Welcome to Uli Walther's Research Page
I am interested in everything that is near algebraic geometry, algebraic topology or commutative algebra. Much of my time I spend thinking about topics related to sheaf theory, in particular local cohomology, de Rham cohomology, and solution sheaves to A-hypergeometric systems. I used to work on algorithms that compute (co)homological functors and most of the time concentrate on characteristic zero. That is reflected in the fact that I often use Groebner bases in D-modules. I am also interested in toric varieties and singularities in general. Invariants that fascinate me are Bernstein--Sato polynomials, zeta functions, Milnor fibers, and logarithmic vector fields.

Until July 1999 I was a graduate student at the University of Minnesota, where it is very pretty. There I obtained a PhD. My advisor was Gennady Lyubeznik. Here is my Thesis .

Before I came to Minnesota I was a student at the Martin Luther University in Halle. Halle is near Leipzig in the former Sowjetische Besatzungszone, which in the democratic world was known as East Germany. People you might have heard about in mathematics that worked in Halle include Pfaff, Heine, Roch, Jung, Scherk, Cantor, Keller, Koebe, Pluecker, Hasse, Herrmann, Vogel and Schenzel. I got a "Diplom in Mathematik" there and as part of this I wrote a thesis called Singularities on Generalized Flag Manifolds under the supervision of Konrad Drechsler. It deals with flag manifolds where the flags are not necessarily in strict order under inclusion.

The fourth year of my Diplom degree I actually spent at Sheffield University in the UK where I earned a Master's degree in Pure Mathematics. For this I wrote a Master's thesis called Local Cohomology: A Survey . My advisor in Sheffield was Rodney Sharp.

In the fall of 1998 I was a guest at the Mathematical Sciences Research Institute, MSRI. In 1999-2000 I was at MSRI as a postdoctoral fellow where my mentor was David Eisenbud.

2002/03 I was a guest at the Universitaet Leipzig and the Max-Planck-Institut in the same city where I was hosted by Juergen Stueckrad and Juergen Jost.

I am now faculty at Purdue University.

Publications/Preprints:

Algorithmic computation of local cohomology modules and the local cohomological dimension of algebraic varieties (JPAA (139), 1999.)

Algorithmic computation of de Rham cohomology of complements of complex affine varieties (JSC (29), 2000)

A localization algorithm for D-modules, (JSC (29), 2000). With T. Oaku, N. Takayama.

Switching Surfaces and Groebner bases With A. Tannenbaum and T. Georgiou.(IEEE Trans. Autom. Contr. (46), 2001).

On the Lyubeznik numbers of a local ring (PAMS (129:6), 2001.)

Computing the cup product structure for complements of complex affine varieties. (JPAA (164), 2001).

D-modules on smooth toric varieties With M. Mustata, G. Smith, H. Tsai. (JA (240), 2001)

Computing the rational cohomology of complex varieties (Contemp. Math. (286), 2001)

Computing homomorphisms between holonomic $D$-modules With H. Tsai. (JSC (32), 2001)

D-modules and cohomology of varieties (in "Computations in algebraic geometry with Macaulay 2", Eisenbud, Grayson, Sturmfels, Stillman edts., Springer, Algorithms + Computations in Mathematics 8, 2001)

Cohomology, stratifications and parametric Groebner bases (JSC (35), 2003)

Bernstein-Sato polynomials and Milnor fibers for generic hyperplane arrangements (Compositio Math. (141), 2005)

Algorithmic Stratification of RHom(M,N) for regular holonomic modules on C^n (JSC (39), 2005)

Arbitrary rank jumps of $A$-hypergeometric systems through Laurent monomials With L. Matusevich (J. LMS (75), 2007)

Homological methods for hypergeometric families With L. Matusevich and E. Miller (JAMS (18), 2005)

On the arithmetic rank of certain Segre products With A.~Singh (Contemp. Math. (390), 2005)

Experiments with the restriction functor (preprint)

Duality and reducibility of hypergeometric systems (Math. Ann. (338), 2007)

A connectedness result in positive characteristic With A. Singh (TAMS (360), 2008)

Irregularity of hypergeometric systems via slopes along coordinate subspaces With M. Schulze (Duke MJ (142.3), 2008).

Local cohomology and pure morphisms With A. Singh (Ill. J. Math., the Griffith issue (51), 2007)

24 hours of local cohomology: the Snowbird lecture notes With S. Iyengar, G. Leuschke, A. Leykin, C. Miller, E. Miller, A. Singh (AMS Graduate Studies in Math (87), 2007)

Cohen--Macaulayness and computation of Newton graded toric rings With M. Schulze (JPAA (213), 2009)

Hypergeometric D-modules and twisted Gauss--Manin systems With M. Schulze (JA (223), 2009)

Bockstein homomorphisms in local cohomology With A.K. Singh (Crelle (655) 2011 )

Restriction of $A$-hypergeometric systems to coordinate subspaces With M. Fernandez-Fernandez (PAMS (139) 2011)

Resonance equals reducibility for $A$-hypergeometric systems With M. Schulze (ANT (6) 2012)

Local cohomology of logarithmic forms With G. Denham, H. Schenck, M. Schulze and M. Wakefield (Ann. Inst. Fourier (63) 2013)

Local cohomology modules supported at determinantal ideals With G. Lyubeznik, A.K. Singh (JEMS (18) 2016)

Survey on the D-module generated by f^s with an appendix by A. Leykin (Math. Sci. Res. Inst. Publ., 67 on the special year in commutative algebra and noncommutative algebraic geometry 2012/13)

Singularities and holonomicity of binomial D-modules With C.B. Zamaere and L.F. Matusevich (JA (439) 2015)

The Jacobian, the Milnor fiber, and the $D$-module generated by $f^s$ (Inventiones Math. (207) 2017)

Bernstein--Sato polynomials for maximal minors and sub-maximal Pfaffians With A. Lorincz, C. Raicu and J. Weyman (Advances Math. (307) 2017)

On B-functions of hypergeometric systems With T. Reichelt and C. Sevenheck (IMRN (21) 2018)

On categories of equivariant D-modules With A. Lorincz (Advances Math. (351) 2019)

Gauss--Manin systems of families of Laurent polynomials and $A$-hypergeometric systems With T. Reichelt (Comm. in Alg. (47) 2019)

Torus equivariant D-modules and hypergeometric systems With C.~Berkesch and L.~Matusevich (Advances Math. (350) 2019)

On normalized Horn systems With C.~Berkesch and L.~Matusevich (Coll. Math. (71) 2020)

Matroid connectivity and singularities of configuration hypersurfaces With G.~Denham and M.~Schulze (Lett.\ in Math.\ Phys. (to appear))

On a conjecture of Lynch With A.K. Singh (Comm. Alg (48) 2020)

Koszul and local cohomology, and a question of Dutta With L. Ma and A.K. Singh (Math. Zeitschrift () 2021)

Weight filtrations on GKZ-systems With T. Reichelt (submitted)

Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings With T. Reichelt and M. Saito (Selecta Math. () 2021)

Graph hypersurfaces with torus action and a conjecture of Aluffi With G.~Denham and D.~Pol and M.~Schulze(Comm. Numb. Th. + Phys () 2021)

Algebraic aspects of hypergeometric dufferential equations With T. Reichelt M. Schulze and C. Sevenheck (Beitr. Alg.+Geom.() 2021)

Current work is on local cohomology; hypergeometric systems; b-functions and homological properties of $D$-modules; zeta-functions.

Some notes and talks

REU Problems

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walther@math.purdue.edu