Anatomically Diverse Butterfly Scales Produce Structural Colors by Coherent Scattering. *Journal of Experimental Biology*, *209*, 748-765.

Bényi, &. Nahmod, A. & Torres, R. H. (2006). Sobolev space estimate and symbolic calculus for bilinear pseudodifferential operators. *J. Geom. Anal.*, *16.3*, 431-453.

Prum, R. & Torres, R. H. (2005). Análisis espectral de nanoestructuras en tejidos biológicos. *Matematicalia*, *1*(2). http://www.matematicalia.net/

Bényi, &. & Torres, R. H. (2004). Almost orthogonality and a class of bounded bilinear pseudodifferential operators. *Math. Res. Lett.*, *11*, 1-11.

Cole, J. Prum, R. & Torres, R. H. (2004). Blue integumentary structural colours in dragonflies (Odonata) are not produced by incoherent Tyndall scattering. *Journal of Experimental Biology*, *207*, 3999-4009.

Stefanov, A. & Torres, R. H. (2004). Calderón-Zygmund operators on mixed Lebesgue spaces and applications to null forms. *J. London Math. Soc.*, *70*, 447-462.

Prum, R. & Torres, R. H. (2004). Structural colouration of mammalian skin: convergent evolution of coherently scattering dermal collagen arrays. *Journal of Experimental Biology*, *207*, 2157-2172.

Prum, R. & Torres, R. H. (2003). A Fourier tool for the analysis of coherent light scattering by bio-optical nanostructures. *Integr. Comp. Biol.*, *43*, 591-602.

Andersson, S. Prum, R. & Torres, R. H. (2003). Coherent light scattering of ultraviolet light by avian feather barbs. *Auk*, *120*, 163-170.

Boza, S. & Torres, R. H. (2003). Decomposition of Ḃ10,1(ℤ) into special atoms. *Math. Nachr.*, *254/255*, 3-10.

Pérez, C. & Torres, R. H. (2003). Sharp maximal function estimates for multilinear singular integrals. *Contemp. Math.*, *320*, 323-331.

Prum, R. & Torres, R. H. (2003). Structural colouration of avian skin: convergent evolution of coherently scattering dermal collagen arrays. *Journal of Experimental Biology*, *206*, 2409-2429.

Bényi, &. & Torres, R. H. (2003). Symbolic calculus for the transposes of bilinear pseudodifferential operators. *Comm. Partial Diff. Eq.*, *28*, 1161-1181.

Brown, R. & Torres, R. H. (2003). Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives in Lᵖ, p > 2n. *J. Fourier Anal. and Appl.*, *9*, 563–574.

Prum, R. & Torres, R. H. (2002). Azul de ave. Un color estructural. *Temas Investigación y Ciencia: El color*, *27*, 30-31.

Grafakos, L. & Torres, R. H. (2002). Discrete decomposition for bilinear operators and almost diagonal conditions. *Trans. Amer. Math. Soc.*, *354*, 1153-1176.

Grafakos, L. & Torres, R. H. (2002). Maximal operator and weighted norm inequalities for multilinear singular integrals. *Indiana Univ. Math. J.*, *51*, 1261-1276.

Grafakos, L. & Torres, R. H. (2002). Multilinear Calderón-Zygmund theory. *Adv. in Math.*, *165*, 124-164.

Grafakos, L. & Torres, R. H. (2002). On multilinear singular integrals of Calderón-Zygmund type. *Publ. Mat.*, *extra*, 57-91.

Torres, R. H. (2001). The analysis of oscillatory behavior in signals through their samples. In J. Benedetto & P. Ferreira (Eds.), *Modern Sampling Theory: Mathematics and Applications*. Birkhauser.

Grafakos, L. & Torres, R. H. (2001). A multilinear Schur test and multiplier operators. *J. Func. Anal.*, *187*, 1-24.

Prum, R. & Torres, R. H. (2001). Azul de ave. Un color estructural. *Investigación y Ciencia (Spanish edition of Scientific American)*, *299*, 36-37.

Goodman, S. Kovach, C. Prum, R. Torres, R. H., & Williamson, S. (1999). Coherent light scattering by nanostructures collagen arrays in the cranucles of the Malagasy Asities (Eurylaimidae:
Aves). *Journal of Experimental Biology*, *202*, 3507-3522.

Grafakos, L. & Torres, R. H. (1999). Pseudodifferential operators with homogeneous symbols. *Michigan Math. J.*, *46*, 261-269.

Dyck, J. Prum, R. Torres, R. H., & Williamson, S. (1999). Two-dimensional Fourier analysis of the spongy medullary keratin of structurally coloured feather barbs. *Proc. Royal Society: Biological Sciences (B)*, *266*, 13-22.

Dyck, J. Prum, R. Torres, R. H., & Williamson, S. (1998). Coherent light scattering by blue feather barbs. *Nature*, *396*, 28-29.

Torres, R. H. (1998). Maxwell's equations and dielectric obstacles with Lipschitz boundaries. *J. London Math. Soc.*, *57*, 157-169.

Torres, R. H. (1998). Mean oscillation of functions and the Paley–Wiener space. *J. Fourier Anal. and Appl.*, *4*, 283-297.

Torres, R. H. (1997). Sampling and nonlinear approximation of band limited signals in mean oscillation spaces. In *Proc. SAMPTA 97, Aviero, Portugal*.

Torres, R. H. (1996). A transmission problem in the scattering of electromagnetic waves by a penetrable object. *SIAM J. Math. Anal.*, *27*, 1406-1423.

Torres, R. H., & Welland, G. (1996). Layer potential operators and a space of boundary data for electromagnetism in nonsmooth domains. *Michigan Math. J.*, *43*, 189-206.

Torres, R. H., & Welland, G. (1995). Open Problem. In . (Ed.), *Clifford Algebras in Analysis and Related Topics*. Boca Raton, FL: CRC Press.

Mitrea, M. Torres, R. H., & Welland, G. (1995). Regularity and approximations results for the Maxwell problem on C1 and Lipschitz domain. In . (Ed.), *Clifford Algebras in Analysis and Related Topics*. Boca Raton, FL: CRC Press.

Adolfsson, V. Jawerth, B. & Torres, R. H. (1994). A boundary integral method for parabolic equations in nonsmooth domains. *Comm. Pure Appl. Math.*, *47*, 861-892.

Frazier, M. & Torres, R. H. (1993). The sampling theorem, φ–transform, and Shannon Wavelets for R, Z, T and ZN. In . (Ed.), *Wavelets: Mathematics and Applications*. Boca Raton, FL: CRC Press.

Torres, R. H., & Welland, G. (1993). The Helmholtz equation and transmission problems with Lipschitz interfaces. *Indiana Univ. Math. J.*, *42*, 1457-1485.

Torres, R. H. (1991). *Mem. Amer. Math. Soc., 442*. *Boundedness results for operators with singular kernels on distribution spaces*, 178.

Adolfsson, V. Jawerth, B. & Torres, R. H. (1991). Singular integral equations, spaces of homogeneous type, and boundary elements in non–smooth domains. *Rev. U. Math. Arg.*, *37*, 163-183.

Torres, R. H. (1991). Spaces of sequences, sampling theorem and functions of exponential type. *Studia Math.*, *100*, 51-74.

Torres, R. H. (1990). Continuity properties of pseudodifferential operators of type 1,1. *Comm. Partial Diff. Eq.*, *15*, 1313-1328.

Frazier, M. Torres, R. H., & Weiss, G. (1988). The Boundedness of Calderón–Zygmund operators on the spaces Ḟpα, q. *Rev. Mat. Iberoamericana*, *4*, 41-72.

### Selected Grants

Torres, Rodolfo, (Principal), Huan, Jun, (Co-Investigator), Collaborative Research: EAGER: Automating HERD Reporting Using Machine Learning and Administrative Data, National Science Foundation, $176,071, Submitted 06/02/2015 (09/01/2015 - 08/31/2018) . Federal. Status: Funded.

Torres, Rodolfo H., (Principal), Gavosto, E., (Co-Principal), Korten, M., Moore, C., Collaborative Research: Prairie Analysis Seminar 2015–2017, DMS-1519016, National Science Foundation, Analysis Program, $16,660, (09/01/2015 - 08/31/2018) . Federal. Status: Funded.

Torres, Rodolfo H., (Principal), Fourier Analysis and Multilinear Operators, DMS-1069015, National Science Foundation, Analysis Program, $229,999, (08/15/2011 - 07/31/2015) . Federal. Status: Funded.

Pérez, C., (Principal), Torres, Rodolfo H., Análisis Real y Armόnico (Real and Harmonic Analysis), Consejería de Innovaciόn, Ciencia y Empresa, Junta de Andalucia, $389,859, (01/01/2010 - 12/31/2014) . Foreign (company, govt, agencies, etc.). Status: Funded.

Torres, Rodolfo H., (Principal), Gavosto, E., Korten, M., Moore, C., Collaborative Research: Prairie Analysis Seminar 2009–2011, DMS-0942775, National Science Foundation, Analysis Program, $16,321, (10/01/2009 - 09/30/2013) . Federal. Status: Funded.

Pérez, C, (Principal), Torres, Rodolfo H, Análisis Armόnico y Espacios de Banach (Harmonic Analysis and Banach Spaces), Ministerio de Ciencia e Innovaciόn, Gobierno de España (Spain), $101,665, (01/01/2010 - 12/31/2012) . Foreign (company, govt, agencies, etc.). Status: Funded.

Torres, Rodolfo H, (Principal), Fourier Analysis Techniques, Operators, and Function Spaces, National Science Foundation, Analysis Program, $192,341, (01/01/2008 - 12/31/2012) . Federal. Status: Funded.

Torres, Rodolfo H., New Estimates for Multilinear Singular Integrals, University of Kansas, (01/01/2008 - 12/31/2009) . University (KU or KUMC). Status: Funded.

Torres, Rodolfo H, (Principal), Gavosto, E, Korten, M, Moore, C, Prairie Analysis Seminar 2008, National Science Foundation, Analysis Program, $15,400, - 12/31/2008) . Federal. Status: Funded.

Torres, Rodolfo H, (Principal), Fourier Analysis Techniques and Applications, National Science Foundation, Analysis Program and Computational Neuroscience Program, $140,000, (01/01/2004 - 12/31/2008) . Federal. Status: Funded.

Torres, Rodolfo H, (Co-Principal), Gavosto, E, Korten, M, Moore, C, Prairie Analysis Seminar 2007, National Science Foundation, Analysis Program, $15,034, - 12/31/2007) . Federal. Status: Funded.

Torres, Rodolfo H., (Co-Principal), Nochetto, R., (Principal), Benedetto, J., Harmonic Analysis and Numerical Analysis Problems in MRI and PDE, National Science Foundation, US-Argentina Program, $24,000, (01/01/2002 - 12/31/2006) . Federal. Status: Funded.

Torres, Rodolfo H, (Co-Principal), Gavosto, E, Korten, M, Moore, C, Prairie Analysis Seminar 2005, National Science Foundation, Analysis Program, $9,580, - 12/31/2005) . Federal. Status: Funded.

Torres, Rodolfo H, (Principal), Gavosto, E, Korten, M, Moore, C, Prairie Analysis Seminar 2004, National Science Foundation, Analysis Program, $7,428, - 12/31/2004) . Federal. Status: Funded.

Torres, Rodolfo H, (Co-Principal), Huneke, C, (Principal), Byers, R, Gavosto, E, Huang, W, Scientific Computing Research Environments for the Mathematical Sciences (SCREMS), National Science Foundation, Infrastructure Program, $49,927, (01/01/2001 - 12/31/2004) . Federal. Status: Funded.

Torres, Rodolfo H, (Principal), Fourier Analysis Tool for Biological Nano-optics, National Science Foundation, Biological Databases and Information Program, $512,107, (01/01/2000 - 12/31/2004) . Federal. Status: Funded.

Torres, Rodolfo H, (Principal), Multilinear Operators, Discrete Decompositions, and Spectral Resolution of Nanostructures, National Science Foundation, Analysis Program, $100,998, (01/01/2000 - 12/31/2004) . Federal. Status: Funded.

Torres, Rodolfo H, (Co-Principal), Gavosto, E, Korten, M, Moore, C, Prairie Analysis Seminar 2003, National Science Foundation, Analysis Program, $6,000, - 12/31/2003) . Federal. Status: Funded.

Torres, Rodolfo H, (Principal), MRI: Acquisition of High Performance Multiprocessors for Computational Research in Sciences and Engineering, National Science Foundation, Experimental Program to Stimulate Computational Research, (01/01/1999 - 12/31/2002) . Federal. Status: Funded.

Torres, Rodolfo H, Harmonic Analysis of Multilinear Operators, University of Kansas, - 12/31/2000) . University (KU or KUMC). Status: Funded.

Torres, Rodolfo H, (Principal), Real Variable Techniques in the Approximation of Functions and Boundary Value Problems in Nonsmooth Domains, National Science Foundation, Analysis Program, $78,841, (01/01/1996 - 12/31/2000) . Federal. Status: Funded.

Torres, Rodolfo H, (Principal), Singular Integral Operators, Decompositions of Function Spaces Related to Wavelets, and Boundary Elements in Nonsmooth Domains, National Science Foundation, Analysis Program, $25,000, (01/01/1993 - 12/31/1995) . Federal. Status: Funded.

Torres, Rodolfo H, Travel Award to attend the ICM in Zürich, Switzerland, NSF-AMS, - 12/31/1994) . Federal. Status: Funded.

Torres, Rodolfo H, Travel Award to attend the ICM in Kyoto, Japan, NSF-AMS, - 12/31/1990) . Federal. Status: Funded.