Please contact cadi.howell@maine.edu for a zoom link.

�վ��ٱ���:�� COMBINATORIAL RECIPROCITY FOR EQUIVARIANT CHROMATIC
POLYNOMIALS

�������ٰ�������:��The chromatic polynomial of a graph counts the number of ways in which the graph can be colored, with an input being the number of colors at hand. A combinatorial reciprocity theorem of Stanley’s tells us that evaluating the chromatic polynomial at -1 counts the number of acyclic orientations of the graph. Hanlon has proved an analogue of the aforementioned reciprocity theorem for the orbital chromatic polynomial, which counts colorings that differ by a rotation or symmetry as the same. In this thesis we establish a reciprocity theorem for a new family of chromatic polynomials, defined in terms of group representations, that generalizes both the standard and the orbital chromatic polynomials.

Title: A connection between zeros and central values of L-functions

Abstract: L-functions appear as generating functions encapsulating information about various objects such as Galois representations, elliptic curves, arithmetic functions, modular forms, Maass forms, etc. Studying L-functions is therefore of utmost importance in number theory at large. Two of their attached data carry critical information: their zeros, which govern the distributional behavior of underlying objects; and their central values, which are related to invariants such as the class number of a field extension.

We will discuss the important conjectures, one concerning the distribution of the zeros and one concerning the distribution of the central values, and explain a general principle that any restricted result towards the first conjecture can be refined to show that most corresponding central values have the typical distribution predicted by the second conjecture. We will instantiate this general principle for a wide class of L-functions, and provide a more detailed discussion in the case of L-functions attached to modular forms.

More information to come.

Speaker: Jack Buttcane, 91����

Abstract:

Rotations in three dimensions depend on three angles, which we call the Euler angles. A function of the Euler angles can be expressed in terms of Wigner-D matrices, which are representations of the rotation group SO(3). These matrix functions are strongly tied to quantum mechanics where they measure the probability of a state transition under a rotation, but as a number theorist, I am interested in their connection to automorphic forms: Suppose we have a vector-valued function of the 3×3 invertible matrices that transforms by a Wigner-D matrix under rotations. Then an interesting quantity to study is the “sup norm” sup_A ||f(A)||, i.e. the maximum value of the length of the vector f(A) over 3×3 invertible matrices A.

We can also think of the function f as a vector of scalar-valued functions f=(f_{-d},…f_d) and for each entry f_j in the vector, we can consider sup_A |f_j(A)|. The question I want to consider is, to what extent does the ratio of these two sup norms depend exclusively on the Wigner-D matrix, independent of the particular function f? As an undergraduate summer research project Andrii Obertas did some computations on this project and I’ll report on the outcome of those, as well.

Speaker: James Dibble, University of Southern Maine

Abstract: A classical family of problems in differential geometry involves bounding certain geometric quantities, such as volume or the length of the shortest closed geodesic, in terms of others. In this talk, we will discuss a variety of such inequalities, including new ones for convex hypersurfaces. The focus will be on results that can be stated without making reference to how quickly a space curves. This is joint work with Joseph Hoisington.

Abstract: Character varieties arise as spaces that parametrize group homomorphisms from a finitely presented group Γ into an “nice” algebraic group G, up to a natural notion of equivalence. In this talk, we explore these spaces by studying their points over finite fields Fq and examining how symmetries of the group Γ, encoded by its outer automorphism group Out(Γ), act on them.
We introduce a natural stratification of the space and count the number of points in each stratum, using these counts to gain insight into the asymptotic behavior (when q goes to ∞) of the action as the size of the field grows. In particular, we consider the case where Γ = Zr and G = SL3(C).

The talk will be self-contained and aimed at a broad audience, with the necessary background developed along the way.

Speaker: Matt Jones, Colby College

Abstract: For hundreds of years, graphs have been used by mathematicians to study the connections between things. I will begin by introducing graphs and graph colorings, and briefly talk through a classical result from the field of graph coloring problems. After that, I will show how graph colorings can also be used to study group behavior by introducing a new kind of graph coloring. These “locally-optimal’’ colorings are relevant when a group tries to reach consensus, and I will describe an algorithm to count them. Finally, I’ll talk about how to think of consensus as a stochastic process and how to apply these ideas to real-world scenarios.