Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
In a paper from 1996, Knuth took a combinatorial approach to
Pfaffians, i.e. determinants of skew-symmetric matrices. It was
immediately noticed that this approach facilitates generalizations and
simplified proofs of several known identities involving Pfaffians. I
will discuss one particular case: a formula that expresses arbitrary
minors of a skew symmetric matrix in terms of Pfaffians; it first
appeared in a 1904 paper by Brill.
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We study a theory of support varieties over a skew complete
intersection $R$. Color differential graded homological algebra is put
to use to compute the derived braided Hochschild cohomology of $R$;
moreover, its action on $\mathrm{Ext}_R(M,N)$ is shown to be
noetherian for any pair of finitely generated color $R$-modules $M$
and $N$. When the parameters defining $R$ are roots of unity we use
these cohomology operators to define the support variety for a pair of
color $R$-modules. Applications include a proof of the Generalized
Auslander-Reiten Conjecture and that $R$ possesses symmetric
complexity when the defining parameters of $R$ are roots of unity.
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In this talk I will discuss an investigation of a certain class of
complete intersection homomorphisms. I will present some of their
well-behaved behavior, strongly linking the homological algebra over
the source and target rings, while also presenting examples that
reveal some surprising properties. This is ongoing joint work with
Srikanth Iyengar, Janina Letz, and Jian Liu.
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In their 1987 paper, Eliahou and Kervaire constructed a minimal
resolution of a class of monomial ideals of a polynomial ring, and
they showed that this resolution has the structure of a differential
graded algebra. I will discuss how the minimal resolution they
constructed can be extended to a skew polynomial ring, giving the
minimal resolution the structure of a color commutative differential
graded algebra.
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It is known that the Betti numbers for any finitely generated module
over a local complete intersection ring grow on the order of a
polynomial. Further, it can be shown that, for large enough degree,
there are two polynomials of interest: one explicitly giving the even
Betti numbers and one giving the odd Betti numbers. The aim of this
talk is to show a bound on the discrepancy of these two polynomials
for every finitely generated module over a complete intersection with
respect to an invariant of the ring called its "quadratic
codimension". This is joint work with Lucho Avramov and Mark Walker.
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In the first part of the talk, we will study about the distribution of
gaps between eigenvalues of Hecke operators in both horizontal and
vertical settings. As an application of this we will obtain a strong
multiplicity one theorem and evidence towards Maeda conjecture. The
horizontal setting is a joint work with M. Ram Murty. In the second
part of the talk, using recent developments in the theory of l-adic
Galois representations we will study the normal number of prime
factors of sums of Fourier coefficients of eigenforms. Moreover, we
will see an all purpose Erd ̈os-Kac theorem. The final part is joint
work with M. Ram Murty and V. Kumar Murty.
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Let A^e and B^e be the enveloping algebra of k-algebras A and B, and
Mod(A^e) the category of A^e-modules. It is natural to ask when an
exact functor from Mod(A^e) to Mod(B^e) gives rise to a graded
homomorphism between the Hochschild cohomologies of A and B. A
recollement of module categories can be thought of as a ``short exact
sequence" of categories with maps being adjunct functors. Reiner
Hermann showed that recollements of module categories give rise to
homomorphisms between the associated Hochschild cohomology algebras
preserving the strict Gerstenhaber structure. This led to a
formulation of another variation of the Snashall-Solberg finite
generation conjecture which asks whether the Hochschild cohomology
modulo the weak Gerstenhaber ideal generated by homogeneous nilpotent
elements is finitely generated. We present an answer to this question
using Nicole Snashall's counterexample to the Snashall-Solberg finite
generation conjecture.
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Let $f$ be a polynomial with integer coefficients and consider the
diophantine equation $f(x)=by^l$ for some integers $l, b\geq 1$. If
all the irreducible factors of $f(x)$ are linear then Erd\"os and
Selfridge provided the conditions for the solvability of the above
diophantine equation. For some other polynomials, some results are
also known. In this talk, we consider the above diophantine equation
with $l$ is the largest possible integer so that the equation has
integral solutions in the interval $[1, N]$. For some suitable
polynomial, we provide a non-trivial bound of $l$ in the term of
$N$. In the course of the proof, we introduce a group-theoretic
invariant which is a natural generalization of the Davenport
constant. We provide a non-trivial upper bound of this new
constant. This is joint work with Eshita Mazumdar.
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In the 1950s Serre introduced an algebraic invariant for regular local
rings to measure the geometric notion of intersection multiplicity of
curves, and proposed a number of conjectures about its
behavior. Hochster, and later Dao, studied generalizations of this
invariant for hypersurface and complete intersection rings. This talk
will focus on the case of a graded complete intersection ring, where
we will show vanishing of a closely related invariant under
assumptions on the complexity of a pair of modules. Our main tool is
to draw a connection between this invariant and work of Avramov,
Buchweitz, and Sally involving orders of Laurent series. This is on
joint work with David Jorgensen and Liana Sega.
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In this talk, I will mostly focus on large homomorphisms of local
rings introduced by Gerson Levin in 1979. Some examples and
characterization of large homomorphisms in terms of Koszul homologies
over complete intersection and Golod local rings will be
addressed. Next, I will discuss some conditions under which a
homomorphism from (or in to) a Koszul algebra is large, small, or
Golod. In the last part of the talk, I will focus on a specific class
of local rings namely minimal intersections. We will see that the
largeness or smallness of the natural maps to a minimal intersection
(R,m,k) enables us to give a formula for the Poincare series of k.
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For a given sequence one can associate a power series and a Dirichlet
series. We investigate the relationship between possible singularities
that appear when we analytically continue both of these series. The
most basic case, when the power series has a pole singularity at $z=1$
is analyzed in detail by employing some (infinite order) discrete
derivative operator (associated to the power series) that we call
Bernoulli operator. Its main property is that it naturally acts on the
vector space of analytic functions in the plane (with possible
isolated singularities) that fall in the image of the Laplace-Mellin
transform (for the variable in some half-plane). The action of the
Bernoulli operator on the function $t^s$, provides the analytic
continuation of the associated Dirichlet series and also detailed
information about the location of poles, their residues, and special
values. Using examples of arithmetic origin, I will attempt to
illustrate what is reasonable to expect when the power series has a
non-pole singularity at $z=1$.
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Meeting ID: 962 1712 8540
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