Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
Lifting theory of modules was studied intensively by Auslander, Ding,
and Solberg. This notion is tightly connected to the deformation
theory of modules and has important applications in the theory of
maximal Cohen-Macaulay approximations. Later, Yoshino generalized the
lifting results for modules to the case of complexes. Further
generalization was given by Nasseh and Sather-Wagstaff for DG modules
(a notion from rational homotopy theory) in order to obtain a clearer
insight on a conjecture of Vasconcelos in commutative algebra.
In this talk, I will survey recent developments on lifting theory of
DG modules and describe the relationship between this notion and
another conjecture in commutative algebra, namely, the
Auslander-Reiten conjecture. The talk is based on several joint works
with Maiko Ono and Yuji Yoshino.
Join Zoom Meeting https://texastech.zoom.us/j/91729629174?pwd=TFJHbDk1ZS9KeTBRaldNL1hUbVNlQT09
Meeting ID: 971 2962 9174
Passcode: 914040
Ghost ideals were firstly introduced by Dan Christensen in the stable
homotopy category. We will present an abelian version of the ghost
ideal associated to a class of objects. Then we study powers of the
ghost ideal and show that, under mild assumptions, any (finite or
infinite) power of a ghost ideal appears as the half of a complete
ideal cotorsion pair, and in fact, it agrees with a nice complete
cotorsion pair of objects for some infinite inductive power.
Applications will be given in the category of modules and unbounded
chain complexes of modules.
The talk is based on a work in progress with XianHui Fu, Ivo Herzog
and Sinem Odabasi
Join Zoom Meeting https://texastech.zoom.us/j/91729629174?pwd=TFJHbDk1ZS9KeTBRaldNL1hUbVNlQT09
Meeting ID: 971 2962 9174
Passcode: 914040
Motivated by an open question of Dyson and Serre we define Jacobi
forms with CM analogously to modular forms with CM. We will mainly
focus on the history of this problem and some beautiful sum-to-product
identities that result from this new construction. As applications we
will discuss Jacobi forms associated to elliptic curves with CM and a
process to construct partition statistics which explain Ramanujan-type
congruences for integer partitions.
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The Hecke eigenvalues of classical modular forms are realized in the
Fourier expansions of eigenforms. For Siegel modular forms, the
relationship between Hecke eigenvalues and Fourier coefficients is
more complicated. In this talk, I will present joint work with Brooks
Roberts and Ralf Schmidt in which we develop a theory of stable
Klingen vectors inside of paramodular representations with sufficient
ramification that has many beautiful consequences, including a
generalization of Andrianov's rationality result for a certain
Dirichlet series of Fourier coefficients.
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Meeting ID: 952 4925 1386
Passcode: 853016
The origins of the Improved New Intersection Theorem can be traced
back to the following linear algebra exercise: let U,V and W be vector
spaces over a field such that U and V are contained in W, then the
dimension of of the intersection of U and V is at least dim U+dim
V-dim W. In their most modern forms, the intersection theorems are
concerned with bounding the length of finite free complexes over local
rings. In this talk, we will explore the history of these theorems,
culminating in a result due to L. Christensen and me.
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Meeting ID: 990 5902 7169
Passcode: 648002
The Ramanujan congruences provide beautiful and
explicit results on the arithmetic of the unrestricted partition
function, while Euler's pentagonal number theorem furnishes
a complete description of the parity of the number of partitions
into distinct parts. Motivated by the latter result, in this talk
we will consider the arithmetic of k-regular partitions, which
are partitions in which no part is divisible by k. We give
particular attention to cases where the theory of modular
forms can be used to give a complete (or nearly so)
characterization of this arithmetic.
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Meeting ID: 990 5902 7169
Passcode: 648002