Thu, April 18, 2019
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April 2019
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10:00am [10:00am] Kirill Cherednichenko, University of Bath
Lecture Series on Partial Differential Equations. Speaker: Kirill Cherednichenko. Affiliation: University of Bath. Title: Periodic PDEs with micro-resonators: unified approach to homogenisation and time-dispersive media. Venue: Ramanujan Hall, Department of Mathematics. Thursday 18 April, 10.00 am - 12.00 pm Lecture III: Periodic media with resonant components (“high contrast composites”). Gelfand transform and direct integral: a reduction of the full-space problem to a family of “transmission” problems on the period cell. A reformulation in terms of the M-operator on the interface. Diagonalisation of the M-operator on the nonresonant component: Steklov eigenvalue problem.

5:00pm [5:15pm] Priyamvad Srivastav, IMSc, Chennai
CACAAG seminar. Speaker: Priyamvad Srivastav. Affiliation: IMSc, Chennai. Date and Time: Thursday 18 April, 5.15 pm - 6.15 pm. Venue: Ramanujan Hall, Department of Mathematics. Title: Product of primes in arithmetic progression. Abstract: Let $q$ be a positive integer and let $(a,q)=1$ be a given residue class. Let $p(a,q)$ denote the least prime congruent to $a \mod{q}$. Linnik's theorem tells us that there is a constant $L>0$, such that the $p(a,q) \ll q^L$. The best known value today is $L = 5.18$. A conjecture of Erdos asks if there exist primes $p_1$ and $p_2$, both less than $q$, such that $p_1 p_2 \equiv a \mod{q}$. Recently, Ramar\'{e} and Walker proved that for all $q \geq 2$, there are primes $p_1, p_2, p_3$, each less than $q^{16/3}$, such that $p_1 p_2 p_3 \equiv a \mod{q}$. Their proof combines additive combinatorics with sieve theoretic techniques. We sketch the ideas involved in their proof and talk about a joint work with Olivier Ramar\'{e}, where we refine this method and obtain an improved exponent of $q$.