July 21, 2025

Our morning began with an engaging and thought-provoking lecture by Dr. Victor Barranca, a professor of applied mathematics at Swarthmore College. Returning for his second summer with us, Dr. Barranca spoke on the topic “Applied Mathematics – Puzzles from Past to Present.” He opened with a seemingly simple question: “What is mathematics?” After hearing several student responses, he offered his own definition:

Mathematics is the study of patterns using abstraction.

He then explored how these patterns appear not only in numbers but also in shapes, physical objects, and even in how we perceive the world. A circle, for instance, can represent a tabletop, a water bottle, or a wheel. He went on to show how the brain interprets visual information using mathematical principles. Students examined optical illusions such as the Hermann grid and binocular rivalry, which reveal the brain’s natural tendency to search for patterns. Dr. Barranca explained how mathematical models of these perceptual processes have contributed to advances in neuroscience, artificial intelligence, and computer vision.

To conclude his lecture, Dr. Barranca introduced the classic Birthday Problem, a probability puzzle that often defies our expectations. In a group of just 23 people, the chance that two individuals share a birthday is greater than fifty percent. He then recounted a famous moment on The Tonight Show when Johnny Carson misunderstood the problem entirely. Instead of looking for any shared birthday among audience members, Carson looked for someone who shared his own birthday. Dr. Barranca asked our students why Carson’s approach was incorrect, and they quickly provided the correct explanation. Mathematician Steven Strogatz wrote a thoughtful and entertaining piece about this moment in The New York Times.

This wide-ranging and thought-provoking lecture highlighted how mathematics helps us recognize structure in the world around us, while also sharpening our ability to reason clearly when our intuition falls short.

After a short break, we welcomed Dr. Mourya Narasareddygari from Rider College for our second lecture of the day. Dr. Narasareddygari spoke about cybersecurity, encryption, and how to protect our digital data—a topic that felt both timely and important.

She began by asking students how they secure their phones and introduced the idea of encryption as a way to keep data safe during transmission. Using the classic Caesar cipher, she showed how letters can be shifted to create secret messages. Students eagerly worked in teams to decrypt a message using a backward shift. She then introduced keyword ciphers, which use a scrambled alphabet based on a chosen word, demonstrating how small changes in the key can transform the entire message.

The highlight of the session was a team-based decryption challenge using a Caesar cipher with a known key. Students raced the clock to crack the code, working together with impressive focus and energy.

In the second half of her talk, Dr. Narasareddygari discussed digital footprints and the risks of cyberbullying and identity theft. She reminded students that deleted photos and messages aren’t always truly gone, and that kindness, caution, and speaking up are key to staying safe online.

It was a thoughtful and engaging session that combined hands-on fun with real-world lessons in digital responsibility.


In the afternoon the students heard two lectures from Professor Shapiro. The first lecture was “Introduction to Mathematical Physics.” It was quite a challenging and wide-ranging presentation that started with a definition of a function (essentially a list of directions as to how to get from an input value to a specific output value). The discussion progressed through examinations of the graphs of functions and identifying one-to-one functions. (One-to-one functions present a situation in which the output of the function can always be traced back to figure out a singular input.) This new familiarity with functions was then applied to classical mechanics and the calculation of trajectory, a function with an input of a moment in time and an output of a position in space.

After a short break, Professor Shapiro continued with a discussion of sequences and limits, which he defined as the answer to the hypothetical question, “what if we could do a procedure infinitely many times?” Students were asked to consider many examples of given numerical sequences to figure out their limits, some of which were numbers like 0 or 1 and others infinite or non-existent. Students then were presented with recursive algebraic definitions of numerical sequences to consider. The discussion was definitely getting close to the basics of calculus, and beyond into analysis!

We all gathered in the Brower Student Center for our opening evening dinner.
After dinner, all our staff and participants walked over to the Decker Social Space for our evening activities. Our counselors have received training in leading improv games, which they taught to our participants. These games give our students a chance to get acquainted with one another through some fun interactions. For most of these games, the team members form a circle and do activities that emphasize learning one another’s names, collaborating, listening, and in some cases a little friendly competing. There was a lot of laughter - it’s a great way to release tensions and begin making friends. After that it was off to the dorms for the traditional fire drill and then to bed.