Friday, November 17 — Holt 175, 4:00pm
Jordan Schettler — San Jose State UniversityTitle: Small Gaps Between Semiprimes and Generalizations of the Erdős-Mirsky Conjecture
Abstract: The Erdős-Mirsky conjecture (proven by Heath-Brown in 1984) states that there are infinitely many positive integers x such that d(x) = d(x+1) where d(x) denotes the number of divisors of x. In contrast, the still unproven twin prime conjecture is equivalent to the statement that d(x) = d(x+2) = 2 for infinitely many x. Using modifications of the GPY sieve, we are able to obtain small gaps between semiprimes (products of two primes) to prove generalizations of the Erdős-Mirsky conjecture. For example, the smallest gap known to contain infinitely many pairs of primes is 246, while the smallest gap known to contain infinitely many pairs of squarefree semiprimes is 6, and this can be used to show that d(x) = d(x+1) = 24 has infinitely many solutions. In fact, for any positive integer n there is a constant C such that d(x) = d(x+n) = C for infinitely many x. Pushing the combinatorial and sieve theoretic methods further, we can get results of the form d(x) = d(x+a) = d(x+b) = C for infinitely many x.
Friday, October 27 — Holt 175, 4:00pm
Orsola Capovilla-Searle — UC, DavisTitle: Exact Lagrangian fillings of Legendrian links
Abstract: An important problem in contact topology is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Legendrian links arise as wavefronts in optics, and can sometimes be used to distinguish contact structures. Legendrian links can also arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4-ball which are called fillings of the link. In the last seven years, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. I will talk about recent work establishing connections between fillings and Newton polytopes, as well as applications to higher dimensional Legendrian submanifolds and non-orientable fillings.
Friday, October 13 — Holt 175, 4:00pm
Fu Liu — UC, DavisTitle: On Ehrhart positivity
Abstract: The Ehrhart polynomial counts the number of lattice points inside dilations of an integral polytope, that is, a polytope whose vertices are lattice points. We say a polytope is Ehrhart positive if its Ehrhart polynomial has positive coefficients. In the literature, different families of polytopes have been shown to be Ehrhart positive using different techniques. We will survey these results in the first part of the talk, after giving a brief introduction to polytopes and Ehrhart polynomials.
Through work of Danilov/McMullen, there is an interpretation of Ehrhart coefficients relating to the normalized volumes of faces. In the second part of the talk, I will discuss collaborative work with Castillo in which we try to make this relation more explicit in the context of regular permutohedra. Our motivation is to establish Ehrhart positivity for generalized permutohedra. Although it turns out not all generalized permutohedra are Ehrhart positive, we are able to show that the third and fourth Ehrhart coefficients of generalized permutohedra are always
positive.
If time permits, I will also discuss some other related questions.
Friday, September 29 — Holt 175, 4:00pm
Yesim Demiroglu — CSU, SacramentoTitle: A Short Journey to Waring's Problem and Sum-Product Formulas
Abstract: Since Edward Waring stated his famous conjecture in his book "Meditationes Algebraicae" in 1770, Waring's problem has been of particular interest to mathematicians. As Charles Small put it in his survey "Indeed, it is one of those nasty gems, like Fermat's Last Theorem, which begins with a simply-stated assertion about natural numbers, and leads quickly into deep water." The first half of our journey will be a quick introduction to Waring's problem together with our contribution. We present some new proofs (using Cayley digraphs and spectral graph theory) for Waring's problem over finite fields, and explain how in the process of re-proving these results, we obtain another result that provides an analogue of Sárközy's theorem in the finite field setting (showing that any subset E of a finite field Fq for which |E| > qk/(q - 1)1/2 must contain at least two distinct elements whose difference is a kth power). Once we have our results for finite fields, one can apply some classical mathematics to extend our Waring's problem results to the context of general (not necessarily commutative) finite rings. In the second half of our talk, we present some sum-product formula results related to matrix rings over finite fields and explain the connection between these two interesting problems.
Friday, September 22 — Holt 175, 4:00pm
András Domokos — CSU, SacramentoTitle: Mathematical structures exhibiting a fundamental scale
Abstract: The existence of a fundamental scale (or Planck scale), a lower bound to any positive measurement, it is assumed by various quantum theories. In this talk we will explain an interesting geometrical structure emerging from the hyperspaces of decomposable and not locally connected homogeneous continua, in particular the circle of pseudo-arcs. In these structures the boundary between the smooth and non-smooth partitions could be interpreted as a fundamental scale.
Friday, September 15 — Holt 175, 4:00pm
Rusiru Gambheera Arachchige — UC, Santa BarbaraTitle: Main Conjectures in Iwasawa theory
Abstract: One of the main goals in number theory is to study various aspects of number fields (finite extensions of the field of rationals) such as their ring of integers, group of units, class groups etc. However, most of the time, studying individual number fields is hard and not very helpful. So, in Iwasawa theory we study infinite towers of number fields as a whole. That sometimes gives us valuable information about each individual number field and the growth of its invariants when we go up in the tower. Main conjectures in Iwasawa theory relate certain modules defined at the top of the Iwasawa tower to p-adic L functions. We proved an equivariant main conjecture and as an application computed the Fitting ideal of a naturally arising Iwasawa module. This is joint work with Cristian Popescu.