I am in the year of mourning for my mother who pass away in December. I am a "modern orhtodox" Jew, shomer shabbat, and a Zionist etc. When my father passed away I was still living in the USA, now I am happy to be living in Israel for over five years. My question is what activites, if any , can I participate in on this day. I have been generally following Rabbi Lamm’s book, but he is silent about this specific date.

Shalom, Thank you for your question. Rav Aviner once told me that a person in the year of mourning should treat Yom Ha'Aztmaut in a similar fashion to Chanukah. Based on this you your celebration this year should include, of course, going to synagogue and taking part in all the special parts of the service that are added. Also, if you normally have a festive meal (in the house or outside in nature) with your close family, this is permitted, on the condition that there is no live music. Attending a large party or festive gathering in honor of the day is forbidden (- Rav Dov Leor, on our Hebrew site). It is certainly permitted to go to Torah lectures and also to see the international Bible quiz. May you merit to celebrate many more Independence Days in total happiness and joy – together with the entire Jewish people! Blessings.