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Frobenius coin-exchange generating functions (Matthias Beck, San Francisco State University)

Roberts North 102, CMC

We study variants of the Frobenius coin-exchange problem: Given n positive relatively prime parameters, what is the largest integer that cannot be represented as a nonnegative integral linear combination of the given integers? This problem and its siblings can be understood through generating functions with 0/1 coefficients according to whether or not an integer is representable. […]

f^*-vectors of lattice polytopes (Max Hlavacek, Pomona College)

Roberts North 102, CMC

The Ehrhart polynomial of a lattice polytope P counts the number of integer points in the nth integral dilate of P. The f^* -vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of the Ehrhart polynomial with respect to the binomial coefficient basis . Similarly to h and h^* -vectors, […]

On the Cox ring of a weighted projective plane blown-up at a point (Javier Gonzalez Anaya, HMC)

Roberts North 102, CMC

The Cox ring of a projective variety is the ring of all its meromorphic functions, together with a grading of geometric origin. Determining whether this ring is finitely generated is a challenging task, even for simple examples. In this talk, we will discuss our efforts to tackle this problem for a specific class of varieties, […]

What can chicken nuggets tell us about symmetric functions, positive polynomials, random norms, and AF algebras? (Stephan Garcia, Pomona)

Roberts North 102, CMC

A simple question about chicken nuggets connects everything from analysis and combinatorics to probability theory and computer-aided design.  With tools from complex, harmonic, and functional analysis, probability theory, algebraic combinatorics, and spline theory, we answer many asymptotic questions about factorization lengths in numerical semigroups.  Our results yield uncannily accurate predictions, along with unexpected results about […]

Skein algebra of a punctured surface (Helen Wong, CMC)

Roberts North 102, CMC

The Kauffman bracket skein algebra of a surface is at once related to quantum topology and to hyperbolic geometry. In this talk, we consider a generalization of the skein algebra due to Roger and Yang for surfaces with punctures. In joint work with Han-Bom Moon, we show that the generalized skein algebra is a quantization […]

Using quantum statistical mechanical systems to study real quadratic fields (Jane Panangaden, Pitzer College)

Estella 2099

The original Bost-Connes system was constructed in 1990 and is a QSM system with deep connections to the field of rationals. In particular, its partition function is the Riemann-zeta function and its ground states evaluated on certain arithmetic objects yield generators of the maximal Abelian extension of the rationals. In this talk we describe the […]

Quiver categorification of quandle invariants (Sam Nelson, CMC)

Estella 2099

Quiver structures are naturally associated to subsets of the endomorphism sets of quandles and other knot-coloring structures, providing a natural form of categorification of homset invariants and their enhancements. In this talk we will survey recent work in this area.

Point-counting and topology of algebraic varieties (Siddarth Kannan, UCLA)

Estella 2099

A projective algebraic variety X is the zero locus of a collection of homogeneous polynomials, in projective space. When the polynomials have integer coefficients, we can think of the k-valued points X(k) of the variety, for any field k. Now suppose we have two different fields k and k'. How does the behavior of X(k) […]

The restricted variable Kakeya problem (Pete Clark, University of Georgia)

Estella 2099

For a finite field F_q, a subset of F_q^N is a Kakeya set if it contains a line in every direction (i.e., a coset of every one-dimensional linear subspace).  The finite field Kakeya problem is to determine the minimal size K(N,q) of a Kakeya set in F_q^N.  This problem was posed by Wolff in 1999 as […]

Homological mirror symmetry, curve counting, and a classical example: 27 lines on a nonsingular cubic surface (Reggie Anderson, CMC)

Estella 2099

Though mirror symmetry requires much technical background, it gained traction in the mathematical community when physicists Candelas-de la Ossa-Green-Parkes discovered enumerative invariants counting the number of rational degree d curves inside of certain space called a ``quintic threefold." This answered longstanding problems in enumerative geometry from antiquity. In particular, the number of rational degree d=1 […]

Almost-prime times in horospherical flows (Taylor McAdam, Pomona)

Estella 2099

There is a rich connection between homogeneous dynamics and number theory.  Often in such applications it is desirable for dynamical results to be effective (i.e. the rates of convergence for dynamical phenomena are known).  In the first part of this talk, I will provide the necessary background and relevant history to state an effective equidistribution […]